id	sid	tid	token	lemma	pos
ejpam-841	1	1	1_841_wang.dvi	1_841_wang.dvi	NUM
ejpam-841	1	2	european	european	ADJ
ejpam-841	1	3	journal	journal	NOUN
ejpam-841	1	4	of	of	ADP
ejpam-841	1	5	pure	pure	ADJ
ejpam-841	1	6	and	and	CCONJ
ejpam-841	1	7	applied	apply	VERB
ejpam-841	1	8	mathematics	mathematic	NOUN
ejpam-841	1	9	vol	vol	NOUN
ejpam-841	1	10	.	.	PUNCT
ejpam-841	2	1	3	3	NUM
ejpam-841	2	2	,	,	PUNCT
ejpam-841	2	3	no	no	INTJ
ejpam-841	2	4	.	.	NOUN
ejpam-841	2	5	4	4	NUM
ejpam-841	2	6	,	,	PUNCT
ejpam-841	2	7	2010	2010	NUM
ejpam-841	2	8	,	,	PUNCT
ejpam-841	2	9	602	602	NUM
ejpam-841	2	10	-	-	SYM
ejpam-841	2	11	632	632	NUM
ejpam-841	2	12	issn	issn	PROPN
ejpam-841	2	13	1307	1307	NUM
ejpam-841	2	14	-	-	SYM
ejpam-841	2	15	5543	5543	NUM
ejpam-841	2	16	–	–	PUNCT
ejpam-841	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-841	2	18	minimal	minimal	ADJ
ejpam-841	2	19	generators	generator	NOUN
ejpam-841	2	20	for	for	ADP
ejpam-841	2	21	the	the	DET
ejpam-841	2	22	rees	rees	PROPN
ejpam-841	2	23	algebra	algebra	NOUN
ejpam-841	2	24	of	of	ADP
ejpam-841	2	25	rational	rational	ADJ
ejpam-841	2	26	space	space	NOUN
ejpam-841	2	27	curves	curve	NOUN
ejpam-841	2	28	of	of	ADP
ejpam-841	2	29	type	type	NOUN
ejpam-841	2	30	(	(	PUNCT
ejpam-841	2	31	1	1	NUM
ejpam-841	2	32	,	,	PUNCT
ejpam-841	2	33	1	1	NUM
ejpam-841	2	34	,	,	PUNCT
ejpam-841	2	35	d	d	NOUN
ejpam-841	2	36	−	−	PROPN
ejpam-841	2	37	2	2	NUM
ejpam-841	2	38	)	)	PUNCT
ejpam-841	2	39	j.	j.	PROPN
ejpam-841	2	40	william	william	PROPN
ejpam-841	2	41	hoffman1	hoffman1	PROPN
ejpam-841	2	42	,	,	PUNCT
ejpam-841	2	43	haohao	haohao	PROPN
ejpam-841	2	44	wang2,∗	wang2,∗	PROPN
ejpam-841	2	45	,	,	PUNCT
ejpam-841	2	46	xiaohong	xiaohong	PROPN
ejpam-841	2	47	jia3	jia3	PROPN
ejpam-841	2	48	,	,	PUNCT
ejpam-841	2	49	ron	ron	PROPN
ejpam-841	2	50	goldman4	goldman4	PROPN
ejpam-841	2	51	1	1	NUM
ejpam-841	2	52	department	department	NOUN
ejpam-841	2	53	of	of	ADP
ejpam-841	2	54	mathematics	mathematics	PROPN
ejpam-841	2	55	,	,	PUNCT
ejpam-841	2	56	louisiana	louisiana	PROPN
ejpam-841	2	57	state	state	PROPN
ejpam-841	2	58	university	university	PROPN
ejpam-841	2	59	,	,	PUNCT
ejpam-841	2	60	baton	baton	NOUN
ejpam-841	2	61	rouge	rouge	NOUN
ejpam-841	2	62	,	,	PUNCT
ejpam-841	2	63	la	la	PROPN
ejpam-841	2	64	,	,	PUNCT
ejpam-841	2	65	usa	usa	PROPN
ejpam-841	2	66	2	2	NUM
ejpam-841	2	67	department	department	NOUN
ejpam-841	2	68	of	of	ADP
ejpam-841	2	69	mathematics	mathematic	NOUN
ejpam-841	2	70	,	,	PUNCT
ejpam-841	2	71	southeast	southeast	PROPN
ejpam-841	2	72	missouri	missouri	PROPN
ejpam-841	2	73	state	state	PROPN
ejpam-841	2	74	university	university	PROPN
ejpam-841	2	75	,	,	PUNCT
ejpam-841	2	76	cape	cape	PROPN
ejpam-841	2	77	girardeau	girardeau	PROPN
ejpam-841	2	78	,	,	PUNCT
ejpam-841	2	79	mo	mo	PROPN
ejpam-841	2	80	,	,	PUNCT
ejpam-841	2	81	usa	usa	PROPN
ejpam-841	2	82	3	3	NUM
ejpam-841	2	83	department	department	NOUN
ejpam-841	2	84	of	of	ADP
ejpam-841	2	85	computer	computer	NOUN
ejpam-841	2	86	science	science	NOUN
ejpam-841	2	87	,	,	PUNCT
ejpam-841	2	88	the	the	DET
ejpam-841	2	89	university	university	NOUN
ejpam-841	2	90	of	of	ADP
ejpam-841	2	91	hong	hong	PROPN
ejpam-841	2	92	kong	kong	PROPN
ejpam-841	2	93	,	,	PUNCT
ejpam-841	2	94	hong	hong	PROPN
ejpam-841	2	95	kong	kong	PROPN
ejpam-841	2	96	,	,	PUNCT
ejpam-841	2	97	p.	p.	PROPN
ejpam-841	2	98	r.	r.	PROPN
ejpam-841	3	1	china	china	PROPN
ejpam-841	3	2	4	4	NUM
ejpam-841	3	3	computer	computer	NOUN
ejpam-841	3	4	science	science	NOUN
ejpam-841	3	5	department	department	PROPN
ejpam-841	3	6	,	,	PUNCT
ejpam-841	3	7	rice	rice	NOUN
ejpam-841	3	8	university	university	NOUN
ejpam-841	3	9	,	,	PUNCT
ejpam-841	3	10	6100	6100	NUM
ejpam-841	3	11	main	main	PROPN
ejpam-841	3	12	st	st	PROPN
ejpam-841	3	13	.	.	PROPN
ejpam-841	3	14	,	,	PUNCT
ejpam-841	3	15	ms-132	ms-132	PROPN
ejpam-841	3	16	,	,	PUNCT
ejpam-841	3	17	houston	houston	PROPN
ejpam-841	3	18	,	,	PUNCT
ejpam-841	3	19	tx	tx	PROPN
ejpam-841	3	20	77251	77251	NUM
ejpam-841	3	21	,	,	PUNCT
ejpam-841	3	22	usa	usa	PROPN
ejpam-841	3	23	abstract	abstract	NOUN
ejpam-841	3	24	.	.	PUNCT
ejpam-841	4	1	we	we	PRON
ejpam-841	4	2	provide	provide	VERB
ejpam-841	4	3	an	an	DET
ejpam-841	4	4	algorithm	algorithm	NOUN
ejpam-841	4	5	to	to	PART
ejpam-841	4	6	find	find	VERB
ejpam-841	4	7	a	a	DET
ejpam-841	4	8	minimal	minimal	ADJ
ejpam-841	4	9	set	set	NOUN
ejpam-841	4	10	of	of	ADP
ejpam-841	4	11	generators	generator	NOUN
ejpam-841	4	12	for	for	ADP
ejpam-841	4	13	the	the	DET
ejpam-841	4	14	rees	rees	PROPN
ejpam-841	4	15	algebra	algebra	NOUN
ejpam-841	4	16	associated	associate	VERB
ejpam-841	4	17	to	to	ADP
ejpam-841	4	18	rational	rational	ADJ
ejpam-841	4	19	space	space	NOUN
ejpam-841	4	20	curves	curve	NOUN
ejpam-841	4	21	of	of	ADP
ejpam-841	4	22	type	type	NOUN
ejpam-841	4	23	(	(	PUNCT
ejpam-841	4	24	1,1	1,1	NUM
ejpam-841	4	25	,	,	PUNCT
ejpam-841	4	26	d−2	d−2	PROPN
ejpam-841	4	27	)	)	PUNCT
ejpam-841	4	28	in	in	ADP
ejpam-841	4	29	projective	projective	ADJ
ejpam-841	4	30	3	3	NUM
ejpam-841	4	31	-	-	PUNCT
ejpam-841	4	32	space	space	NOUN
ejpam-841	4	33	based	base	VERB
ejpam-841	4	34	solely	solely	ADV
ejpam-841	4	35	on	on	ADP
ejpam-841	4	36	a	a	DET
ejpam-841	4	37	µ-basis	µ-basis	NOUN
ejpam-841	4	38	of	of	ADP
ejpam-841	4	39	the	the	DET
ejpam-841	4	40	curve	curve	NOUN
ejpam-841	4	41	.	.	PUNCT
ejpam-841	5	1	we	we	PRON
ejpam-841	5	2	also	also	ADV
ejpam-841	5	3	illustrate	illustrate	VERB
ejpam-841	5	4	the	the	DET
ejpam-841	5	5	geometry	geometry	NOUN
ejpam-841	5	6	behind	behind	ADP
ejpam-841	5	7	the	the	DET
ejpam-841	5	8	generators	generator	NOUN
ejpam-841	5	9	via	via	ADP
ejpam-841	5	10	a	a	DET
ejpam-841	5	11	case	case	NOUN
ejpam-841	5	12	study	study	NOUN
ejpam-841	5	13	of	of	ADP
ejpam-841	5	14	rational	rational	ADJ
ejpam-841	5	15	quartic	quartic	ADJ
ejpam-841	5	16	space	space	NOUN
ejpam-841	5	17	curves	curve	NOUN
ejpam-841	5	18	.	.	PUNCT
ejpam-841	6	1	2000	2000	NUM
ejpam-841	6	2	mathematics	mathematic	NOUN
ejpam-841	6	3	subject	subject	NOUN
ejpam-841	6	4	classifications	classification	NOUN
ejpam-841	6	5	:	:	PUNCT
ejpam-841	6	6	14q05	14q05	NUM
ejpam-841	6	7	(	(	PUNCT
ejpam-841	6	8	primary	primary	NOUN
ejpam-841	6	9	)	)	PUNCT
ejpam-841	6	10	,	,	PUNCT
ejpam-841	6	11	13d02	13d02	NUM
ejpam-841	6	12	,	,	PUNCT
ejpam-841	6	13	14h20(secondary	14h20(secondary	NUM
ejpam-841	6	14	)	)	PUNCT
ejpam-841	6	15	key	key	ADJ
ejpam-841	6	16	words	word	NOUN
ejpam-841	6	17	and	and	CCONJ
ejpam-841	6	18	phrases	phrase	NOUN
ejpam-841	6	19	:	:	PUNCT
ejpam-841	6	20	rees	rees	PROPN
ejpam-841	6	21	algebra	algebra	PROPN
ejpam-841	6	22	,	,	PUNCT
ejpam-841	6	23	syzygy	syzygy	NOUN
ejpam-841	6	24	,	,	PUNCT
ejpam-841	6	25	implicit	implicit	ADJ
ejpam-841	6	26	equations	equation	NOUN
ejpam-841	6	27	,	,	PUNCT
ejpam-841	6	28	resultant	resultant	NOUN
ejpam-841	6	29	,	,	PUNCT
ejpam-841	6	30	µ-basis	µ-basis	NOUN
ejpam-841	6	31	1	1	NUM
ejpam-841	6	32	.	.	PUNCT
ejpam-841	7	1	introduction	introduction	NOUN
ejpam-841	7	2	the	the	DET
ejpam-841	7	3	rees	rees	PROPN
ejpam-841	7	4	algebra	algebra	NOUN
ejpam-841	7	5	of	of	ADP
ejpam-841	7	6	an	an	DET
ejpam-841	7	7	ideal	ideal	NOUN
ejpam-841	8	1	i	i	PRON
ejpam-841	8	2	⊂	⊂	VERB
ejpam-841	8	3	r	r	VERB
ejpam-841	8	4	defined	define	VERB
ejpam-841	8	5	as	as	ADP
ejpam-841	8	6	the	the	DET
ejpam-841	8	7	graded	grade	VERB
ejpam-841	8	8	algebra	algebra	NOUN
ejpam-841	8	9	(	(	PUNCT
ejpam-841	8	10	with	with	ADP
ejpam-841	8	11	the	the	DET
ejpam-841	8	12	elements	element	NOUN
ejpam-841	8	13	of	of	ADP
ejpam-841	8	14	r	r	NOUN
ejpam-841	8	15	having	have	VERB
ejpam-841	8	16	degree	degree	NOUN
ejpam-841	8	17	0	0	PUNCT
ejpam-841	8	18	and	and	CCONJ
ejpam-841	8	19	the	the	DET
ejpam-841	8	20	elements	element	NOUN
ejpam-841	8	21	of	of	ADP
ejpam-841	8	22	i	i	PRON
ejpam-841	8	23	having	have	VERB
ejpam-841	8	24	degree	degree	NOUN
ejpam-841	8	25	1	1	NUM
ejpam-841	8	26	)	)	PUNCT
ejpam-841	8	27	rees(i	rees(i	PROPN
ejpam-841	8	28	)	)	PUNCT
ejpam-841	9	1	=	=	SYM
ejpam-841	9	2	r⊕	r⊕	NOUN
ejpam-841	9	3	i	i	PROPN
ejpam-841	9	4	⊕	⊕	PROPN
ejpam-841	9	5	i2	i2	PROPN
ejpam-841	9	6	⊕	⊕	PROPN
ejpam-841	9	7	...	...	PUNCT
ejpam-841	9	8	is	be	AUX
ejpam-841	9	9	a	a	DET
ejpam-841	9	10	classical	classical	ADJ
ejpam-841	9	11	algebraic	algebraic	ADJ
ejpam-841	9	12	structure	structure	NOUN
ejpam-841	9	13	which	which	PRON
ejpam-841	9	14	has	have	AUX
ejpam-841	9	15	been	be	AUX
ejpam-841	9	16	studied	study	VERB
ejpam-841	9	17	for	for	ADP
ejpam-841	9	18	decades	decade	NOUN
ejpam-841	9	19	by	by	ADP
ejpam-841	9	20	the	the	DET
ejpam-841	9	21	commutative	commutative	ADJ
ejpam-841	9	22	algebra	algebra	PROPN
ejpam-841	9	23	community	community	NOUN
ejpam-841	9	24	,	,	PUNCT
ejpam-841	9	25	see	see	VERB
ejpam-841	9	26	[	[	X
ejpam-841	9	27	25	25	NUM
ejpam-841	9	28	]	]	PUNCT
ejpam-841	9	29	.	.	PUNCT
ejpam-841	10	1	one	one	NUM
ejpam-841	10	2	motivation	motivation	NOUN
ejpam-841	10	3	for	for	ADP
ejpam-841	10	4	this	this	DET
ejpam-841	10	5	study	study	NOUN
ejpam-841	10	6	is	be	AUX
ejpam-841	10	7	that	that	SCONJ
ejpam-841	10	8	it	it	PRON
ejpam-841	10	9	is	be	AUX
ejpam-841	10	10	related	relate	VERB
ejpam-841	10	11	to	to	ADP
ejpam-841	10	12	a	a	DET
ejpam-841	10	13	classical	classical	ADJ
ejpam-841	10	14	problem	problem	NOUN
ejpam-841	10	15	in	in	ADP
ejpam-841	10	16	elimination	elimination	NOUN
ejpam-841	10	17	theory	theory	NOUN
ejpam-841	10	18	:	:	PUNCT
ejpam-841	10	19	the	the	DET
ejpam-841	10	20	implicitization	implicitization	NOUN
ejpam-841	10	21	problem	problem	NOUN
ejpam-841	10	22	.	.	PUNCT
ejpam-841	11	1	the	the	DET
ejpam-841	11	2	implicitization	implicitization	NOUN
ejpam-841	11	3	problem	problem	NOUN
ejpam-841	11	4	is	be	AUX
ejpam-841	11	5	to	to	PART
ejpam-841	11	6	find	find	VERB
ejpam-841	11	7	an	an	DET
ejpam-841	11	8	algorithm	algorithm	NOUN
ejpam-841	11	9	to	to	PART
ejpam-841	11	10	convert	convert	VERB
ejpam-841	11	11	a	a	DET
ejpam-841	11	12	parametrization	parametrization	NOUN
ejpam-841	11	13	given	give	VERB
ejpam-841	11	14	by	by	ADP
ejpam-841	11	15	a	a	DET
ejpam-841	11	16	rational	rational	ADJ
ejpam-841	11	17	map	map	NOUN
ejpam-841	11	18	f	f	NOUN
ejpam-841	11	19	:	:	PUNCT
ejpam-841	11	20	pm→	pm→	NOUN
ejpam-841	11	21	pn	pn	VERB
ejpam-841	11	22	into	into	ADP
ejpam-841	11	23	defining	define	VERB
ejpam-841	11	24	equations	equation	NOUN
ejpam-841	11	25	for	for	ADP
ejpam-841	11	26	the	the	DET
ejpam-841	11	27	closure	closure	NOUN
ejpam-841	11	28	x	x	PUNCT
ejpam-841	11	29	of	of	ADP
ejpam-841	11	30	the	the	DET
ejpam-841	11	31	image	image	NOUN
ejpam-841	11	32	f	f	PROPN
ejpam-841	11	33	(	(	PUNCT
ejpam-841	11	34	pm	pm	NOUN
ejpam-841	11	35	)	)	PUNCT
ejpam-841	11	36	.	.	PUNCT
ejpam-841	12	1	rational	rational	ADJ
ejpam-841	12	2	curves	curve	NOUN
ejpam-841	12	3	and	and	CCONJ
ejpam-841	12	4	surfaces	surface	NOUN
ejpam-841	12	5	are	be	AUX
ejpam-841	12	6	widely	widely	ADV
ejpam-841	12	7	used	use	VERB
ejpam-841	12	8	in	in	ADP
ejpam-841	12	9	computer	computer	NOUN
ejpam-841	12	10	aided	aid	VERB
ejpam-841	12	11	design	design	NOUN
ejpam-841	12	12	,	,	PUNCT
ejpam-841	12	13	since	since	SCONJ
ejpam-841	12	14	it	it	PRON
ejpam-841	12	15	is	be	AUX
ejpam-841	12	16	easy	easy	ADJ
ejpam-841	12	17	to	to	PART
ejpam-841	12	18	describe	describe	VERB
ejpam-841	12	19	the	the	DET
ejpam-841	12	20	points	point	NOUN
ejpam-841	12	21	on	on	ADP
ejpam-841	12	22	these	these	DET
ejpam-841	12	23	∗corresponding	∗corresponde	VERB
ejpam-841	12	24	author	author	NOUN
ejpam-841	12	25	.	.	PUNCT
ejpam-841	13	1	email	email	NOUN
ejpam-841	13	2	addresses	address	NOUN
ejpam-841	13	3	:	:	PUNCT
ejpam-841	13	4	hoffman�math.lsu.edu	hoffman�math.lsu.edu	PROPN
ejpam-841	13	5	(	(	PUNCT
ejpam-841	13	6	j.	j.	PROPN
ejpam-841	13	7	hoffman	hoffman	PROPN
ejpam-841	13	8	)	)	PUNCT
ejpam-841	13	9	,	,	PUNCT
ejpam-841	13	10	hwang�semo.edu	hwang�semo.edu	PROPN
ejpam-841	13	11	(	(	PUNCT
ejpam-841	13	12	h.	h.	PROPN
ejpam-841	13	13	wang	wang	PROPN
ejpam-841	13	14	)	)	PUNCT
ejpam-841	13	15	,	,	PUNCT
ejpam-841	13	16	xhjia	xhjia	PROPN
ejpam-841	13	17	�	�	PROPN
ejpam-841	13	18	s.hku.hk	s.hku.hk	PROPN
ejpam-841	13	19	(	(	PUNCT
ejpam-841	13	20	x.	x.	NOUN
ejpam-841	13	21	jia	jia	PROPN
ejpam-841	13	22	)	)	PUNCT
ejpam-841	13	23	,	,	PUNCT
ejpam-841	13	24	rng	rng	PROPN
ejpam-841	13	25	�	�	PROPN
ejpam-841	13	26	ri	ri	PROPN
ejpam-841	14	1	e.edu	e.edu	X
ejpam-841	14	2	(	(	PUNCT
ejpam-841	14	3	r.	r.	PROPN
ejpam-841	14	4	goldman	goldman	PROPN
ejpam-841	14	5	)	)	PUNCT
ejpam-841	14	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-841	15	1	602	602	NUM
ejpam-841	15	2	c	c	X
ejpam-841	15	3	©	©	VERB
ejpam-841	15	4	2010	2010	NUM
ejpam-841	15	5	ejpam	ejpam	NOUN
ejpam-841	15	6	all	all	DET
ejpam-841	15	7	rights	right	NOUN
ejpam-841	15	8	reserved	reserve	VERB
ejpam-841	15	9	.	.	PUNCT
ejpam-841	16	1	j.	j.	PROPN
ejpam-841	16	2	hoffman	hoffman	PROPN
ejpam-841	16	3	,	,	PUNCT
ejpam-841	16	4	h.	h.	PROPN
ejpam-841	16	5	wang	wang	PROPN
ejpam-841	16	6	,	,	PUNCT
ejpam-841	16	7	x.	x.	PROPN
ejpam-841	16	8	jia	jia	PROPN
ejpam-841	16	9	,	,	PUNCT
ejpam-841	16	10	r.	r.	PROPN
ejpam-841	16	11	goldman	goldman	PROPN
ejpam-841	16	12	/	/	SYM
ejpam-841	16	13	eur	eur	PROPN
ejpam-841	16	14	.	.	PUNCT
ejpam-841	17	1	j.	j.	PROPN
ejpam-841	17	2	pure	pure	PROPN
ejpam-841	17	3	appl	appl	PROPN
ejpam-841	17	4	.	.	PROPN
ejpam-841	17	5	math	math	PROPN
ejpam-841	17	6	,	,	PUNCT
ejpam-841	17	7	3	3	NUM
ejpam-841	17	8	(	(	PUNCT
ejpam-841	17	9	2010	2010	NUM
ejpam-841	17	10	)	)	PUNCT
ejpam-841	17	11	,	,	PUNCT
ejpam-841	17	12	602	602	NUM
ejpam-841	17	13	-	-	SYM
ejpam-841	17	14	632	632	NUM
ejpam-841	17	15	603	603	NUM
ejpam-841	17	16	curves	curve	NOUN
ejpam-841	17	17	and	and	CCONJ
ejpam-841	17	18	surfaces	surface	NOUN
ejpam-841	17	19	by	by	ADP
ejpam-841	17	20	means	mean	NOUN
ejpam-841	17	21	of	of	ADP
ejpam-841	17	22	their	their	PRON
ejpam-841	17	23	parameter	parameter	NOUN
ejpam-841	17	24	values	value	NOUN
ejpam-841	17	25	.	.	PUNCT
ejpam-841	18	1	however	however	ADV
ejpam-841	18	2	,	,	PUNCT
ejpam-841	18	3	it	it	PRON
ejpam-841	18	4	is	be	AUX
ejpam-841	18	5	not	not	PART
ejpam-841	18	6	convenient	convenient	ADJ
ejpam-841	18	7	to	to	PART
ejpam-841	18	8	use	use	VERB
ejpam-841	18	9	the	the	DET
ejpam-841	18	10	parametric	parametric	ADJ
ejpam-841	18	11	representations	representation	NOUN
ejpam-841	18	12	to	to	PART
ejpam-841	18	13	describe	describe	VERB
ejpam-841	18	14	the	the	DET
ejpam-841	18	15	set	set	NOUN
ejpam-841	18	16	of	of	ADP
ejpam-841	18	17	points	point	NOUN
ejpam-841	18	18	that	that	PRON
ejpam-841	18	19	are	be	AUX
ejpam-841	18	20	common	common	ADJ
ejpam-841	18	21	to	to	ADP
ejpam-841	18	22	two	two	NUM
ejpam-841	18	23	different	different	ADJ
ejpam-841	18	24	parametrically	parametrically	ADV
ejpam-841	18	25	defined	define	VERB
ejpam-841	18	26	curves	curve	NOUN
ejpam-841	18	27	or	or	CCONJ
ejpam-841	18	28	surfaces	surface	NOUN
ejpam-841	18	29	.	.	PUNCT
ejpam-841	19	1	thus	thus	ADV
ejpam-841	19	2	there	there	PRON
ejpam-841	19	3	is	be	VERB
ejpam-841	19	4	a	a	DET
ejpam-841	19	5	need	need	NOUN
ejpam-841	19	6	to	to	PART
ejpam-841	19	7	go	go	VERB
ejpam-841	19	8	back	back	ADV
ejpam-841	19	9	and	and	CCONJ
ejpam-841	19	10	forth	forth	ADV
ejpam-841	19	11	between	between	ADP
ejpam-841	19	12	a	a	DET
ejpam-841	19	13	parametric	parametric	NOUN
ejpam-841	19	14	and	and	CCONJ
ejpam-841	19	15	an	an	DET
ejpam-841	19	16	implicit	implicit	ADJ
ejpam-841	19	17	description	description	NOUN
ejpam-841	19	18	of	of	ADP
ejpam-841	19	19	a	a	DET
ejpam-841	19	20	curve	curve	NOUN
ejpam-841	19	21	or	or	CCONJ
ejpam-841	19	22	surface	surface	NOUN
ejpam-841	19	23	.	.	PUNCT
ejpam-841	20	1	this	this	PRON
ejpam-841	20	2	is	be	AUX
ejpam-841	20	3	,	,	PUNCT
ejpam-841	20	4	in	in	ADP
ejpam-841	20	5	essence	essence	NOUN
ejpam-841	20	6	,	,	PUNCT
ejpam-841	20	7	the	the	DET
ejpam-841	20	8	implicitization	implicitization	NOUN
ejpam-841	20	9	problem	problem	NOUN
ejpam-841	20	10	:	:	PUNCT
ejpam-841	20	11	to	to	PART
ejpam-841	20	12	develop	develop	VERB
ejpam-841	20	13	efficient	efficient	ADJ
ejpam-841	20	14	algorithms	algorithm	NOUN
ejpam-841	20	15	to	to	PART
ejpam-841	20	16	generate	generate	VERB
ejpam-841	20	17	implicit	implicit	ADJ
ejpam-841	20	18	equations	equation	NOUN
ejpam-841	20	19	for	for	ADP
ejpam-841	20	20	a	a	DET
ejpam-841	20	21	curve	curve	NOUN
ejpam-841	20	22	or	or	CCONJ
ejpam-841	20	23	surface	surface	NOUN
ejpam-841	20	24	for	for	ADP
ejpam-841	20	25	which	which	PRON
ejpam-841	20	26	one	one	PRON
ejpam-841	20	27	knows	know	VERB
ejpam-841	20	28	a	a	DET
ejpam-841	20	29	parametric	parametric	ADJ
ejpam-841	20	30	representation	representation	NOUN
ejpam-841	20	31	.	.	PUNCT
ejpam-841	21	1	implicitization	implicitization	NOUN
ejpam-841	21	2	algorithms	algorithm	NOUN
ejpam-841	21	3	have	have	AUX
ejpam-841	21	4	been	be	AUX
ejpam-841	21	5	most	most	ADV
ejpam-841	21	6	highly	highly	ADV
ejpam-841	21	7	developed	develop	VERB
ejpam-841	21	8	in	in	ADP
ejpam-841	21	9	the	the	DET
ejpam-841	21	10	case	case	NOUN
ejpam-841	21	11	that	that	SCONJ
ejpam-841	21	12	x	x	PRON
ejpam-841	21	13	is	be	AUX
ejpam-841	21	14	a	a	DET
ejpam-841	21	15	hypersurface	hypersurface	NOUN
ejpam-841	21	16	,	,	PUNCT
ejpam-841	21	17	so	so	ADV
ejpam-841	21	18	x	x	X
ejpam-841	21	19	is	be	AUX
ejpam-841	21	20	defined	define	VERB
ejpam-841	21	21	by	by	ADP
ejpam-841	21	22	only	only	ADV
ejpam-841	21	23	one	one	NUM
ejpam-841	21	24	equation	equation	NOUN
ejpam-841	21	25	f	f	NOUN
ejpam-841	21	26	=	=	SYM
ejpam-841	21	27	0	0	X
ejpam-841	21	28	.	.	PUNCT
ejpam-841	22	1	it	it	PRON
ejpam-841	22	2	turns	turn	VERB
ejpam-841	22	3	out	out	ADP
ejpam-841	22	4	that	that	SCONJ
ejpam-841	22	5	f	f	PROPN
ejpam-841	22	6	is	be	AUX
ejpam-841	22	7	related	relate	VERB
ejpam-841	22	8	to	to	ADP
ejpam-841	22	9	the	the	DET
ejpam-841	22	10	structure	structure	NOUN
ejpam-841	22	11	of	of	ADP
ejpam-841	22	12	the	the	DET
ejpam-841	22	13	rees	rees	PROPN
ejpam-841	22	14	algebra	algebra	NOUN
ejpam-841	22	15	of	of	ADP
ejpam-841	22	16	i	i	PRON
ejpam-841	22	17	.	.	PUNCT
ejpam-841	23	1	for	for	ADP
ejpam-841	23	2	instance	instance	NOUN
ejpam-841	23	3	consider	consider	VERB
ejpam-841	23	4	the	the	DET
ejpam-841	23	5	implicitization	implicitization	NOUN
ejpam-841	23	6	problem	problem	NOUN
ejpam-841	23	7	for	for	ADP
ejpam-841	23	8	rational	rational	ADJ
ejpam-841	23	9	surfaces	surface	NOUN
ejpam-841	23	10	in	in	ADP
ejpam-841	23	11	p3	p3	PROPN
ejpam-841	23	12	.	.	PUNCT
ejpam-841	24	1	algebraically	algebraically	ADV
ejpam-841	24	2	the	the	DET
ejpam-841	24	3	problem	problem	NOUN
ejpam-841	24	4	is	be	AUX
ejpam-841	24	5	this	this	DET
ejpam-841	24	6	:	:	PUNCT
ejpam-841	24	7	given	give	VERB
ejpam-841	24	8	an	an	DET
ejpam-841	24	9	ideal	ideal	NOUN
ejpam-841	24	10	i	i	NOUN
ejpam-841	24	11	=	=	SYM
ejpam-841	24	12	〈	〈	PROPN
ejpam-841	24	13	f0	f0	PROPN
ejpam-841	24	14	,	,	PUNCT
ejpam-841	24	15	f1	f1	NOUN
ejpam-841	24	16	,	,	PUNCT
ejpam-841	24	17	f2	f2	PROPN
ejpam-841	24	18	,	,	PUNCT
ejpam-841	24	19	f3	f3	ADJ
ejpam-841	24	20	〉	〉	NOUN
ejpam-841	24	21	⊂	⊂	X
ejpam-841	24	22	r	r	NOUN
ejpam-841	24	23	where	where	SCONJ
ejpam-841	24	24	the	the	DET
ejpam-841	24	25	fi	fi	NOUN
ejpam-841	24	26	are	be	AUX
ejpam-841	24	27	homogeneous	homogeneous	ADJ
ejpam-841	24	28	polynomials	polynomial	NOUN
ejpam-841	24	29	of	of	ADP
ejpam-841	24	30	degree	degree	NOUN
ejpam-841	24	31	d	d	NOUN
ejpam-841	24	32	in	in	ADP
ejpam-841	24	33	the	the	DET
ejpam-841	24	34	standard	standard	ADJ
ejpam-841	24	35	z	z	ADV
ejpam-841	24	36	-	-	PUNCT
ejpam-841	24	37	graded	grade	VERB
ejpam-841	24	38	ring	ring	NOUN
ejpam-841	24	39	r	r	NOUN
ejpam-841	24	40	:	:	PUNCT
ejpam-841	24	41	=	=	SYM
ejpam-841	24	42	k[s	k[s	PROPN
ejpam-841	24	43	,	,	PUNCT
ejpam-841	24	44	t	t	PROPN
ejpam-841	24	45	,	,	PUNCT
ejpam-841	24	46	u	u	NOUN
ejpam-841	24	47	]	]	PUNCT
ejpam-841	24	48	over	over	ADP
ejpam-841	24	49	an	an	DET
ejpam-841	24	50	infinite	infinite	ADJ
ejpam-841	24	51	field	field	NOUN
ejpam-841	24	52	k	k	NOUN
ejpam-841	24	53	,	,	PUNCT
ejpam-841	24	54	find	find	VERB
ejpam-841	24	55	a	a	DET
ejpam-841	24	56	minimal	minimal	ADJ
ejpam-841	24	57	set	set	NOUN
ejpam-841	24	58	of	of	ADP
ejpam-841	24	59	generators	generator	NOUN
ejpam-841	24	60	for	for	ADP
ejpam-841	24	61	the	the	DET
ejpam-841	24	62	kernel	kernel	NOUN
ejpam-841	24	63	of	of	ADP
ejpam-841	24	64	the	the	DET
ejpam-841	24	65	map	map	NOUN
ejpam-841	24	66	h	h	NOUN
ejpam-841	24	67	:	:	PUNCT
ejpam-841	24	68	r[x0	r[x0	ADV
ejpam-841	24	69	,	,	PUNCT
ejpam-841	24	70	x1	x1	PROPN
ejpam-841	24	71	,	,	PUNCT
ejpam-841	24	72	x2	x2	PROPN
ejpam-841	24	73	,	,	PUNCT
ejpam-841	24	74	x3]→	x3]→	PROPN
ejpam-841	24	75	rees(i	rees(i	PROPN
ejpam-841	24	76	)	)	PUNCT
ejpam-841	24	77	,	,	PUNCT
ejpam-841	24	78	where	where	SCONJ
ejpam-841	24	79	h(x	h(x	PROPN
ejpam-841	24	80	i	i	PRON
ejpam-841	24	81	)	)	PUNCT
ejpam-841	25	1	=	=	PUNCT
ejpam-841	25	2	fi	fi	NOUN
ejpam-841	25	3	for	for	ADP
ejpam-841	25	4	i	i	PRON
ejpam-841	25	5	=	=	NOUN
ejpam-841	25	6	0,1,2,3	0,1,2,3	NUM
ejpam-841	25	7	.	.	PUNCT
ejpam-841	26	1	under	under	ADP
ejpam-841	26	2	certain	certain	ADJ
ejpam-841	26	3	general	general	ADJ
ejpam-841	26	4	circumstances	circumstance	NOUN
ejpam-841	26	5	,	,	PUNCT
ejpam-841	26	6	the	the	DET
ejpam-841	26	7	implicit	implicit	ADJ
ejpam-841	26	8	equation	equation	NOUN
ejpam-841	26	9	f	f	NOUN
ejpam-841	26	10	is	be	AUX
ejpam-841	26	11	the	the	DET
ejpam-841	26	12	element	element	NOUN
ejpam-841	26	13	of	of	ADP
ejpam-841	26	14	ker(h	ker(h	PROPN
ejpam-841	26	15	)	)	PUNCT
ejpam-841	26	16	of	of	ADP
ejpam-841	26	17	degree	degree	NOUN
ejpam-841	26	18	0	0	NUM
ejpam-841	26	19	in	in	ADP
ejpam-841	26	20	the	the	DET
ejpam-841	26	21	variables	variable	NOUN
ejpam-841	26	22	s	s	PROPN
ejpam-841	26	23	,	,	PUNCT
ejpam-841	26	24	t	t	PROPN
ejpam-841	26	25	,	,	PUNCT
ejpam-841	26	26	u.	u.	NOUN
ejpam-841	26	27	elements	element	NOUN
ejpam-841	26	28	of	of	ADP
ejpam-841	26	29	ker(h	ker(h	PROPN
ejpam-841	26	30	)	)	PUNCT
ejpam-841	26	31	,	,	PUNCT
ejpam-841	26	32	under	under	ADP
ejpam-841	26	33	the	the	DET
ejpam-841	26	34	name	name	NOUN
ejpam-841	26	35	of	of	ADP
ejpam-841	26	36	moving	move	VERB
ejpam-841	26	37	lines	line	NOUN
ejpam-841	26	38	and	and	CCONJ
ejpam-841	26	39	moving	move	VERB
ejpam-841	26	40	planes	plane	NOUN
ejpam-841	26	41	,	,	PUNCT
ejpam-841	26	42	were	be	AUX
ejpam-841	26	43	introduced	introduce	VERB
ejpam-841	26	44	into	into	ADP
ejpam-841	26	45	computer	computer	NOUN
ejpam-841	26	46	aided	aid	VERB
ejpam-841	26	47	geometric	geometric	ADJ
ejpam-841	26	48	design	design	NOUN
ejpam-841	26	49	by	by	ADP
ejpam-841	26	50	sederberg	sederberg	PROPN
ejpam-841	26	51	,	,	PUNCT
ejpam-841	26	52	cox	cox	PROPN
ejpam-841	26	53	and	and	CCONJ
ejpam-841	26	54	their	their	PRON
ejpam-841	26	55	collaborators	collaborator	NOUN
ejpam-841	26	56	in	in	ADP
ejpam-841	26	57	order	order	NOUN
ejpam-841	26	58	to	to	PART
ejpam-841	26	59	develop	develop	VERB
ejpam-841	26	60	robust	robust	ADJ
ejpam-841	26	61	,	,	PUNCT
ejpam-841	26	62	efficient	efficient	ADJ
ejpam-841	26	63	algorithms	algorithm	NOUN
ejpam-841	26	64	for	for	ADP
ejpam-841	26	65	implicitizing	implicitize	VERB
ejpam-841	26	66	rational	rational	ADJ
ejpam-841	26	67	curves	curve	NOUN
ejpam-841	26	68	and	and	CCONJ
ejpam-841	26	69	surfaces	surface	NOUN
ejpam-841	26	70	[	[	X
ejpam-841	26	71	10	10	NUM
ejpam-841	26	72	]	]	PUNCT
ejpam-841	26	73	,	,	PUNCT
ejpam-841	26	74	[	[	X
ejpam-841	26	75	20	20	NUM
ejpam-841	26	76	]	]	PUNCT
ejpam-841	26	77	,	,	PUNCT
ejpam-841	26	78	[	[	X
ejpam-841	26	79	21	21	NUM
ejpam-841	26	80	]	]	PUNCT
ejpam-841	26	81	,	,	PUNCT
ejpam-841	26	82	[	[	X
ejpam-841	26	83	22	22	NUM
ejpam-841	26	84	]	]	PUNCT
ejpam-841	26	85	.	.	PUNCT
ejpam-841	27	1	in	in	ADP
ejpam-841	27	2	the	the	DET
ejpam-841	27	3	past	past	ADJ
ejpam-841	27	4	two	two	NUM
ejpam-841	27	5	years	year	NOUN
ejpam-841	27	6	,	,	PUNCT
ejpam-841	27	7	[	[	X
ejpam-841	27	8	4	4	NUM
ejpam-841	27	9	]	]	PUNCT
ejpam-841	27	10	,	,	PUNCT
ejpam-841	27	11	[	[	X
ejpam-841	27	12	7	7	NUM
ejpam-841	27	13	]	]	PUNCT
ejpam-841	27	14	,	,	PUNCT
ejpam-841	27	15	[	[	X
ejpam-841	27	16	8	8	NUM
ejpam-841	27	17	]	]	PUNCT
ejpam-841	27	18	,	,	PUNCT
ejpam-841	27	19	and	and	CCONJ
ejpam-841	27	20	[	[	X
ejpam-841	27	21	16	16	NUM
ejpam-841	27	22	]	]	PUNCT
ejpam-841	27	23	utilized	utilize	VERB
ejpam-841	27	24	the	the	DET
ejpam-841	27	25	method	method	NOUN
ejpam-841	27	26	of	of	ADP
ejpam-841	27	27	moving	move	VERB
ejpam-841	27	28	curves	curve	NOUN
ejpam-841	27	29	and	and	CCONJ
ejpam-841	27	30	surfaces	surface	NOUN
ejpam-841	27	31	to	to	PART
ejpam-841	27	32	determine	determine	VERB
ejpam-841	27	33	the	the	DET
ejpam-841	27	34	defining	define	VERB
ejpam-841	27	35	equations	equation	NOUN
ejpam-841	27	36	for	for	ADP
ejpam-841	27	37	the	the	DET
ejpam-841	27	38	rees	rees	PROPN
ejpam-841	27	39	algebra	algebra	NOUN
ejpam-841	27	40	of	of	ADP
ejpam-841	27	41	plane	plane	NOUN
ejpam-841	27	42	algebraic	algebraic	ADJ
ejpam-841	27	43	curves	curve	NOUN
ejpam-841	27	44	.	.	PUNCT
ejpam-841	28	1	they	they	PRON
ejpam-841	28	2	each	each	PRON
ejpam-841	28	3	develop	develop	VERB
ejpam-841	28	4	different	different	ADJ
ejpam-841	28	5	methods	method	NOUN
ejpam-841	28	6	and	and	CCONJ
ejpam-841	28	7	algorithms	algorithm	NOUN
ejpam-841	28	8	for	for	ADP
ejpam-841	28	9	finding	find	VERB
ejpam-841	28	10	explicit	explicit	ADJ
ejpam-841	28	11	moving	move	VERB
ejpam-841	28	12	curves	curve	NOUN
ejpam-841	28	13	that	that	PRON
ejpam-841	28	14	are	be	AUX
ejpam-841	28	15	a	a	DET
ejpam-841	28	16	minimal	minimal	ADJ
ejpam-841	28	17	set	set	NOUN
ejpam-841	28	18	of	of	ADP
ejpam-841	28	19	generators	generator	NOUN
ejpam-841	28	20	for	for	ADP
ejpam-841	28	21	the	the	DET
ejpam-841	28	22	associated	associated	ADJ
ejpam-841	28	23	rees	rees	PROPN
ejpam-841	28	24	algebra	algebra	NOUN
ejpam-841	28	25	.	.	PUNCT
ejpam-841	29	1	the	the	DET
ejpam-841	29	2	approach	approach	NOUN
ejpam-841	29	3	in	in	ADP
ejpam-841	29	4	[	[	X
ejpam-841	29	5	7	7	NUM
ejpam-841	29	6	]	]	PUNCT
ejpam-841	29	7	,	,	PUNCT
ejpam-841	29	8	[	[	X
ejpam-841	29	9	8	8	NUM
ejpam-841	29	10	]	]	PUNCT
ejpam-841	29	11	is	be	AUX
ejpam-841	29	12	based	base	VERB
ejpam-841	29	13	on	on	ADP
ejpam-841	29	14	iterations	iteration	NOUN
ejpam-841	29	15	of	of	ADP
ejpam-841	29	16	sylvester	sylvest	ADJ
ejpam-841	29	17	determinants	determinant	NOUN
ejpam-841	29	18	,	,	PUNCT
ejpam-841	29	19	regular	regular	ADJ
ejpam-841	29	20	sequences	sequence	NOUN
ejpam-841	29	21	,	,	PUNCT
ejpam-841	29	22	and	and	CCONJ
ejpam-841	29	23	local	local	ADJ
ejpam-841	29	24	cohomology	cohomology	NOUN
ejpam-841	29	25	computations	computation	NOUN
ejpam-841	29	26	this	this	DET
ejpam-841	29	27	approach	approach	NOUN
ejpam-841	29	28	to	to	ADP
ejpam-841	29	29	the	the	DET
ejpam-841	29	30	implicitization	implicitization	NOUN
ejpam-841	29	31	problem	problem	NOUN
ejpam-841	29	32	,	,	PUNCT
ejpam-841	29	33	which	which	PRON
ejpam-841	29	34	has	have	AUX
ejpam-841	29	35	been	be	AUX
ejpam-841	29	36	developed	develop	VERB
ejpam-841	29	37	especially	especially	ADV
ejpam-841	29	38	by	by	ADP
ejpam-841	29	39	jouanoulou	jouanoulou	PROPN
ejpam-841	29	40	,	,	PUNCT
ejpam-841	29	41	busé	busé	NOUN
ejpam-841	29	42	,	,	PUNCT
ejpam-841	29	43	chardin	chardin	PROPN
ejpam-841	29	44	,	,	PUNCT
ejpam-841	29	45	cox	cox	PROPN
ejpam-841	29	46	,	,	PUNCT
ejpam-841	29	47	d’andrea	d’andrea	PROPN
ejpam-841	29	48	and	and	CCONJ
ejpam-841	29	49	others	other	NOUN
ejpam-841	29	50	,	,	PUNCT
ejpam-841	29	51	utilizes	utilize	VERB
ejpam-841	29	52	the	the	DET
ejpam-841	29	53	structure	structure	NOUN
ejpam-841	29	54	of	of	ADP
ejpam-841	29	55	a	a	DET
ejpam-841	29	56	free	free	ADJ
ejpam-841	29	57	resolution	resolution	NOUN
ejpam-841	29	58	of	of	ADP
ejpam-841	29	59	i	i	PRON
ejpam-841	29	60	as	as	ADP
ejpam-841	29	61	an	an	DET
ejpam-841	29	62	r	r	NOUN
ejpam-841	29	63	-	-	PUNCT
ejpam-841	29	64	module	module	NOUN
ejpam-841	29	65	(	(	PUNCT
ejpam-841	29	66	see	see	VERB
ejpam-841	29	67	[	[	X
ejpam-841	29	68	2	2	NUM
ejpam-841	29	69	]	]	PUNCT
ejpam-841	29	70	,	,	PUNCT
ejpam-841	29	71	[	[	X
ejpam-841	29	72	3	3	NUM
ejpam-841	29	73	]	]	NUM
ejpam-841	29	74	)	)	PUNCT
ejpam-841	29	75	.	.	PUNCT
ejpam-841	30	1	however	however	ADV
ejpam-841	30	2	,	,	PUNCT
ejpam-841	30	3	these	these	DET
ejpam-841	30	4	studies	study	NOUN
ejpam-841	30	5	have	have	AUX
ejpam-841	30	6	been	be	AUX
ejpam-841	30	7	largely	largely	ADV
ejpam-841	30	8	limited	limit	VERB
ejpam-841	30	9	to	to	ADP
ejpam-841	30	10	the	the	DET
ejpam-841	30	11	case	case	NOUN
ejpam-841	30	12	of	of	ADP
ejpam-841	30	13	hypersurface	hypersurface	NOUN
ejpam-841	30	14	parametrizations	parametrization	NOUN
ejpam-841	30	15	,	,	PUNCT
ejpam-841	30	16	and	and	CCONJ
ejpam-841	30	17	they	they	PRON
ejpam-841	30	18	lead	lead	VERB
ejpam-841	30	19	to	to	ADP
ejpam-841	30	20	expressions	expression	NOUN
ejpam-841	30	21	for	for	ADP
ejpam-841	30	22	f	f	PROPN
ejpam-841	30	23	as	as	ADP
ejpam-841	30	24	determinants	determinant	NOUN
ejpam-841	30	25	of	of	ADP
ejpam-841	30	26	certain	certain	ADJ
ejpam-841	30	27	complexes	complex	NOUN
ejpam-841	30	28	.	.	PUNCT
ejpam-841	31	1	see	see	VERB
ejpam-841	31	2	also	also	ADV
ejpam-841	31	3	[	[	X
ejpam-841	31	4	1	1	NUM
ejpam-841	31	5	]	]	PUNCT
ejpam-841	31	6	,	,	PUNCT
ejpam-841	31	7	[	[	X
ejpam-841	31	8	4	4	NUM
ejpam-841	31	9	]	]	PUNCT
ejpam-841	31	10	,	,	PUNCT
ejpam-841	31	11	[	[	X
ejpam-841	31	12	5	5	NUM
ejpam-841	31	13	]	]	PUNCT
ejpam-841	31	14	,	,	PUNCT
ejpam-841	31	15	[	[	X
ejpam-841	31	16	15	15	NUM
ejpam-841	31	17	]	]	PUNCT
ejpam-841	31	18	.	.	PUNCT
ejpam-841	32	1	the	the	DET
ejpam-841	32	2	corresponding	corresponding	ADJ
ejpam-841	32	3	problems	problem	NOUN
ejpam-841	32	4	for	for	ADP
ejpam-841	32	5	codimension	codimension	NOUN
ejpam-841	32	6	two	two	NUM
ejpam-841	32	7	(	(	PUNCT
ejpam-841	32	8	and	and	CCONJ
ejpam-841	32	9	higher	high	ADJ
ejpam-841	32	10	)	)	PUNCT
ejpam-841	32	11	are	be	AUX
ejpam-841	32	12	much	much	ADV
ejpam-841	32	13	more	more	ADV
ejpam-841	32	14	difficult	difficult	ADJ
ejpam-841	32	15	.	.	PUNCT
ejpam-841	33	1	given	give	VERB
ejpam-841	33	2	an	an	DET
ejpam-841	33	3	ideal	ideal	NOUN
ejpam-841	33	4	i	i	NOUN
ejpam-841	33	5	=	=	SYM
ejpam-841	33	6	〈	〈	PROPN
ejpam-841	33	7	f0	f0	PROPN
ejpam-841	33	8	,	,	PUNCT
ejpam-841	33	9	f1	f1	NOUN
ejpam-841	33	10	,	,	PUNCT
ejpam-841	33	11	.	.	PUNCT
ejpam-841	33	12	.	.	PUNCT
ejpam-841	33	13	.	.	PUNCT
ejpam-841	34	1	,	,	PUNCT
ejpam-841	34	2	fn	fn	X
ejpam-841	34	3	〉	〉	NOUN
ejpam-841	34	4	⊂	⊂	X
ejpam-841	34	5	r	r	NOUN
ejpam-841	34	6	of	of	ADP
ejpam-841	34	7	height	height	NOUN
ejpam-841	34	8	two	two	NUM
ejpam-841	34	9	where	where	SCONJ
ejpam-841	34	10	f0	f0	PROPN
ejpam-841	34	11	,	,	PUNCT
ejpam-841	34	12	f1	f1	NOUN
ejpam-841	34	13	,	,	PUNCT
ejpam-841	34	14	.	.	PUNCT
ejpam-841	34	15	.	.	PUNCT
ejpam-841	34	16	.	.	PUNCT
ejpam-841	35	1	,	,	PUNCT
ejpam-841	35	2	fn	fn	PROPN
ejpam-841	35	3	are	be	AUX
ejpam-841	35	4	homogeneous	homogeneous	ADJ
ejpam-841	35	5	polynomials	polynomial	NOUN
ejpam-841	35	6	of	of	ADP
ejpam-841	35	7	degree	degree	NOUN
ejpam-841	35	8	d	d	NOUN
ejpam-841	35	9	in	in	ADP
ejpam-841	35	10	the	the	DET
ejpam-841	35	11	standard	standard	ADJ
ejpam-841	35	12	z	z	ADV
ejpam-841	35	13	-	-	PUNCT
ejpam-841	35	14	graded	grade	VERB
ejpam-841	35	15	ring	ring	NOUN
ejpam-841	35	16	r	r	NOUN
ejpam-841	35	17	:	:	PUNCT
ejpam-841	35	18	=	=	SYM
ejpam-841	35	19	k[s	k[	NOUN
ejpam-841	35	20	,	,	PUNCT
ejpam-841	35	21	t	t	PROPN
ejpam-841	35	22	]	]	PUNCT
ejpam-841	35	23	over	over	ADP
ejpam-841	35	24	an	an	DET
ejpam-841	35	25	infinite	infinite	ADJ
ejpam-841	35	26	field	field	NOUN
ejpam-841	35	27	k	k	NOUN
ejpam-841	35	28	,	,	PUNCT
ejpam-841	35	29	the	the	DET
ejpam-841	35	30	goal	goal	NOUN
ejpam-841	35	31	is	be	AUX
ejpam-841	35	32	to	to	PART
ejpam-841	35	33	find	find	VERB
ejpam-841	35	34	a	a	DET
ejpam-841	35	35	minimal	minimal	ADJ
ejpam-841	35	36	set	set	NOUN
ejpam-841	35	37	of	of	ADP
ejpam-841	35	38	generators	generator	NOUN
ejpam-841	35	39	for	for	ADP
ejpam-841	35	40	the	the	DET
ejpam-841	35	41	kernel	kernel	NOUN
ejpam-841	35	42	of	of	ADP
ejpam-841	35	43	the	the	DET
ejpam-841	35	44	map	map	NOUN
ejpam-841	35	45	h	h	NOUN
ejpam-841	35	46	:	:	PUNCT
ejpam-841	35	47	r[x0	r[x0	ADV
ejpam-841	35	48	,	,	PUNCT
ejpam-841	35	49	x1	x1	PROPN
ejpam-841	35	50	,	,	PUNCT
ejpam-841	35	51	.	.	PUNCT
ejpam-841	35	52	.	.	PUNCT
ejpam-841	36	1	.	.	PUNCT
ejpam-841	37	1	,	,	PUNCT
ejpam-841	37	2	xn]→	xn]→	PROPN
ejpam-841	37	3	rees(i	rees(i	PROPN
ejpam-841	37	4	)	)	PUNCT
ejpam-841	37	5	,	,	PUNCT
ejpam-841	37	6	where	where	SCONJ
ejpam-841	37	7	h(x	h(x	PROPN
ejpam-841	37	8	i	i	PRON
ejpam-841	37	9	)	)	PUNCT
ejpam-841	37	10	=	=	PUNCT
ejpam-841	38	1	fi	fi	NOUN
ejpam-841	38	2	for	for	ADP
ejpam-841	38	3	i	i	PROPN
ejpam-841	38	4	=	=	NOUN
ejpam-841	38	5	0,1	0,1	NUM
ejpam-841	38	6	,	,	PUNCT
ejpam-841	38	7	.	.	PUNCT
ejpam-841	38	8	.	.	PUNCT
ejpam-841	39	1	.	.	PUNCT
ejpam-841	40	1	,	,	PUNCT
ejpam-841	40	2	n.	n.	VERB
ejpam-841	40	3	the	the	DET
ejpam-841	40	4	first	first	ADJ
ejpam-841	40	5	nontrivial	nontrivial	ADJ
ejpam-841	40	6	case	case	NOUN
ejpam-841	40	7	is	be	AUX
ejpam-841	40	8	that	that	PRON
ejpam-841	40	9	of	of	ADP
ejpam-841	40	10	rational	rational	ADJ
ejpam-841	40	11	space	space	NOUN
ejpam-841	40	12	curves	curve	NOUN
ejpam-841	40	13	,	,	PUNCT
ejpam-841	40	14	n	n	NOUN
ejpam-841	40	15	=	=	SYM
ejpam-841	40	16	3	3	X
ejpam-841	40	17	.	.	X
ejpam-841	40	18	finding	find	VERB
ejpam-841	40	19	a	a	DET
ejpam-841	40	20	minimal	minimal	ADJ
ejpam-841	40	21	set	set	NOUN
ejpam-841	40	22	of	of	ADP
ejpam-841	40	23	generators	generator	NOUN
ejpam-841	40	24	for	for	ADP
ejpam-841	40	25	the	the	DET
ejpam-841	40	26	rees	rees	PROPN
ejpam-841	40	27	algebra	algebra	NOUN
ejpam-841	40	28	of	of	ADP
ejpam-841	40	29	the	the	DET
ejpam-841	40	30	ideal	ideal	NOUN
ejpam-841	40	31	i	i	PRON
ejpam-841	40	32	of	of	ADP
ejpam-841	40	33	a	a	DET
ejpam-841	40	34	rational	rational	ADJ
ejpam-841	40	35	space	space	NOUN
ejpam-841	40	36	curve	curve	NOUN
ejpam-841	40	37	solves	solve	VERB
ejpam-841	40	38	the	the	DET
ejpam-841	40	39	implicitization	implicitization	NOUN
ejpam-841	40	40	problem	problem	NOUN
ejpam-841	40	41	for	for	ADP
ejpam-841	40	42	space	space	NOUN
ejpam-841	40	43	curves	curve	NOUN
ejpam-841	40	44	.	.	PUNCT
ejpam-841	41	1	since	since	SCONJ
ejpam-841	41	2	the	the	DET
ejpam-841	41	3	ideal	ideal	NOUN
ejpam-841	41	4	i	i	PRON
ejpam-841	41	5	gives	give	VERB
ejpam-841	41	6	rise	rise	NOUN
ejpam-841	41	7	to	to	ADP
ejpam-841	41	8	a	a	DET
ejpam-841	41	9	rational	rational	ADJ
ejpam-841	41	10	function	function	NOUN
ejpam-841	41	11	p1→	p1→	X
ejpam-841	41	12	pn	pn	NOUN
ejpam-841	41	13	mapping	mapping	NOUN
ejpam-841	41	14	(	(	PUNCT
ejpam-841	41	15	s	s	PROPN
ejpam-841	41	16	,	,	PUNCT
ejpam-841	41	17	t)→	t)→	PROPN
ejpam-841	41	18	(	(	PUNCT
ejpam-841	41	19	x0	x0	PROPN
ejpam-841	41	20	,	,	PUNCT
ejpam-841	41	21	.	.	PUNCT
ejpam-841	41	22	.	.	PUNCT
ejpam-841	41	23	.	.	PUNCT
ejpam-841	42	1	,	,	PUNCT
ejpam-841	42	2	xn	xn	PROPN
ejpam-841	42	3	)	)	PUNCT
ejpam-841	42	4	,	,	PUNCT
ejpam-841	42	5	the	the	DET
ejpam-841	42	6	image	image	NOUN
ejpam-841	42	7	of	of	ADP
ejpam-841	42	8	this	this	DET
ejpam-841	42	9	map	map	NOUN
ejpam-841	42	10	is	be	AUX
ejpam-841	42	11	a	a	DET
ejpam-841	42	12	curve	curve	NOUN
ejpam-841	42	13	c	c	PROPN
ejpam-841	42	14	⊂	⊂	PROPN
ejpam-841	42	15	pn	pn	PROPN
ejpam-841	42	16	with	with	ADP
ejpam-841	42	17	homogeneous	homogeneous	ADJ
ejpam-841	42	18	coordinate	coordinate	NOUN
ejpam-841	42	19	ring	ring	NOUN
ejpam-841	42	20	k[x0	k[x0	ADV
ejpam-841	42	21	,	,	PUNCT
ejpam-841	42	22	.	.	PUNCT
ejpam-841	42	23	.	.	PUNCT
ejpam-841	43	1	.	.	PUNCT
ejpam-841	44	1	,	,	PUNCT
ejpam-841	44	2	xn	xn	PROPN
ejpam-841	44	3	]	]	X
ejpam-841	44	4	.	.	PUNCT
ejpam-841	45	1	the	the	DET
ejpam-841	45	2	ideal	ideal	ADJ
ejpam-841	45	3	theoretic	theoretic	ADJ
ejpam-841	45	4	implicit	implicit	ADJ
ejpam-841	45	5	equations	equation	NOUN
ejpam-841	45	6	of	of	ADP
ejpam-841	45	7	the	the	DET
ejpam-841	45	8	curve	curve	NOUN
ejpam-841	45	9	c	c	PROPN
ejpam-841	45	10	are	be	AUX
ejpam-841	45	11	among	among	ADP
ejpam-841	45	12	the	the	DET
ejpam-841	45	13	minimal	minimal	ADJ
ejpam-841	45	14	generators	generator	NOUN
ejpam-841	45	15	of	of	ADP
ejpam-841	45	16	the	the	DET
ejpam-841	45	17	defining	define	VERB
ejpam-841	45	18	equations	equation	NOUN
ejpam-841	45	19	for	for	ADP
ejpam-841	45	20	the	the	DET
ejpam-841	45	21	rees	rees	PROPN
ejpam-841	45	22	algebra	algebra	NOUN
ejpam-841	45	23	associated	associate	VERB
ejpam-841	45	24	to	to	ADP
ejpam-841	45	25	the	the	DET
ejpam-841	45	26	space	space	NOUN
ejpam-841	45	27	curve	curve	NOUN
ejpam-841	45	28	.	.	PUNCT
ejpam-841	46	1	for	for	ADP
ejpam-841	46	2	example	example	NOUN
ejpam-841	46	3	,	,	PUNCT
ejpam-841	46	4	[	[	X
ejpam-841	46	5	6	6	NUM
ejpam-841	46	6	,	,	PUNCT
ejpam-841	46	7	11	11	NUM
ejpam-841	46	8	,	,	PUNCT
ejpam-841	46	9	12	12	NUM
ejpam-841	46	10	,	,	PUNCT
ejpam-841	46	11	19	19	NUM
ejpam-841	46	12	,	,	PUNCT
ejpam-841	46	13	23	23	NUM
ejpam-841	46	14	]	]	PUNCT
ejpam-841	46	15	,	,	PUNCT
ejpam-841	46	16	have	have	AUX
ejpam-841	46	17	all	all	PRON
ejpam-841	46	18	investigated	investigate	VERB
ejpam-841	46	19	minimal	minimal	ADJ
ejpam-841	46	20	generators	generator	NOUN
ejpam-841	46	21	for	for	ADP
ejpam-841	46	22	the	the	DET
ejpam-841	46	23	rees	rees	PROPN
ejpam-841	46	24	algebra	algebra	NOUN
ejpam-841	46	25	of	of	ADP
ejpam-841	46	26	the	the	DET
ejpam-841	46	27	ideal	ideal	NOUN
ejpam-841	46	28	of	of	ADP
ejpam-841	46	29	a	a	DET
ejpam-841	46	30	rational	rational	ADJ
ejpam-841	46	31	space	space	NOUN
ejpam-841	46	32	curve	curve	NOUN
ejpam-841	46	33	.	.	PUNCT
ejpam-841	47	1	recently	recently	ADV
ejpam-841	47	2	,	,	PUNCT
ejpam-841	47	3	kustin	kustin	PROPN
ejpam-841	47	4	,	,	PUNCT
ejpam-841	47	5	polini	polini	NOUN
ejpam-841	47	6	and	and	CCONJ
ejpam-841	47	7	ulrich	ulrich	PROPN
ejpam-841	48	1	[	[	X
ejpam-841	48	2	18	18	NUM
ejpam-841	48	3	]	]	PUNCT
ejpam-841	48	4	studied	study	VERB
ejpam-841	48	5	minimal	minimal	ADJ
ejpam-841	48	6	generators	generator	NOUN
ejpam-841	48	7	of	of	ADP
ejpam-841	48	8	the	the	DET
ejpam-841	48	9	defining	define	VERB
ejpam-841	48	10	equations	equation	NOUN
ejpam-841	48	11	for	for	ADP
ejpam-841	48	12	the	the	DET
ejpam-841	48	13	rees	rees	PROPN
ejpam-841	48	14	algebra	algebra	NOUN
ejpam-841	48	15	of	of	ADP
ejpam-841	48	16	a	a	DET
ejpam-841	48	17	height	height	NOUN
ejpam-841	48	18	two	two	NUM
ejpam-841	48	19	ideal	ideal	NOUN
ejpam-841	48	20	i	i	PRON
ejpam-841	48	21	with	with	ADP
ejpam-841	48	22	a	a	DET
ejpam-841	48	23	minimal	minimal	ADJ
ejpam-841	48	24	free	free	ADJ
ejpam-841	48	25	resolution	resolution	NOUN
ejpam-841	48	26	of	of	ADP
ejpam-841	48	27	the	the	DET
ejpam-841	48	28	followj	followj	NOUN
ejpam-841	48	29	.	.	PUNCT
ejpam-841	49	1	hoffman	hoffman	PROPN
ejpam-841	49	2	,	,	PUNCT
ejpam-841	49	3	h.	h.	PROPN
ejpam-841	49	4	wang	wang	PROPN
ejpam-841	49	5	,	,	PUNCT
ejpam-841	49	6	x.	x.	PROPN
ejpam-841	49	7	jia	jia	PROPN
ejpam-841	49	8	,	,	PUNCT
ejpam-841	49	9	r.	r.	PROPN
ejpam-841	49	10	goldman	goldman	PROPN
ejpam-841	49	11	/	/	SYM
ejpam-841	49	12	eur	eur	PROPN
ejpam-841	49	13	.	.	PUNCT
ejpam-841	50	1	j.	j.	PROPN
ejpam-841	50	2	pure	pure	PROPN
ejpam-841	50	3	appl	appl	PROPN
ejpam-841	50	4	.	.	PROPN
ejpam-841	50	5	math	math	PROPN
ejpam-841	50	6	,	,	PUNCT
ejpam-841	50	7	3	3	NUM
ejpam-841	50	8	(	(	PUNCT
ejpam-841	50	9	2010	2010	NUM
ejpam-841	50	10	)	)	PUNCT
ejpam-841	50	11	,	,	PUNCT
ejpam-841	50	12	602	602	NUM
ejpam-841	50	13	-	-	SYM
ejpam-841	50	14	632	632	NUM
ejpam-841	50	15	604	604	NUM
ejpam-841	50	16	ing	ing	ADJ
ejpam-841	50	17	form	form	NOUN
ejpam-841	50	18	:	:	PUNCT
ejpam-841	50	19	0→	0→	X
ejpam-841	50	20	r(−d	r(−d	VERB
ejpam-841	50	21	−	−	PROPN
ejpam-841	50	22	1)n−1	1)n−1	NUM
ejpam-841	50	23	⊕	⊕	PROPN
ejpam-841	50	24	r(−2d	r(−2d	NUM
ejpam-841	50	25	+	+	CCONJ
ejpam-841	50	26	n+	n+	NUM
ejpam-841	50	27	1	1	NUM
ejpam-841	50	28	)	)	PUNCT
ejpam-841	50	29	−−−→	−−−→	PROPN
ejpam-841	50	30	rn(−d	rn(−d	NOUN
ejpam-841	50	31	)	)	PUNCT
ejpam-841	50	32	[	[	PUNCT
ejpam-841	50	33	f0	f0	PROPN
ejpam-841	50	34	,	,	PUNCT
ejpam-841	50	35	...	...	PUNCT
ejpam-841	50	36	,	,	PUNCT
ejpam-841	50	37	fn]−−−−→	fn]−−−−→	PROPN
ejpam-841	50	38	i	i	PRON
ejpam-841	50	39	→	→	PUNCT
ejpam-841	50	40	0	0	X
ejpam-841	50	41	.	.	PUNCT
ejpam-841	51	1	since	since	SCONJ
ejpam-841	51	2	the	the	DET
ejpam-841	51	3	homogeneous	homogeneous	ADJ
ejpam-841	51	4	coordinate	coordinate	NOUN
ejpam-841	51	5	ring	ring	NOUN
ejpam-841	51	6	k[x0	k[x0	ADV
ejpam-841	51	7	,	,	PUNCT
ejpam-841	51	8	.	.	PUNCT
ejpam-841	51	9	.	.	PUNCT
ejpam-841	52	1	.	.	PUNCT
ejpam-841	53	1	,	,	PUNCT
ejpam-841	53	2	xn	xn	PROPN
ejpam-841	53	3	]	]	X
ejpam-841	53	4	is	be	AUX
ejpam-841	53	5	isomorphic	isomorphic	ADJ
ejpam-841	53	6	to	to	ADP
ejpam-841	53	7	the	the	DET
ejpam-841	53	8	special	special	ADJ
ejpam-841	53	9	fiber	fiber	NOUN
ejpam-841	53	10	ring	ring	NOUN
ejpam-841	53	11	of	of	ADP
ejpam-841	53	12	i	i	PRON
ejpam-841	53	13	,	,	PUNCT
ejpam-841	53	14	their	their	PRON
ejpam-841	53	15	approach	approach	NOUN
ejpam-841	53	16	to	to	ADP
ejpam-841	53	17	determining	determine	VERB
ejpam-841	53	18	the	the	DET
ejpam-841	53	19	generators	generator	NOUN
ejpam-841	53	20	for	for	ADP
ejpam-841	53	21	the	the	DET
ejpam-841	53	22	rees	rees	PROPN
ejpam-841	53	23	algebra	algebra	NOUN
ejpam-841	53	24	of	of	ADP
ejpam-841	53	25	the	the	DET
ejpam-841	53	26	curve	curve	NOUN
ejpam-841	53	27	c	c	PROPN
ejpam-841	53	28	is	be	AUX
ejpam-841	53	29	to	to	PART
ejpam-841	53	30	find	find	VERB
ejpam-841	53	31	the	the	DET
ejpam-841	53	32	defining	define	VERB
ejpam-841	53	33	ideal	ideal	NOUN
ejpam-841	53	34	of	of	ADP
ejpam-841	53	35	this	this	DET
ejpam-841	53	36	special	special	ADJ
ejpam-841	53	37	fiber	fiber	NOUN
ejpam-841	53	38	ring	ring	NOUN
ejpam-841	53	39	.	.	PUNCT
ejpam-841	54	1	they	they	PRON
ejpam-841	54	2	pay	pay	VERB
ejpam-841	54	3	close	close	ADJ
ejpam-841	54	4	attention	attention	NOUN
ejpam-841	54	5	to	to	ADP
ejpam-841	54	6	the	the	DET
ejpam-841	54	7	depth	depth	NOUN
ejpam-841	54	8	and	and	CCONJ
ejpam-841	54	9	algebraic	algebraic	ADJ
ejpam-841	54	10	properties	property	NOUN
ejpam-841	54	11	of	of	ADP
ejpam-841	54	12	this	this	DET
ejpam-841	54	13	fiber	fiber	NOUN
ejpam-841	54	14	ring	ring	NOUN
ejpam-841	54	15	of	of	ADP
ejpam-841	54	16	i	i	PRON
ejpam-841	54	17	,	,	PUNCT
ejpam-841	54	18	and	and	CCONJ
ejpam-841	54	19	give	give	VERB
ejpam-841	54	20	an	an	DET
ejpam-841	54	21	explicit	explicit	ADJ
ejpam-841	54	22	description	description	NOUN
ejpam-841	54	23	of	of	ADP
ejpam-841	54	24	the	the	DET
ejpam-841	54	25	minimal	minimal	ADJ
ejpam-841	54	26	generators	generator	NOUN
ejpam-841	54	27	.	.	PUNCT
ejpam-841	55	1	in	in	ADP
ejpam-841	55	2	this	this	DET
ejpam-841	55	3	paper	paper	NOUN
ejpam-841	55	4	,	,	PUNCT
ejpam-841	55	5	we	we	PRON
ejpam-841	55	6	specialize	specialize	VERB
ejpam-841	55	7	the	the	DET
ejpam-841	55	8	setting	setting	NOUN
ejpam-841	55	9	in	in	ADP
ejpam-841	55	10	kustin	kustin	PROPN
ejpam-841	55	11	,	,	PUNCT
ejpam-841	55	12	polini	polini	NOUN
ejpam-841	55	13	and	and	CCONJ
ejpam-841	55	14	ulrich	ulrich	PROPN
ejpam-841	55	15	[	[	X
ejpam-841	55	16	18	18	NUM
ejpam-841	55	17	]	]	PUNCT
ejpam-841	55	18	to	to	ADP
ejpam-841	55	19	rational	rational	ADJ
ejpam-841	55	20	space	space	NOUN
ejpam-841	55	21	curves	curve	NOUN
ejpam-841	55	22	of	of	ADP
ejpam-841	55	23	type	type	NOUN
ejpam-841	55	24	(	(	PUNCT
ejpam-841	55	25	1,1	1,1	NUM
ejpam-841	55	26	,	,	PUNCT
ejpam-841	55	27	d−2	d−2	PROPN
ejpam-841	55	28	)	)	PUNCT
ejpam-841	55	29	in	in	ADP
ejpam-841	55	30	projective	projective	ADJ
ejpam-841	55	31	3	3	NUM
ejpam-841	55	32	-	-	PUNCT
ejpam-841	55	33	space	space	NOUN
ejpam-841	55	34	.	.	PUNCT
ejpam-841	56	1	we	we	PRON
ejpam-841	56	2	do	do	AUX
ejpam-841	56	3	not	not	PART
ejpam-841	56	4	prove	prove	VERB
ejpam-841	56	5	any	any	DET
ejpam-841	56	6	new	new	ADJ
ejpam-841	56	7	theorems	theorem	NOUN
ejpam-841	56	8	about	about	ADP
ejpam-841	56	9	the	the	DET
ejpam-841	56	10	structure	structure	NOUN
ejpam-841	56	11	of	of	ADP
ejpam-841	56	12	rees	ree	NOUN
ejpam-841	56	13	algebras	algebra	NOUN
ejpam-841	56	14	;	;	PUNCT
ejpam-841	56	15	rather	rather	ADV
ejpam-841	56	16	our	our	PRON
ejpam-841	56	17	primary	primary	ADJ
ejpam-841	56	18	goal	goal	NOUN
ejpam-841	56	19	is	be	AUX
ejpam-841	56	20	to	to	PART
ejpam-841	56	21	provide	provide	VERB
ejpam-841	56	22	an	an	DET
ejpam-841	56	23	algorithmic	algorithmic	ADJ
ejpam-841	56	24	approach	approach	NOUN
ejpam-841	56	25	and	and	CCONJ
ejpam-841	56	26	to	to	PART
ejpam-841	56	27	give	give	VERB
ejpam-841	56	28	elementary	elementary	ADJ
ejpam-841	56	29	constructions	construction	NOUN
ejpam-841	56	30	for	for	ADP
ejpam-841	56	31	the	the	DET
ejpam-841	56	32	minimal	minimal	ADJ
ejpam-841	56	33	generators	generator	NOUN
ejpam-841	56	34	for	for	ADP
ejpam-841	56	35	the	the	DET
ejpam-841	56	36	rees	rees	PROPN
ejpam-841	56	37	algebra	algebra	NOUN
ejpam-841	56	38	associated	associate	VERB
ejpam-841	56	39	to	to	ADP
ejpam-841	56	40	these	these	DET
ejpam-841	56	41	curves	curve	NOUN
ejpam-841	56	42	based	base	VERB
ejpam-841	56	43	solely	solely	ADV
ejpam-841	56	44	on	on	ADP
ejpam-841	56	45	their	their	PRON
ejpam-841	56	46	µ-basis	µ-basis	NOUN
ejpam-841	56	47	.	.	PUNCT
ejpam-841	57	1	our	our	PRON
ejpam-841	57	2	approach	approach	NOUN
ejpam-841	57	3	is	be	AUX
ejpam-841	57	4	to	to	PART
ejpam-841	57	5	study	study	VERB
ejpam-841	57	6	separately	separately	ADV
ejpam-841	57	7	the	the	DET
ejpam-841	57	8	cases	case	NOUN
ejpam-841	57	9	when	when	SCONJ
ejpam-841	57	10	the	the	DET
ejpam-841	57	11	rational	rational	ADJ
ejpam-841	57	12	curve	curve	NOUN
ejpam-841	57	13	is	be	AUX
ejpam-841	57	14	either	either	CCONJ
ejpam-841	57	15	singular	singular	ADJ
ejpam-841	57	16	or	or	CCONJ
ejpam-841	57	17	non	non	ADJ
ejpam-841	57	18	-	-	ADJ
ejpam-841	57	19	singular	singular	ADJ
ejpam-841	57	20	.	.	PUNCT
ejpam-841	58	1	if	if	SCONJ
ejpam-841	58	2	the	the	DET
ejpam-841	58	3	rational	rational	ADJ
ejpam-841	58	4	space	space	NOUN
ejpam-841	58	5	curve	curve	NOUN
ejpam-841	58	6	is	be	AUX
ejpam-841	58	7	singular	singular	ADJ
ejpam-841	58	8	,	,	PUNCT
ejpam-841	58	9	then	then	ADV
ejpam-841	58	10	we	we	PRON
ejpam-841	58	11	study	study	VERB
ejpam-841	58	12	separately	separately	ADV
ejpam-841	58	13	the	the	DET
ejpam-841	58	14	cases	case	NOUN
ejpam-841	58	15	when	when	SCONJ
ejpam-841	58	16	the	the	DET
ejpam-841	58	17	degree	degree	NOUN
ejpam-841	58	18	of	of	ADP
ejpam-841	58	19	the	the	DET
ejpam-841	58	20	curve	curve	NOUN
ejpam-841	58	21	is	be	AUX
ejpam-841	58	22	either	either	CCONJ
ejpam-841	58	23	even	even	ADV
ejpam-841	58	24	or	or	CCONJ
ejpam-841	58	25	odd	odd	ADJ
ejpam-841	58	26	.	.	PUNCT
ejpam-841	59	1	the	the	DET
ejpam-841	59	2	generators	generator	NOUN
ejpam-841	59	3	of	of	ADP
ejpam-841	59	4	the	the	DET
ejpam-841	59	5	rees	rees	PROPN
ejpam-841	59	6	algebra	algebra	NOUN
ejpam-841	59	7	are	be	AUX
ejpam-841	59	8	all	all	PRON
ejpam-841	59	9	expressed	express	VERB
ejpam-841	59	10	entirely	entirely	ADV
ejpam-841	59	11	in	in	ADP
ejpam-841	59	12	terms	term	NOUN
ejpam-841	59	13	of	of	ADP
ejpam-841	59	14	the	the	DET
ejpam-841	59	15	three	three	NUM
ejpam-841	59	16	elements	element	NOUN
ejpam-841	59	17	of	of	ADP
ejpam-841	59	18	the	the	DET
ejpam-841	59	19	µ-basis	µ-basis	NOUN
ejpam-841	59	20	.	.	PUNCT
ejpam-841	60	1	we	we	PRON
ejpam-841	60	2	will	will	AUX
ejpam-841	60	3	prove	prove	VERB
ejpam-841	60	4	our	our	PRON
ejpam-841	60	5	results	result	NOUN
ejpam-841	60	6	by	by	ADP
ejpam-841	60	7	comparing	compare	VERB
ejpam-841	60	8	the	the	DET
ejpam-841	60	9	generators	generator	NOUN
ejpam-841	60	10	produced	produce	VERB
ejpam-841	60	11	by	by	ADP
ejpam-841	60	12	our	our	PRON
ejpam-841	60	13	algorithm	algorithm	NOUN
ejpam-841	60	14	with	with	ADP
ejpam-841	60	15	those	those	PRON
ejpam-841	60	16	described	describe	VERB
ejpam-841	60	17	in	in	ADP
ejpam-841	60	18	[	[	X
ejpam-841	60	19	18	18	NUM
ejpam-841	60	20	]	]	PUNCT
ejpam-841	60	21	.	.	PUNCT
ejpam-841	61	1	our	our	PRON
ejpam-841	61	2	algorithm	algorithm	NOUN
ejpam-841	61	3	shows	show	VERB
ejpam-841	61	4	that	that	SCONJ
ejpam-841	61	5	the	the	DET
ejpam-841	61	6	very	very	ADV
ejpam-841	61	7	complicated	complicated	ADJ
ejpam-841	61	8	description	description	NOUN
ejpam-841	61	9	of	of	ADP
ejpam-841	61	10	the	the	DET
ejpam-841	61	11	generators	generator	NOUN
ejpam-841	61	12	of	of	ADP
ejpam-841	61	13	the	the	DET
ejpam-841	61	14	rees	rees	PROPN
ejpam-841	61	15	algebra	algebra	NOUN
ejpam-841	61	16	given	give	VERB
ejpam-841	61	17	in	in	ADP
ejpam-841	61	18	[	[	X
ejpam-841	61	19	18	18	NUM
ejpam-841	61	20	]	]	PUNCT
ejpam-841	61	21	can	can	AUX
ejpam-841	61	22	be	be	AUX
ejpam-841	61	23	simplified	simplify	VERB
ejpam-841	61	24	considerably	considerably	ADV
ejpam-841	61	25	in	in	ADP
ejpam-841	61	26	the	the	DET
ejpam-841	61	27	case	case	NOUN
ejpam-841	61	28	of	of	ADP
ejpam-841	61	29	rational	rational	ADJ
ejpam-841	61	30	space	space	NOUN
ejpam-841	61	31	curves	curve	NOUN
ejpam-841	61	32	of	of	ADP
ejpam-841	61	33	type	type	NOUN
ejpam-841	61	34	(	(	PUNCT
ejpam-841	61	35	1,1	1,1	NUM
ejpam-841	61	36	,	,	PUNCT
ejpam-841	61	37	d	d	NOUN
ejpam-841	61	38	−	−	PROPN
ejpam-841	61	39	2	2	NUM
ejpam-841	61	40	)	)	PUNCT
ejpam-841	61	41	.	.	PUNCT
ejpam-841	62	1	the	the	DET
ejpam-841	62	2	second	second	ADJ
ejpam-841	62	3	goal	goal	NOUN
ejpam-841	62	4	of	of	ADP
ejpam-841	62	5	this	this	DET
ejpam-841	62	6	paper	paper	NOUN
ejpam-841	62	7	is	be	AUX
ejpam-841	62	8	to	to	PART
ejpam-841	62	9	illustrate	illustrate	VERB
ejpam-841	62	10	the	the	DET
ejpam-841	62	11	geometry	geometry	NOUN
ejpam-841	62	12	behind	behind	ADP
ejpam-841	62	13	the	the	DET
ejpam-841	62	14	generators	generator	NOUN
ejpam-841	62	15	via	via	ADP
ejpam-841	62	16	a	a	DET
ejpam-841	62	17	case	case	NOUN
ejpam-841	62	18	study	study	NOUN
ejpam-841	62	19	of	of	ADP
ejpam-841	62	20	rational	rational	ADJ
ejpam-841	62	21	quartic	quartic	ADJ
ejpam-841	62	22	space	space	NOUN
ejpam-841	62	23	curves	curve	NOUN
ejpam-841	62	24	.	.	PUNCT
ejpam-841	63	1	we	we	PRON
ejpam-841	63	2	will	will	AUX
ejpam-841	63	3	construct	construct	VERB
ejpam-841	63	4	the	the	DET
ejpam-841	63	5	implicit	implicit	ADJ
ejpam-841	63	6	equations	equation	NOUN
ejpam-841	63	7	of	of	ADP
ejpam-841	63	8	the	the	DET
ejpam-841	63	9	curve	curve	NOUN
ejpam-841	63	10	and	and	CCONJ
ejpam-841	63	11	the	the	DET
ejpam-841	63	12	defining	define	VERB
ejpam-841	63	13	equations	equation	NOUN
ejpam-841	63	14	for	for	ADP
ejpam-841	63	15	the	the	DET
ejpam-841	63	16	rees	rees	PROPN
ejpam-841	63	17	algebra	algebra	NOUN
ejpam-841	63	18	in	in	ADP
ejpam-841	63	19	a	a	DET
ejpam-841	63	20	simple	simple	ADJ
ejpam-841	63	21	manner	manner	NOUN
ejpam-841	63	22	from	from	ADP
ejpam-841	63	23	the	the	DET
ejpam-841	63	24	elements	element	NOUN
ejpam-841	63	25	of	of	ADP
ejpam-841	63	26	the	the	DET
ejpam-841	63	27	µ-basis	µ-basis	NOUN
ejpam-841	63	28	.	.	PUNCT
ejpam-841	64	1	we	we	PRON
ejpam-841	64	2	proceed	proceed	VERB
ejpam-841	64	3	in	in	ADP
ejpam-841	64	4	the	the	DET
ejpam-841	64	5	following	follow	VERB
ejpam-841	64	6	fashion	fashion	NOUN
ejpam-841	64	7	.	.	PUNCT
ejpam-841	65	1	in	in	ADP
ejpam-841	65	2	section	section	NOUN
ejpam-841	65	3	2	2	NUM
ejpam-841	65	4	we	we	PRON
ejpam-841	65	5	review	review	VERB
ejpam-841	65	6	the	the	DET
ejpam-841	65	7	basic	basic	ADJ
ejpam-841	65	8	notion	notion	NOUN
ejpam-841	65	9	of	of	ADP
ejpam-841	65	10	moving	move	VERB
ejpam-841	65	11	surfaces	surface	NOUN
ejpam-841	65	12	and	and	CCONJ
ejpam-841	65	13	µ-bases	µ-base	NOUN
ejpam-841	65	14	,	,	PUNCT
ejpam-841	65	15	and	and	CCONJ
ejpam-841	65	16	recall	recall	VERB
ejpam-841	65	17	some	some	DET
ejpam-841	65	18	results	result	NOUN
ejpam-841	65	19	concerning	concern	VERB
ejpam-841	65	20	how	how	SCONJ
ejpam-841	65	21	to	to	PART
ejpam-841	65	22	detect	detect	VERB
ejpam-841	65	23	the	the	DET
ejpam-841	65	24	singularities	singularity	NOUN
ejpam-841	65	25	of	of	ADP
ejpam-841	65	26	rational	rational	ADJ
ejpam-841	65	27	space	space	NOUN
ejpam-841	65	28	curves	curve	NOUN
ejpam-841	65	29	using	use	VERB
ejpam-841	65	30	µ-bases	µ-base	NOUN
ejpam-841	65	31	.	.	PUNCT
ejpam-841	66	1	in	in	ADP
ejpam-841	66	2	section	section	NOUN
ejpam-841	66	3	3	3	NUM
ejpam-841	66	4	we	we	PRON
ejpam-841	66	5	provide	provide	VERB
ejpam-841	66	6	an	an	DET
ejpam-841	66	7	algorithm	algorithm	NOUN
ejpam-841	66	8	to	to	PART
ejpam-841	66	9	find	find	VERB
ejpam-841	66	10	minimal	minimal	ADJ
ejpam-841	66	11	generators	generator	NOUN
ejpam-841	66	12	for	for	ADP
ejpam-841	66	13	the	the	DET
ejpam-841	66	14	rees	rees	PROPN
ejpam-841	66	15	algebra	algebra	NOUN
ejpam-841	66	16	of	of	ADP
ejpam-841	66	17	the	the	DET
ejpam-841	66	18	ideal	ideal	NOUN
ejpam-841	66	19	of	of	ADP
ejpam-841	66	20	a	a	DET
ejpam-841	66	21	rational	rational	ADJ
ejpam-841	66	22	space	space	NOUN
ejpam-841	66	23	curve	curve	NOUN
ejpam-841	66	24	based	base	VERB
ejpam-841	66	25	solely	solely	ADV
ejpam-841	66	26	on	on	ADP
ejpam-841	66	27	the	the	DET
ejpam-841	66	28	three	three	NUM
ejpam-841	66	29	elements	element	NOUN
ejpam-841	66	30	of	of	ADP
ejpam-841	66	31	a	a	DET
ejpam-841	66	32	µ-basis	µ-basis	NOUN
ejpam-841	66	33	of	of	ADP
ejpam-841	66	34	the	the	DET
ejpam-841	66	35	rational	rational	ADJ
ejpam-841	66	36	space	space	NOUN
ejpam-841	66	37	curve	curve	NOUN
ejpam-841	66	38	.	.	PUNCT
ejpam-841	67	1	in	in	ADP
ejpam-841	67	2	section	section	NOUN
ejpam-841	67	3	4	4	NUM
ejpam-841	67	4	we	we	PRON
ejpam-841	67	5	illustrate	illustrate	VERB
ejpam-841	67	6	the	the	DET
ejpam-841	67	7	geometry	geometry	NOUN
ejpam-841	67	8	behind	behind	ADP
ejpam-841	67	9	the	the	DET
ejpam-841	67	10	generators	generator	NOUN
ejpam-841	67	11	with	with	ADP
ejpam-841	67	12	a	a	DET
ejpam-841	67	13	case	case	NOUN
ejpam-841	67	14	study	study	NOUN
ejpam-841	67	15	of	of	ADP
ejpam-841	67	16	rational	rational	ADJ
ejpam-841	67	17	quartic	quartic	ADJ
ejpam-841	67	18	space	space	NOUN
ejpam-841	67	19	curves	curve	NOUN
ejpam-841	67	20	.	.	PUNCT
ejpam-841	68	1	2	2	X
ejpam-841	68	2	.	.	X
ejpam-841	68	3	moving	move	VERB
ejpam-841	68	4	planes	plane	NOUN
ejpam-841	68	5	and	and	CCONJ
ejpam-841	68	6	µ-bases	µ-base	VERB
ejpam-841	68	7	throughout	throughout	ADP
ejpam-841	68	8	this	this	DET
ejpam-841	68	9	paper	paper	NOUN
ejpam-841	68	10	,	,	PUNCT
ejpam-841	68	11	we	we	PRON
ejpam-841	68	12	shall	shall	AUX
ejpam-841	68	13	consider	consider	VERB
ejpam-841	68	14	rational	rational	ADJ
ejpam-841	68	15	space	space	NOUN
ejpam-841	68	16	curves	curve	NOUN
ejpam-841	68	17	c	c	NOUN
ejpam-841	68	18	in	in	ADP
ejpam-841	68	19	three	three	NUM
ejpam-841	68	20	-	-	PUNCT
ejpam-841	68	21	dimensional	dimensional	ADJ
ejpam-841	68	22	projective	projective	ADJ
ejpam-841	68	23	space	space	NOUN
ejpam-841	68	24	over	over	ADP
ejpam-841	68	25	a	a	DET
ejpam-841	68	26	field	field	NOUN
ejpam-841	68	27	k	k	NOUN
ejpam-841	68	28	of	of	ADP
ejpam-841	68	29	characteristic	characteristic	ADJ
ejpam-841	68	30	0	0	NUM
ejpam-841	68	31	,	,	PUNCT
ejpam-841	68	32	given	give	VERB
ejpam-841	68	33	as	as	ADP
ejpam-841	68	34	the	the	DET
ejpam-841	68	35	image	image	NOUN
ejpam-841	68	36	of	of	ADP
ejpam-841	68	37	a	a	DET
ejpam-841	68	38	generic	generic	ADJ
ejpam-841	68	39	1	1	NUM
ejpam-841	68	40	-	-	SYM
ejpam-841	68	41	1	1	NUM
ejpam-841	68	42	rational	rational	ADJ
ejpam-841	68	43	parametrization	parametrization	NOUN
ejpam-841	68	44	:	:	PUNCT
ejpam-841	68	45	f(s	f(s	PROPN
ejpam-841	68	46	,	,	PUNCT
ejpam-841	68	47	t	t	PROPN
ejpam-841	68	48	)	)	PUNCT
ejpam-841	68	49	=	=	SYM
ejpam-841	68	50	(	(	PUNCT
ejpam-841	68	51	f0(s	f0(s	PROPN
ejpam-841	68	52	,	,	PUNCT
ejpam-841	68	53	t	t	PROPN
ejpam-841	68	54	)	)	PUNCT
ejpam-841	68	55	,	,	PUNCT
ejpam-841	68	56	f1(s	f1(s	PROPN
ejpam-841	68	57	,	,	PUNCT
ejpam-841	68	58	t	t	PROPN
ejpam-841	68	59	)	)	PUNCT
ejpam-841	68	60	,	,	PUNCT
ejpam-841	68	61	f2(s	f2(s	PROPN
ejpam-841	68	62	,	,	PUNCT
ejpam-841	68	63	t	t	PROPN
ejpam-841	68	64	)	)	PUNCT
ejpam-841	68	65	,	,	PUNCT
ejpam-841	68	66	f3(s	f3(s	PROPN
ejpam-841	68	67	,	,	PUNCT
ejpam-841	68	68	t	t	PROPN
ejpam-841	68	69	)	)	PUNCT
ejpam-841	68	70	)	)	PUNCT
ejpam-841	68	71	,	,	PUNCT
ejpam-841	68	72	(	(	PUNCT
ejpam-841	68	73	s	s	X
ejpam-841	68	74	,	,	PUNCT
ejpam-841	68	75	t	t	PROPN
ejpam-841	68	76	)	)	PUNCT
ejpam-841	68	77	6=	6=	ADP
ejpam-841	68	78	(	(	PUNCT
ejpam-841	68	79	0,0	0,0	NOUN
ejpam-841	68	80	)	)	PUNCT
ejpam-841	68	81	,	,	PUNCT
ejpam-841	68	82	(	(	PUNCT
ejpam-841	68	83	1	1	X
ejpam-841	68	84	)	)	PUNCT
ejpam-841	68	85	where	where	SCONJ
ejpam-841	68	86	f0	f0	PROPN
ejpam-841	68	87	,	,	PUNCT
ejpam-841	68	88	f1	f1	NOUN
ejpam-841	68	89	,	,	PUNCT
ejpam-841	68	90	f2	f2	PROPN
ejpam-841	68	91	,	,	PUNCT
ejpam-841	68	92	f3	f3	PROPN
ejpam-841	68	93	are	be	AUX
ejpam-841	68	94	linearly	linearly	ADV
ejpam-841	68	95	independent	independent	ADJ
ejpam-841	68	96	homogeneous	homogeneous	ADJ
ejpam-841	68	97	polynomials	polynomial	NOUN
ejpam-841	68	98	of	of	ADP
ejpam-841	68	99	the	the	DET
ejpam-841	68	100	same	same	ADJ
ejpam-841	68	101	degree	degree	NOUN
ejpam-841	68	102	d	d	X
ejpam-841	68	103	≥	≥	NUM
ejpam-841	68	104	3	3	NUM
ejpam-841	68	105	in	in	ADP
ejpam-841	68	106	the	the	DET
ejpam-841	68	107	standard	standard	ADJ
ejpam-841	68	108	z	z	ADV
ejpam-841	68	109	-	-	PUNCT
ejpam-841	68	110	graded	grade	VERB
ejpam-841	68	111	ring	ring	NOUN
ejpam-841	68	112	r	r	NOUN
ejpam-841	68	113	:	:	PUNCT
ejpam-841	68	114	=	=	SYM
ejpam-841	68	115	k[s	k[s	PROPN
ejpam-841	68	116	,	,	PUNCT
ejpam-841	68	117	t	t	X
ejpam-841	68	118	]	]	PUNCT
ejpam-841	68	119	,	,	PUNCT
ejpam-841	68	120	and	and	CCONJ
ejpam-841	68	121	gcd	gcd	PROPN
ejpam-841	68	122	(	(	PUNCT
ejpam-841	68	123	f0	f0	PROPN
ejpam-841	68	124	,	,	PUNCT
ejpam-841	68	125	f1	f1	NOUN
ejpam-841	68	126	,	,	PUNCT
ejpam-841	68	127	f2	f2	PROPN
ejpam-841	68	128	,	,	PUNCT
ejpam-841	68	129	f3	f3	NOUN
ejpam-841	68	130	)	)	PUNCT
ejpam-841	68	131	=	=	SYM
ejpam-841	69	1	1	1	X
ejpam-841	69	2	.	.	X
ejpam-841	70	1	we	we	PRON
ejpam-841	70	2	begin	begin	VERB
ejpam-841	70	3	by	by	ADP
ejpam-841	70	4	briefly	briefly	ADV
ejpam-841	70	5	recalling	recall	VERB
ejpam-841	70	6	some	some	DET
ejpam-841	70	7	basic	basic	ADJ
ejpam-841	70	8	definitions	definition	NOUN
ejpam-841	70	9	.	.	PUNCT
ejpam-841	71	1	definition	definition	NOUN
ejpam-841	71	2	1	1	NUM
ejpam-841	71	3	.	.	PUNCT
ejpam-841	72	1	a	a	DET
ejpam-841	72	2	moving	move	VERB
ejpam-841	72	3	surface	surface	NOUN
ejpam-841	72	4	of	of	ADP
ejpam-841	72	5	degree	degree	NOUN
ejpam-841	72	6	r	r	NOUN
ejpam-841	72	7	is	be	AUX
ejpam-841	72	8	a	a	DET
ejpam-841	72	9	polynomial	polynomial	ADJ
ejpam-841	72	10	∑	∑	PUNCT
ejpam-841	72	11	i+	i+	NUM
ejpam-841	72	12	j+ℓ+k	j+ℓ+k	NOUN
ejpam-841	72	13	=	=	NOUN
ejpam-841	72	14	r	r	NOUN
ejpam-841	72	15	ai	ai	VERB
ejpam-841	72	16	jℓk(s	jℓk(s	PROPN
ejpam-841	72	17	,	,	PUNCT
ejpam-841	72	18	t)x	t)x	PUNCT
ejpam-841	72	19	i	i	PROPN
ejpam-841	72	20	y	y	PROPN
ejpam-841	72	21	jzℓwk	jzℓwk	PROPN
ejpam-841	72	22	,	,	PUNCT
ejpam-841	72	23	ai	ai	VERB
ejpam-841	72	24	jℓk	jℓk	PROPN
ejpam-841	72	25	∈	∈	PROPN
ejpam-841	72	26	r.	r.	PROPN
ejpam-841	72	27	j.	j.	PROPN
ejpam-841	72	28	hoffman	hoffman	PROPN
ejpam-841	72	29	,	,	PUNCT
ejpam-841	72	30	h.	h.	PROPN
ejpam-841	72	31	wang	wang	PROPN
ejpam-841	72	32	,	,	PUNCT
ejpam-841	72	33	x.	x.	PROPN
ejpam-841	72	34	jia	jia	PROPN
ejpam-841	72	35	,	,	PUNCT
ejpam-841	72	36	r.	r.	PROPN
ejpam-841	72	37	goldman	goldman	PROPN
ejpam-841	72	38	/	/	SYM
ejpam-841	72	39	eur	eur	PROPN
ejpam-841	72	40	.	.	PUNCT
ejpam-841	73	1	j.	j.	PROPN
ejpam-841	73	2	pure	pure	PROPN
ejpam-841	73	3	appl	appl	PROPN
ejpam-841	73	4	.	.	PROPN
ejpam-841	73	5	math	math	PROPN
ejpam-841	73	6	,	,	PUNCT
ejpam-841	73	7	3	3	NUM
ejpam-841	73	8	(	(	PUNCT
ejpam-841	73	9	2010	2010	NUM
ejpam-841	73	10	)	)	PUNCT
ejpam-841	73	11	,	,	PUNCT
ejpam-841	73	12	602	602	NUM
ejpam-841	73	13	-	-	SYM
ejpam-841	73	14	632	632	NUM
ejpam-841	73	15	605	605	NUM
ejpam-841	73	16	this	this	DET
ejpam-841	73	17	polynomial	polynomial	NOUN
ejpam-841	73	18	is	be	AUX
ejpam-841	73	19	said	say	VERB
ejpam-841	73	20	to	to	PART
ejpam-841	73	21	follow	follow	VERB
ejpam-841	73	22	the	the	DET
ejpam-841	73	23	parametrization	parametrization	NOUN
ejpam-841	73	24	(	(	PUNCT
ejpam-841	73	25	1	1	X
ejpam-841	73	26	)	)	PUNCT
ejpam-841	73	27	if	if	SCONJ
ejpam-841	73	28	∑	∑	PUNCT
ejpam-841	73	29	i+	i+	NUM
ejpam-841	73	30	j+ℓ+k	j+ℓ+k	NOUN
ejpam-841	74	1	=	=	NOUN
ejpam-841	74	2	r	r	NOUN
ejpam-841	74	3	ai	ai	VERB
ejpam-841	74	4	jℓk(s	jℓk(s	PROPN
ejpam-841	74	5	,	,	PUNCT
ejpam-841	74	6	t	t	PROPN
ejpam-841	74	7	)	)	PUNCT
ejpam-841	74	8	f0(s	f0(s	PROPN
ejpam-841	74	9	,	,	PUNCT
ejpam-841	74	10	t)i	t)i	X
ejpam-841	74	11	f1(s	f1(s	PROPN
ejpam-841	74	12	,	,	PUNCT
ejpam-841	74	13	t	t	PROPN
ejpam-841	74	14	)	)	PUNCT
ejpam-841	74	15	j	j	PROPN
ejpam-841	74	16	f2(s	f2(s	PROPN
ejpam-841	74	17	,	,	PUNCT
ejpam-841	74	18	t)ℓ	t)ℓ	X
ejpam-841	74	19	f3(s	f3(s	PROPN
ejpam-841	74	20	,	,	PUNCT
ejpam-841	74	21	t)k	t)k	ADJ
ejpam-841	74	22	≡	≡	PROPN
ejpam-841	74	23	0	0	PUNCT
ejpam-841	74	24	.	.	PUNCT
ejpam-841	75	1	hence	hence	ADV
ejpam-841	75	2	a	a	DET
ejpam-841	75	3	moving	move	VERB
ejpam-841	75	4	surface	surface	NOUN
ejpam-841	75	5	of	of	ADP
ejpam-841	75	6	degree	degree	NOUN
ejpam-841	75	7	r	r	NOUN
ejpam-841	75	8	follows	follow	VERB
ejpam-841	75	9	the	the	DET
ejpam-841	75	10	parametrization	parametrization	NOUN
ejpam-841	75	11	(	(	PUNCT
ejpam-841	75	12	1	1	X
ejpam-841	75	13	)	)	PUNCT
ejpam-841	75	14	if	if	SCONJ
ejpam-841	75	15	and	and	CCONJ
ejpam-841	75	16	only	only	ADV
ejpam-841	75	17	if	if	SCONJ
ejpam-841	75	18	(	(	PUNCT
ejpam-841	75	19	ai	ai	VERB
ejpam-841	75	20	jℓk)i+	jℓk)i+	ADV
ejpam-841	75	21	j+ℓ+k	j+ℓ+k	X
ejpam-841	75	22	=	=	NOUN
ejpam-841	75	23	r	r	NOUN
ejpam-841	75	24	∈	∈	NOUN
ejpam-841	75	25	syz(i	syz(i	NOUN
ejpam-841	75	26	r	r	NOUN
ejpam-841	75	27	)	)	PUNCT
ejpam-841	75	28	,	,	PUNCT
ejpam-841	75	29	where	where	SCONJ
ejpam-841	75	30	i	i	PRON
ejpam-841	75	31	is	be	AUX
ejpam-841	75	32	the	the	DET
ejpam-841	75	33	ideal	ideal	ADJ
ejpam-841	75	34	〈	〈	PROPN
ejpam-841	75	35	f0	f0	PROPN
ejpam-841	75	36	,	,	PUNCT
ejpam-841	75	37	f1	f1	NOUN
ejpam-841	75	38	,	,	PUNCT
ejpam-841	75	39	f2	f2	PROPN
ejpam-841	75	40	,	,	PUNCT
ejpam-841	75	41	f3	f3	ADJ
ejpam-841	75	42	〉	〉	NOUN
ejpam-841	75	43	⊂	⊂	PROPN
ejpam-841	75	44	r.	r.	VERB
ejpam-841	75	45	the	the	DET
ejpam-841	75	46	set	set	NOUN
ejpam-841	75	47	of	of	ADP
ejpam-841	75	48	all	all	DET
ejpam-841	75	49	moving	move	VERB
ejpam-841	75	50	surfaces	surface	NOUN
ejpam-841	75	51	that	that	PRON
ejpam-841	75	52	follow	follow	VERB
ejpam-841	75	53	the	the	DET
ejpam-841	75	54	parametrization	parametrization	NOUN
ejpam-841	75	55	(	(	PUNCT
ejpam-841	75	56	1	1	X
ejpam-841	75	57	)	)	PUNCT
ejpam-841	75	58	is	be	AUX
ejpam-841	75	59	an	an	DET
ejpam-841	75	60	ideal	ideal	NOUN
ejpam-841	75	61	in	in	ADP
ejpam-841	75	62	r[x	r[x	NOUN
ejpam-841	75	63	,	,	PUNCT
ejpam-841	75	64	y	y	PROPN
ejpam-841	75	65	,	,	PUNCT
ejpam-841	75	66	z	z	PROPN
ejpam-841	75	67	,	,	PUNCT
ejpam-841	75	68	w	w	PROPN
ejpam-841	75	69	]	]	X
ejpam-841	75	70	,	,	PUNCT
ejpam-841	75	71	called	call	VERB
ejpam-841	75	72	the	the	DET
ejpam-841	75	73	moving	move	VERB
ejpam-841	75	74	surface	surface	NOUN
ejpam-841	75	75	ideal	ideal	NOUN
ejpam-841	75	76	.	.	PUNCT
ejpam-841	76	1	remark	remark	NOUN
ejpam-841	76	2	1	1	NUM
ejpam-841	76	3	.	.	PUNCT
ejpam-841	77	1	if	if	SCONJ
ejpam-841	77	2	f	f	PROPN
ejpam-841	77	3	is	be	AUX
ejpam-841	77	4	a	a	DET
ejpam-841	77	5	moving	move	VERB
ejpam-841	77	6	surface	surface	NOUN
ejpam-841	77	7	,	,	PUNCT
ejpam-841	77	8	then	then	ADV
ejpam-841	77	9	f	f	PROPN
ejpam-841	77	10	is	be	AUX
ejpam-841	77	11	in	in	ADP
ejpam-841	77	12	the	the	DET
ejpam-841	77	13	bigraded	bigraded	ADJ
ejpam-841	77	14	ringk[s	ringk[	NOUN
ejpam-841	77	15	,	,	PUNCT
ejpam-841	77	16	t	t	PROPN
ejpam-841	77	17	;	;	PUNCT
ejpam-841	77	18	x	x	X
ejpam-841	77	19	,	,	PUNCT
ejpam-841	77	20	y	y	PROPN
ejpam-841	77	21	,	,	PUNCT
ejpam-841	77	22	z	z	PROPN
ejpam-841	77	23	,	,	PUNCT
ejpam-841	77	24	w	w	PROPN
ejpam-841	77	25	]	]	X
ejpam-841	77	26	=	=	SYM
ejpam-841	77	27	r[x	r[x	NOUN
ejpam-841	77	28	,	,	PUNCT
ejpam-841	77	29	y	y	PROPN
ejpam-841	77	30	,	,	PUNCT
ejpam-841	77	31	z	z	PROPN
ejpam-841	77	32	,	,	PUNCT
ejpam-841	77	33	w	w	NOUN
ejpam-841	77	34	]	]	X
ejpam-841	77	35	,	,	PUNCT
ejpam-841	77	36	and	and	CCONJ
ejpam-841	77	37	deg	deg	PROPN
ejpam-841	77	38	(	(	PUNCT
ejpam-841	77	39	f	f	PROPN
ejpam-841	77	40	)	)	PUNCT
ejpam-841	78	1	=	=	SYM
ejpam-841	78	2	(	(	PUNCT
ejpam-841	78	3	d1	d1	PROPN
ejpam-841	78	4	,	,	PUNCT
ejpam-841	78	5	d2	d2	PROPN
ejpam-841	78	6	)	)	PUNCT
ejpam-841	78	7	where	where	SCONJ
ejpam-841	78	8	d1	d1	PROPN
ejpam-841	78	9	is	be	AUX
ejpam-841	78	10	the	the	DET
ejpam-841	78	11	degree	degree	NOUN
ejpam-841	78	12	in	in	ADP
ejpam-841	78	13	s	s	PROPN
ejpam-841	78	14	,	,	PUNCT
ejpam-841	78	15	t	t	PROPN
ejpam-841	78	16	,	,	PUNCT
ejpam-841	78	17	and	and	CCONJ
ejpam-841	78	18	d2	d2	PROPN
ejpam-841	78	19	is	be	AUX
ejpam-841	78	20	the	the	DET
ejpam-841	78	21	degree	degree	NOUN
ejpam-841	78	22	in	in	ADP
ejpam-841	78	23	x	x	SYM
ejpam-841	78	24	,	,	PUNCT
ejpam-841	78	25	y	y	PROPN
ejpam-841	78	26	,	,	PUNCT
ejpam-841	78	27	z	z	PROPN
ejpam-841	78	28	,	,	PUNCT
ejpam-841	78	29	w.	w.	PROPN
ejpam-841	78	30	definition	definition	NOUN
ejpam-841	78	31	2	2	NUM
ejpam-841	78	32	.	.	PUNCT
ejpam-841	79	1	a	a	DET
ejpam-841	79	2	moving	move	VERB
ejpam-841	79	3	plane	plane	NOUN
ejpam-841	79	4	is	be	AUX
ejpam-841	79	5	a	a	DET
ejpam-841	79	6	moving	move	VERB
ejpam-841	79	7	surface	surface	NOUN
ejpam-841	79	8	of	of	ADP
ejpam-841	79	9	degree	degree	NOUN
ejpam-841	79	10	d2	d2	NOUN
ejpam-841	79	11	=	=	SYM
ejpam-841	79	12	1	1	NUM
ejpam-841	79	13	.	.	PUNCT
ejpam-841	80	1	an	an	DET
ejpam-841	80	2	axial	axial	ADJ
ejpam-841	80	3	moving	move	VERB
ejpam-841	80	4	plane	plane	NOUN
ejpam-841	80	5	is	be	AUX
ejpam-841	80	6	a	a	DET
ejpam-841	80	7	moving	move	VERB
ejpam-841	80	8	plane	plane	NOUN
ejpam-841	80	9	where	where	SCONJ
ejpam-841	80	10	all	all	DET
ejpam-841	80	11	the	the	DET
ejpam-841	80	12	planes	plane	NOUN
ejpam-841	80	13	of	of	ADP
ejpam-841	80	14	the	the	DET
ejpam-841	80	15	family	family	NOUN
ejpam-841	80	16	pass	pass	VERB
ejpam-841	80	17	through	through	ADP
ejpam-841	80	18	either	either	CCONJ
ejpam-841	80	19	a	a	DET
ejpam-841	80	20	common	common	ADJ
ejpam-841	80	21	point	point	NOUN
ejpam-841	80	22	a	a	PRON
ejpam-841	80	23	or	or	CCONJ
ejpam-841	80	24	a	a	DET
ejpam-841	80	25	common	common	ADJ
ejpam-841	80	26	line	line	NOUN
ejpam-841	80	27	←→	←→	PROPN
ejpam-841	80	28	ab	ab	PROPN
ejpam-841	80	29	.	.	PUNCT
ejpam-841	81	1	the	the	DET
ejpam-841	81	2	point	point	NOUN
ejpam-841	81	3	a	a	PRON
ejpam-841	81	4	is	be	AUX
ejpam-841	81	5	called	call	VERB
ejpam-841	81	6	an	an	DET
ejpam-841	81	7	axis	axis	ADJ
ejpam-841	81	8	point	point	NOUN
ejpam-841	81	9	,	,	PUNCT
ejpam-841	81	10	and	and	CCONJ
ejpam-841	81	11	the	the	DET
ejpam-841	81	12	line	line	NOUN
ejpam-841	81	13	←→	←→	PROPN
ejpam-841	81	14	ab	ab	PROPN
ejpam-841	81	15	is	be	AUX
ejpam-841	81	16	called	call	VERB
ejpam-841	81	17	the	the	DET
ejpam-841	81	18	axis	axis	ADJ
ejpam-841	81	19	line	line	NOUN
ejpam-841	81	20	or	or	CCONJ
ejpam-841	81	21	axis	axis	NOUN
ejpam-841	81	22	of	of	ADP
ejpam-841	81	23	the	the	DET
ejpam-841	81	24	moving	move	VERB
ejpam-841	81	25	plane	plane	NOUN
ejpam-841	81	26	.	.	PUNCT
ejpam-841	82	1	similarly	similarly	ADV
ejpam-841	82	2	,	,	PUNCT
ejpam-841	82	3	a	a	DET
ejpam-841	82	4	moving	move	VERB
ejpam-841	82	5	quadric	quadric	ADJ
ejpam-841	82	6	is	be	AUX
ejpam-841	82	7	a	a	DET
ejpam-841	82	8	moving	move	VERB
ejpam-841	82	9	surface	surface	NOUN
ejpam-841	82	10	of	of	ADP
ejpam-841	82	11	degree	degree	NOUN
ejpam-841	82	12	d2	d2	NOUN
ejpam-841	82	13	=	=	SYM
ejpam-841	82	14	2	2	NUM
ejpam-841	82	15	.	.	PUNCT
ejpam-841	82	16	when	when	SCONJ
ejpam-841	82	17	we	we	PRON
ejpam-841	82	18	refer	refer	VERB
ejpam-841	82	19	to	to	ADP
ejpam-841	82	20	the	the	DET
ejpam-841	82	21	degree	degree	NOUN
ejpam-841	82	22	of	of	ADP
ejpam-841	82	23	a	a	DET
ejpam-841	82	24	moving	move	VERB
ejpam-841	82	25	plane	plane	NOUN
ejpam-841	82	26	(	(	PUNCT
ejpam-841	82	27	or	or	CCONJ
ejpam-841	82	28	moving	move	VERB
ejpam-841	82	29	quadric	quadric	ADJ
ejpam-841	82	30	)	)	PUNCT
ejpam-841	82	31	,	,	PUNCT
ejpam-841	82	32	we	we	PRON
ejpam-841	82	33	are	be	AUX
ejpam-841	82	34	referring	refer	VERB
ejpam-841	82	35	to	to	ADP
ejpam-841	82	36	the	the	DET
ejpam-841	82	37	degree	degree	NOUN
ejpam-841	82	38	in	in	ADP
ejpam-841	82	39	s	s	PROPN
ejpam-841	82	40	,	,	PUNCT
ejpam-841	82	41	t	t	PROPN
ejpam-841	82	42	,	,	PUNCT
ejpam-841	82	43	i.e.	i.e.	X
ejpam-841	82	44	,	,	PUNCT
ejpam-841	82	45	d1	d1	PROPN
ejpam-841	82	46	.	.	PUNCT
ejpam-841	83	1	for	for	ADP
ejpam-841	83	2	example	example	NOUN
ejpam-841	83	3	,	,	PUNCT
ejpam-841	83	4	consider	consider	VERB
ejpam-841	83	5	the	the	DET
ejpam-841	83	6	rational	rational	ADJ
ejpam-841	83	7	quintic	quintic	ADJ
ejpam-841	83	8	space	space	NOUN
ejpam-841	83	9	curve	curve	NOUN
ejpam-841	83	10	where	where	SCONJ
ejpam-841	83	11	(	(	PUNCT
ejpam-841	83	12	f0	f0	PROPN
ejpam-841	83	13	,	,	PUNCT
ejpam-841	83	14	f1	f1	NOUN
ejpam-841	83	15	,	,	PUNCT
ejpam-841	83	16	f2	f2	PROPN
ejpam-841	83	17	,	,	PUNCT
ejpam-841	83	18	f3	f3	NOUN
ejpam-841	83	19	)	)	PUNCT
ejpam-841	83	20	=	=	PUNCT
ejpam-841	84	1	(	(	PUNCT
ejpam-841	84	2	s	s	NOUN
ejpam-841	84	3	5	5	NUM
ejpam-841	84	4	,	,	PUNCT
ejpam-841	84	5	s3	s3	PROPN
ejpam-841	84	6	t2	t2	NOUN
ejpam-841	84	7	,	,	PUNCT
ejpam-841	84	8	s2	s2	NOUN
ejpam-841	84	9	t3	t3	PROPN
ejpam-841	84	10	,	,	PUNCT
ejpam-841	84	11	t5	t5	PROPN
ejpam-841	84	12	)	)	PUNCT
ejpam-841	84	13	.	.	PUNCT
ejpam-841	85	1	the	the	DET
ejpam-841	85	2	polynomial	polynomial	ADJ
ejpam-841	85	3	t3	t3	PROPN
ejpam-841	85	4	x−s3z	x−s3z	PROPN
ejpam-841	85	5	is	be	AUX
ejpam-841	85	6	an	an	DET
ejpam-841	85	7	axial	axial	ADJ
ejpam-841	85	8	moving	move	VERB
ejpam-841	85	9	plane	plane	NOUN
ejpam-841	85	10	of	of	ADP
ejpam-841	85	11	degree	degree	NOUN
ejpam-841	85	12	3	3	NUM
ejpam-841	85	13	that	that	PRON
ejpam-841	85	14	follows	follow	VERB
ejpam-841	85	15	the	the	DET
ejpam-841	85	16	curve	curve	NOUN
ejpam-841	85	17	with	with	ADP
ejpam-841	85	18	axis	axis	NOUN
ejpam-841	85	19	←→	←→	PROPN
ejpam-841	85	20	ab	ab	PROPN
ejpam-841	85	21	where	where	SCONJ
ejpam-841	85	22	a=	a=	X
ejpam-841	85	23	(	(	PUNCT
ejpam-841	85	24	0,0,0,1	0,0,0,1	NOUN
ejpam-841	85	25	)	)	PUNCT
ejpam-841	85	26	and	and	CCONJ
ejpam-841	85	27	b	b	X
ejpam-841	85	28	=	=	SYM
ejpam-841	85	29	(	(	PUNCT
ejpam-841	85	30	0,1,0,1	0,1,0,1	NOUN
ejpam-841	85	31	)	)	PUNCT
ejpam-841	85	32	.	.	PUNCT
ejpam-841	86	1	remark	remark	NOUN
ejpam-841	86	2	2	2	NUM
ejpam-841	86	3	.	.	PUNCT
ejpam-841	86	4	a	a	DET
ejpam-841	86	5	moving	move	VERB
ejpam-841	86	6	plane	plane	NOUN
ejpam-841	86	7	of	of	ADP
ejpam-841	86	8	degree	degree	NOUN
ejpam-841	86	9	one	one	NUM
ejpam-841	86	10	in	in	ADP
ejpam-841	86	11	s	s	PROPN
ejpam-841	86	12	,	,	PUNCT
ejpam-841	86	13	t	t	PROPN
ejpam-841	86	14	always	always	ADV
ejpam-841	86	15	has	have	VERB
ejpam-841	86	16	an	an	DET
ejpam-841	86	17	axis	axis	ADJ
ejpam-841	86	18	line	line	NOUN
ejpam-841	86	19	;	;	PUNCT
ejpam-841	86	20	a	a	DET
ejpam-841	86	21	moving	move	VERB
ejpam-841	86	22	plane	plane	NOUN
ejpam-841	86	23	of	of	ADP
ejpam-841	86	24	degree	degree	NOUN
ejpam-841	86	25	two	two	NUM
ejpam-841	86	26	in	in	ADP
ejpam-841	86	27	s	s	PROPN
ejpam-841	86	28	,	,	PUNCT
ejpam-841	86	29	t	t	PROPN
ejpam-841	86	30	always	always	ADV
ejpam-841	86	31	has	have	VERB
ejpam-841	86	32	an	an	DET
ejpam-841	86	33	axis	axis	ADJ
ejpam-841	86	34	point	point	NOUN
ejpam-841	86	35	,	,	PUNCT
ejpam-841	86	36	and	and	CCONJ
ejpam-841	86	37	may	may	AUX
ejpam-841	86	38	have	have	VERB
ejpam-841	86	39	an	an	DET
ejpam-841	86	40	axis	axis	ADJ
ejpam-841	86	41	line	line	NOUN
ejpam-841	86	42	.	.	PUNCT
ejpam-841	87	1	indeed	indeed	ADV
ejpam-841	87	2	if	if	SCONJ
ejpam-841	87	3	we	we	PRON
ejpam-841	87	4	write	write	VERB
ejpam-841	87	5	a	a	DET
ejpam-841	87	6	moving	move	VERB
ejpam-841	87	7	plane	plane	NOUN
ejpam-841	87	8	of	of	ADP
ejpam-841	87	9	degree	degree	NOUN
ejpam-841	87	10	one	one	NUM
ejpam-841	87	11	in	in	ADP
ejpam-841	87	12	s	s	PROPN
ejpam-841	87	13	,	,	PUNCT
ejpam-841	87	14	t	t	PROPN
ejpam-841	87	15	as	as	ADP
ejpam-841	87	16	f	f	PROPN
ejpam-841	87	17	(	(	PUNCT
ejpam-841	87	18	x	x	PROPN
ejpam-841	87	19	,	,	PUNCT
ejpam-841	87	20	y	y	PROPN
ejpam-841	87	21	,	,	PUNCT
ejpam-841	87	22	z	z	PROPN
ejpam-841	87	23	,	,	PUNCT
ejpam-841	87	24	w)s	w)s	PUNCT
ejpam-841	88	1	+	+	CCONJ
ejpam-841	88	2	g(x	g(x	PROPN
ejpam-841	88	3	,	,	PUNCT
ejpam-841	88	4	y	y	PROPN
ejpam-841	88	5	,	,	PUNCT
ejpam-841	88	6	z	z	PROPN
ejpam-841	88	7	,	,	PUNCT
ejpam-841	88	8	w)t	w)t	NOUN
ejpam-841	88	9	,	,	PUNCT
ejpam-841	88	10	where	where	SCONJ
ejpam-841	88	11	deg	deg	PROPN
ejpam-841	88	12	(	(	PUNCT
ejpam-841	88	13	f	f	PROPN
ejpam-841	88	14	)	)	PUNCT
ejpam-841	88	15	=	=	PUNCT
ejpam-841	88	16	deg(g	deg(g	PROPN
ejpam-841	88	17	)	)	PUNCT
ejpam-841	88	18	=	=	SYM
ejpam-841	88	19	1	1	NUM
ejpam-841	88	20	,	,	PUNCT
ejpam-841	88	21	gcd	gcd	PROPN
ejpam-841	88	22	(	(	PUNCT
ejpam-841	88	23	f	f	PROPN
ejpam-841	88	24	,	,	PUNCT
ejpam-841	88	25	g	g	NOUN
ejpam-841	88	26	)	)	PUNCT
ejpam-841	88	27	=	=	SYM
ejpam-841	88	28	1	1	NUM
ejpam-841	88	29	,	,	PUNCT
ejpam-841	88	30	then	then	ADV
ejpam-841	88	31	the	the	DET
ejpam-841	88	32	variety	variety	NOUN
ejpam-841	88	33	v	v	PROPN
ejpam-841	88	34	(	(	PUNCT
ejpam-841	88	35	f	f	PROPN
ejpam-841	88	36	,	,	PUNCT
ejpam-841	88	37	g	g	PROPN
ejpam-841	88	38	)	)	PUNCT
ejpam-841	88	39	is	be	AUX
ejpam-841	88	40	the	the	DET
ejpam-841	88	41	axis	axis	NOUN
ejpam-841	88	42	of	of	ADP
ejpam-841	88	43	the	the	DET
ejpam-841	88	44	moving	move	VERB
ejpam-841	88	45	plane	plane	NOUN
ejpam-841	88	46	.	.	PUNCT
ejpam-841	89	1	it	it	PRON
ejpam-841	89	2	is	be	AUX
ejpam-841	89	3	easy	easy	ADJ
ejpam-841	89	4	to	to	PART
ejpam-841	89	5	see	see	VERB
ejpam-841	89	6	that	that	DET
ejpam-841	89	7	v	v	NOUN
ejpam-841	89	8	(	(	PUNCT
ejpam-841	89	9	f	f	PROPN
ejpam-841	89	10	,	,	PUNCT
ejpam-841	89	11	g	g	PROPN
ejpam-841	89	12	)	)	PUNCT
ejpam-841	89	13	is	be	AUX
ejpam-841	89	14	a	a	DET
ejpam-841	89	15	linear	linear	ADJ
ejpam-841	89	16	variety	variety	NOUN
ejpam-841	89	17	of	of	ADP
ejpam-841	89	18	dimension	dimension	NOUN
ejpam-841	89	19	at	at	ADV
ejpam-841	89	20	least	least	ADJ
ejpam-841	89	21	one	one	NUM
ejpam-841	89	22	;	;	PUNCT
ejpam-841	89	23	hence	hence	ADV
ejpam-841	89	24	a	a	DET
ejpam-841	89	25	moving	move	VERB
ejpam-841	89	26	plane	plane	NOUN
ejpam-841	89	27	of	of	ADP
ejpam-841	89	28	degree	degree	NOUN
ejpam-841	89	29	one	one	NUM
ejpam-841	89	30	in	in	ADP
ejpam-841	89	31	s	s	PROPN
ejpam-841	89	32	,	,	PUNCT
ejpam-841	89	33	t	t	PROPN
ejpam-841	89	34	always	always	ADV
ejpam-841	89	35	has	have	VERB
ejpam-841	89	36	an	an	DET
ejpam-841	89	37	axis	axis	ADJ
ejpam-841	89	38	line	line	NOUN
ejpam-841	89	39	.	.	PUNCT
ejpam-841	90	1	in	in	ADP
ejpam-841	90	2	the	the	DET
ejpam-841	90	3	language	language	NOUN
ejpam-841	90	4	of	of	ADP
ejpam-841	90	5	commutative	commutative	ADJ
ejpam-841	90	6	algebra	algebra	NOUN
ejpam-841	90	7	,	,	PUNCT
ejpam-841	90	8	the	the	DET
ejpam-841	90	9	collection	collection	NOUN
ejpam-841	90	10	of	of	ADP
ejpam-841	90	11	moving	move	VERB
ejpam-841	90	12	planes	plane	NOUN
ejpam-841	90	13	that	that	PRON
ejpam-841	90	14	follow	follow	VERB
ejpam-841	90	15	a	a	DET
ejpam-841	90	16	parametrization	parametrization	NOUN
ejpam-841	90	17	f(s	f(	VERB
ejpam-841	90	18	,	,	PUNCT
ejpam-841	90	19	t	t	PROPN
ejpam-841	90	20	)	)	PUNCT
ejpam-841	90	21	=	=	PRON
ejpam-841	90	22	(	(	PUNCT
ejpam-841	90	23	f0	f0	PROPN
ejpam-841	90	24	,	,	PUNCT
ejpam-841	90	25	f1	f1	NOUN
ejpam-841	90	26	,	,	PUNCT
ejpam-841	90	27	f2	f2	PROPN
ejpam-841	90	28	,	,	PUNCT
ejpam-841	90	29	f3	f3	PROPN
ejpam-841	90	30	)	)	PUNCT
ejpam-841	90	31	is	be	AUX
ejpam-841	90	32	exactly	exactly	ADV
ejpam-841	90	33	syz(i	syz(i	NOUN
ejpam-841	90	34	)	)	PUNCT
ejpam-841	90	35	,	,	PUNCT
ejpam-841	90	36	where	where	SCONJ
ejpam-841	90	37	i	i	PRON
ejpam-841	90	38	=	=	SYM
ejpam-841	90	39	〈	〈	PROPN
ejpam-841	90	40	f0	f0	PROPN
ejpam-841	90	41	,	,	PUNCT
ejpam-841	90	42	f1	f1	NOUN
ejpam-841	90	43	,	,	PUNCT
ejpam-841	90	44	f2	f2	PROPN
ejpam-841	90	45	,	,	PUNCT
ejpam-841	90	46	f3	f3	ADJ
ejpam-841	90	47	〉	〉	NOUN
ejpam-841	90	48	.	.	PUNCT
ejpam-841	91	1	the	the	DET
ejpam-841	91	2	hilbert	hilbert	PROPN
ejpam-841	91	3	-	-	PUNCT
ejpam-841	91	4	burch	burch	PROPN
ejpam-841	91	5	theorem	theorem	NOUN
ejpam-841	91	6	[	[	PUNCT
ejpam-841	91	7	9	9	NUM
ejpam-841	91	8	,	,	PUNCT
ejpam-841	91	9	chapter	chapter	NOUN
ejpam-841	91	10	6	6	NUM
ejpam-841	91	11	]	]	PUNCT
ejpam-841	91	12	says	say	VERB
ejpam-841	91	13	that	that	SCONJ
ejpam-841	91	14	the	the	DET
ejpam-841	91	15	minimal	minimal	ADJ
ejpam-841	91	16	free	free	ADJ
ejpam-841	91	17	resolution	resolution	NOUN
ejpam-841	91	18	of	of	ADP
ejpam-841	91	19	the	the	DET
ejpam-841	91	20	ideal	ideal	NOUN
ejpam-841	91	21	i	i	PRON
ejpam-841	91	22	has	have	VERB
ejpam-841	91	23	the	the	DET
ejpam-841	91	24	following	follow	VERB
ejpam-841	91	25	form	form	NOUN
ejpam-841	91	26	:	:	PUNCT
ejpam-841	91	27	0→	0→	X
ejpam-841	91	28	r(−d	r(−d	NOUN
ejpam-841	91	29	−µ1)⊕	−µ1)⊕	NOUN
ejpam-841	91	30	r(−d	r(−d	VERB
ejpam-841	91	31	−µ2)⊕	−µ2)⊕	ADV
ejpam-841	91	32	r(−d	r(−d	VERB
ejpam-841	91	33	−µ3	−µ3	PROPN
ejpam-841	91	34	)	)	PUNCT
ejpam-841	92	1	p	p	X
ejpam-841	92	2	,	,	PUNCT
ejpam-841	92	3	q	q	ADJ
ejpam-841	92	4	,	,	PUNCT
ejpam-841	92	5	r	r	PROPN
ejpam-841	92	6	−−−→	−−−→	VERB
ejpam-841	92	7	r4(−d	r4(−d	PROPN
ejpam-841	92	8	)	)	PUNCT
ejpam-841	92	9	f0	f0	PROPN
ejpam-841	92	10	,	,	PUNCT
ejpam-841	92	11	f1	f1	NOUN
ejpam-841	92	12	,	,	PUNCT
ejpam-841	92	13	f2	f2	PROPN
ejpam-841	92	14	,	,	PUNCT
ejpam-841	92	15	f3	f3	PROPN
ejpam-841	92	16	−−−−−→	−−−−−→	NUM
ejpam-841	92	17	i	i	PROPN
ejpam-841	92	18	→	→	PROPN
ejpam-841	92	19	0	0	PROPN
ejpam-841	92	20	,	,	PUNCT
ejpam-841	92	21	where	where	SCONJ
ejpam-841	92	22	µ1	µ1	PROPN
ejpam-841	92	23	≤	≤	X
ejpam-841	92	24	µ2	µ2	PROPN
ejpam-841	92	25	≤	≤	ADJ
ejpam-841	92	26	µ3	µ3	NOUN
ejpam-841	92	27	,	,	PUNCT
ejpam-841	92	28	µ1	µ1	PROPN
ejpam-841	92	29	+	+	CCONJ
ejpam-841	93	1	µ2	µ2	PROPN
ejpam-841	93	2	+	+	CCONJ
ejpam-841	93	3	µ3	µ3	NOUN
ejpam-841	94	1	=	=	SYM
ejpam-841	94	2	d	d	PROPN
ejpam-841	94	3	=	=	SYM
ejpam-841	94	4	deg	deg	PROPN
ejpam-841	94	5	(	(	PUNCT
ejpam-841	94	6	fi	fi	NOUN
ejpam-841	94	7	)	)	PUNCT
ejpam-841	94	8	.	.	PUNCT
ejpam-841	95	1	thus	thus	ADV
ejpam-841	95	2	the	the	DET
ejpam-841	95	3	syzygies	syzygy	NOUN
ejpam-841	95	4	syz	syz	PROPN
ejpam-841	95	5	(	(	PUNCT
ejpam-841	95	6	f0	f0	PROPN
ejpam-841	95	7	,	,	PUNCT
ejpam-841	95	8	f1	f1	NOUN
ejpam-841	95	9	,	,	PUNCT
ejpam-841	95	10	f2	f2	PROPN
ejpam-841	95	11	,	,	PUNCT
ejpam-841	95	12	f3	f3	PROPN
ejpam-841	95	13	)	)	PUNCT
ejpam-841	95	14	are	be	AUX
ejpam-841	95	15	generated	generate	VERB
ejpam-841	95	16	by	by	ADP
ejpam-841	95	17	p	p	X
ejpam-841	95	18	,	,	PUNCT
ejpam-841	95	19	q	q	PROPN
ejpam-841	95	20	,	,	PUNCT
ejpam-841	95	21	r.	r.	VERB
ejpam-841	95	22	the	the	DET
ejpam-841	95	23	generators	generator	NOUN
ejpam-841	95	24	p	p	X
ejpam-841	95	25	,	,	PUNCT
ejpam-841	95	26	q	q	ADJ
ejpam-841	95	27	,	,	PUNCT
ejpam-841	95	28	r	r	NOUN
ejpam-841	95	29	are	be	AUX
ejpam-841	95	30	called	call	VERB
ejpam-841	95	31	a	a	DET
ejpam-841	95	32	µ-basis	µ-basis	NOUN
ejpam-841	95	33	.	.	PUNCT
ejpam-841	96	1	we	we	PRON
ejpam-841	96	2	sometimes	sometimes	ADV
ejpam-841	96	3	write	write	VERB
ejpam-841	96	4	p	p	PRON
ejpam-841	96	5	,	,	PUNCT
ejpam-841	96	6	q	q	ADJ
ejpam-841	96	7	,	,	PUNCT
ejpam-841	96	8	r	r	NOUN
ejpam-841	96	9	as	as	ADP
ejpam-841	96	10	j.	j.	PROPN
ejpam-841	96	11	hoffman	hoffman	PROPN
ejpam-841	96	12	,	,	PUNCT
ejpam-841	96	13	h.	h.	PROPN
ejpam-841	96	14	wang	wang	PROPN
ejpam-841	96	15	,	,	PUNCT
ejpam-841	96	16	x.	x.	PROPN
ejpam-841	96	17	jia	jia	PROPN
ejpam-841	96	18	,	,	PUNCT
ejpam-841	96	19	r.	r.	PROPN
ejpam-841	96	20	goldman	goldman	PROPN
ejpam-841	96	21	/	/	SYM
ejpam-841	96	22	eur	eur	PROPN
ejpam-841	96	23	.	.	PUNCT
ejpam-841	97	1	j.	j.	PROPN
ejpam-841	97	2	pure	pure	PROPN
ejpam-841	97	3	appl	appl	PROPN
ejpam-841	97	4	.	.	PROPN
ejpam-841	97	5	math	math	PROPN
ejpam-841	97	6	,	,	PUNCT
ejpam-841	97	7	3	3	NUM
ejpam-841	97	8	(	(	PUNCT
ejpam-841	97	9	2010	2010	NUM
ejpam-841	97	10	)	)	PUNCT
ejpam-841	97	11	,	,	PUNCT
ejpam-841	97	12	602	602	NUM
ejpam-841	97	13	-	-	SYM
ejpam-841	97	14	632	632	NUM
ejpam-841	97	15	606	606	NUM
ejpam-841	97	16	three	three	NUM
ejpam-841	97	17	independent	independent	ADJ
ejpam-841	97	18	moving	move	VERB
ejpam-841	97	19	planes	plane	NOUN
ejpam-841	97	20	p	p	X
ejpam-841	97	21	,	,	PUNCT
ejpam-841	97	22	q	q	ADJ
ejpam-841	97	23	,	,	PUNCT
ejpam-841	97	24	r	r	NOUN
ejpam-841	97	25	,	,	PUNCT
ejpam-841	97	26	where	where	SCONJ
ejpam-841	97	27	p	p	NOUN
ejpam-841	97	28	=	=	NOUN
ejpam-841	97	29	p·x	p·x	NOUN
ejpam-841	97	30	,	,	PUNCT
ejpam-841	97	31	q	q	NOUN
ejpam-841	97	32	=	=	PUNCT
ejpam-841	97	33	q·x	q·x	NOUN
ejpam-841	97	34	,	,	PUNCT
ejpam-841	97	35	r	r	NOUN
ejpam-841	97	36	=	=	SYM
ejpam-841	97	37	r·x	r·x	NOUN
ejpam-841	97	38	and	and	CCONJ
ejpam-841	97	39	x=	x=	PUNCT
ejpam-841	97	40	(	(	PUNCT
ejpam-841	97	41	x	x	X
ejpam-841	97	42	,	,	PUNCT
ejpam-841	97	43	y	y	PROPN
ejpam-841	97	44	,	,	PUNCT
ejpam-841	97	45	z	z	PROPN
ejpam-841	97	46	,	,	PUNCT
ejpam-841	97	47	w	w	PROPN
ejpam-841	97	48	)	)	PUNCT
ejpam-841	97	49	,	,	PUNCT
ejpam-841	97	50	of	of	ADP
ejpam-841	97	51	homogeneous	homogeneous	ADJ
ejpam-841	97	52	degrees	degree	NOUN
ejpam-841	97	53	µ1	µ1	NOUN
ejpam-841	97	54	≤	≤	NUM
ejpam-841	97	55	µ2	µ2	PROPN
ejpam-841	97	56	≤	≤	ADJ
ejpam-841	97	57	µ3	µ3	NOUN
ejpam-841	97	58	in	in	ADP
ejpam-841	97	59	s	s	PROPN
ejpam-841	97	60	,	,	PUNCT
ejpam-841	97	61	t	t	PROPN
ejpam-841	97	62	:	:	PUNCT
ejpam-841	97	63	p	p	X
ejpam-841	97	64	=	=	PUNCT
ejpam-841	97	65	px	px	X
ejpam-841	97	66	x	x	PROPN
ejpam-841	98	1	+	+	CCONJ
ejpam-841	98	2	py	py	PROPN
ejpam-841	98	3	y	y	PROPN
ejpam-841	98	4	+	+	CCONJ
ejpam-841	98	5	pzz	pzz	PROPN
ejpam-841	98	6	+	+	CCONJ
ejpam-841	98	7	pww	pww	ADJ
ejpam-841	98	8	,	,	PUNCT
ejpam-841	98	9	q	q	PUNCT
ejpam-841	99	1	=	=	PUNCT
ejpam-841	99	2	qx	qx	PROPN
ejpam-841	99	3	x	x	PROPN
ejpam-841	99	4	+	+	NUM
ejpam-841	99	5	qy	qy	PROPN
ejpam-841	99	6	y	y	NOUN
ejpam-841	99	7	+	+	NUM
ejpam-841	99	8	qzz	qzz	PROPN
ejpam-841	99	9	+	+	NUM
ejpam-841	99	10	qww	qww	NOUN
ejpam-841	99	11	,	,	PUNCT
ejpam-841	99	12	r	r	NOUN
ejpam-841	99	13	=	=	SYM
ejpam-841	99	14	rx	rx	NOUN
ejpam-841	99	15	x	x	PUNCT
ejpam-841	100	1	+	+	PUNCT
ejpam-841	100	2	ry	ry	NOUN
ejpam-841	100	3	y	y	PROPN
ejpam-841	100	4	+	+	CCONJ
ejpam-841	100	5	rzz	rzz	PROPN
ejpam-841	100	6	+	+	CCONJ
ejpam-841	100	7	rww	rww	NOUN
ejpam-841	100	8	,	,	PUNCT
ejpam-841	100	9	and	and	CCONJ
ejpam-841	100	10	these	these	DET
ejpam-841	100	11	three	three	NUM
ejpam-841	100	12	elements	element	NOUN
ejpam-841	100	13	p	p	X
ejpam-841	100	14	,	,	PUNCT
ejpam-841	100	15	q	q	INTJ
ejpam-841	100	16	,	,	PUNCT
ejpam-841	100	17	r	r	NOUN
ejpam-841	100	18	are	be	AUX
ejpam-841	100	19	also	also	ADV
ejpam-841	100	20	called	call	VERB
ejpam-841	100	21	a	a	DET
ejpam-841	100	22	µ-basis	µ-basis	NOUN
ejpam-841	100	23	.	.	PUNCT
ejpam-841	101	1	there	there	PRON
ejpam-841	101	2	is	be	VERB
ejpam-841	101	3	a	a	DET
ejpam-841	101	4	simple	simple	ADJ
ejpam-841	101	5	algorithm	algorithm	NOUN
ejpam-841	101	6	to	to	PART
ejpam-841	101	7	compute	compute	VERB
ejpam-841	101	8	a	a	DET
ejpam-841	101	9	µ-basis	µ-basis	NOUN
ejpam-841	101	10	from	from	ADP
ejpam-841	101	11	the	the	DET
ejpam-841	101	12	polynomials	polynomial	NOUN
ejpam-841	101	13	f0	f0	PROPN
ejpam-841	101	14	,	,	PUNCT
ejpam-841	101	15	f1	f1	NOUN
ejpam-841	101	16	,	,	PUNCT
ejpam-841	101	17	f2	f2	PROPN
ejpam-841	101	18	,	,	PUNCT
ejpam-841	101	19	f3	f3	PROPN
ejpam-841	101	20	using	use	VERB
ejpam-841	101	21	only	only	ADJ
ejpam-841	101	22	gaussian	gaussian	ADJ
ejpam-841	101	23	elimination	elimination	NOUN
ejpam-841	101	24	[	[	X
ejpam-841	101	25	24	24	NUM
ejpam-841	101	26	]	]	PUNCT
ejpam-841	101	27	.	.	PUNCT
ejpam-841	102	1	although	although	SCONJ
ejpam-841	102	2	the	the	DET
ejpam-841	102	3	µ-basis	µ-basis	NOUN
ejpam-841	102	4	elements	element	NOUN
ejpam-841	102	5	are	be	AUX
ejpam-841	102	6	not	not	PART
ejpam-841	102	7	unique	unique	ADJ
ejpam-841	102	8	,	,	PUNCT
ejpam-841	102	9	the	the	DET
ejpam-841	102	10	degrees	degree	NOUN
ejpam-841	102	11	µ1,µ2,µ3	µ1,µ2,µ3	ADV
ejpam-841	102	12	of	of	ADP
ejpam-841	102	13	the	the	DET
ejpam-841	102	14	µ-basis	µ-basis	NOUN
ejpam-841	102	15	elements	element	NOUN
ejpam-841	102	16	are	be	AUX
ejpam-841	102	17	unique	unique	ADJ
ejpam-841	102	18	.	.	PUNCT
ejpam-841	103	1	from	from	ADP
ejpam-841	103	2	now	now	ADV
ejpam-841	103	3	on	on	ADV
ejpam-841	103	4	,	,	PUNCT
ejpam-841	103	5	we	we	PRON
ejpam-841	103	6	will	will	AUX
ejpam-841	103	7	use	use	VERB
ejpam-841	103	8	(	(	PUNCT
ejpam-841	103	9	µ1,µ2,µ3	µ1,µ2,µ3	NOUN
ejpam-841	103	10	)	)	PUNCT
ejpam-841	103	11	to	to	PART
ejpam-841	103	12	denote	denote	VERB
ejpam-841	103	13	the	the	DET
ejpam-841	103	14	type	type	NOUN
ejpam-841	103	15	of	of	ADP
ejpam-841	103	16	a	a	DET
ejpam-841	103	17	rational	rational	ADJ
ejpam-841	103	18	space	space	NOUN
ejpam-841	103	19	curve	curve	NOUN
ejpam-841	103	20	.	.	PUNCT
ejpam-841	104	1	throughout	throughout	ADP
ejpam-841	104	2	this	this	DET
ejpam-841	104	3	paper	paper	NOUN
ejpam-841	104	4	,	,	PUNCT
ejpam-841	104	5	we	we	PRON
ejpam-841	104	6	focus	focus	VERB
ejpam-841	104	7	on	on	ADP
ejpam-841	104	8	rational	rational	ADJ
ejpam-841	104	9	space	space	NOUN
ejpam-841	104	10	curves	curve	NOUN
ejpam-841	104	11	of	of	ADP
ejpam-841	104	12	type	type	NOUN
ejpam-841	104	13	(	(	PUNCT
ejpam-841	104	14	1,1	1,1	NUM
ejpam-841	104	15	,	,	PUNCT
ejpam-841	104	16	d	d	NOUN
ejpam-841	104	17	−	−	PROPN
ejpam-841	104	18	2	2	NUM
ejpam-841	104	19	)	)	PUNCT
ejpam-841	104	20	.	.	PUNCT
ejpam-841	105	1	here	here	ADV
ejpam-841	105	2	we	we	PRON
ejpam-841	105	3	recall	recall	VERB
ejpam-841	105	4	two	two	NUM
ejpam-841	105	5	results	result	NOUN
ejpam-841	105	6	concerning	concern	VERB
ejpam-841	105	7	the	the	DET
ejpam-841	105	8	relation	relation	NOUN
ejpam-841	105	9	between	between	ADP
ejpam-841	105	10	µ-bases	µ-base	NOUN
ejpam-841	105	11	and	and	CCONJ
ejpam-841	105	12	points	point	NOUN
ejpam-841	105	13	on	on	ADP
ejpam-841	105	14	rational	rational	ADJ
ejpam-841	105	15	space	space	NOUN
ejpam-841	105	16	curves	curve	NOUN
ejpam-841	105	17	.	.	PUNCT
ejpam-841	106	1	proposition	proposition	NOUN
ejpam-841	106	2	1	1	NUM
ejpam-841	106	3	.	.	PUNCT
ejpam-841	107	1	[	[	X
ejpam-841	107	2	[	[	X
ejpam-841	107	3	26	26	NUM
ejpam-841	107	4	]	]	X
ejpam-841	107	5	]	]	X
ejpam-841	107	6	let	let	VERB
ejpam-841	107	7	f(s	f(	NOUN
ejpam-841	107	8	,	,	PUNCT
ejpam-841	107	9	t	t	PROPN
ejpam-841	107	10	)	)	PUNCT
ejpam-841	107	11	be	be	AUX
ejpam-841	107	12	a	a	DET
ejpam-841	107	13	rational	rational	ADJ
ejpam-841	107	14	space	space	NOUN
ejpam-841	107	15	curve	curve	NOUN
ejpam-841	107	16	with	with	ADP
ejpam-841	107	17	a	a	DET
ejpam-841	107	18	µ-basis	µ-basis	NOUN
ejpam-841	107	19	p	p	NOUN
ejpam-841	107	20	,	,	PUNCT
ejpam-841	107	21	q	q	PROPN
ejpam-841	107	22	,	,	PUNCT
ejpam-841	107	23	r.	r.	PROPN
ejpam-841	107	24	then	then	ADV
ejpam-841	107	25	a	a	DET
ejpam-841	107	26	point	point	NOUN
ejpam-841	107	27	q	q	NOUN
ejpam-841	107	28	is	be	AUX
ejpam-841	107	29	on	on	ADP
ejpam-841	107	30	the	the	DET
ejpam-841	107	31	space	space	NOUN
ejpam-841	107	32	curve	curve	NOUN
ejpam-841	107	33	f(s	f(s	PROPN
ejpam-841	107	34	,	,	PUNCT
ejpam-841	107	35	t	t	PROPN
ejpam-841	107	36	)	)	PUNCT
ejpam-841	107	37	if	if	SCONJ
ejpam-841	108	1	and	and	CCONJ
ejpam-841	108	2	only	only	ADV
ejpam-841	108	3	if	if	SCONJ
ejpam-841	108	4	deg(gcd(p	deg(gcd(p	PROPN
ejpam-841	108	5	·	·	SYM
ejpam-841	108	6	q	q	NOUN
ejpam-841	108	7	,	,	PUNCT
ejpam-841	108	8	q	q	X
ejpam-841	108	9	·	·	PUNCT
ejpam-841	108	10	q	q	NOUN
ejpam-841	108	11	,	,	PUNCT
ejpam-841	108	12	r	r	NOUN
ejpam-841	108	13	·	·	PUNCT
ejpam-841	108	14	q))≥	q))≥	PROPN
ejpam-841	108	15	1	1	NUM
ejpam-841	108	16	.	.	PUNCT
ejpam-841	109	1	moreover	moreover	ADV
ejpam-841	109	2	,	,	PUNCT
ejpam-841	109	3	the	the	DET
ejpam-841	109	4	roots	root	NOUN
ejpam-841	109	5	of	of	ADP
ejpam-841	109	6	this	this	DET
ejpam-841	109	7	gcd	gcd	NOUN
ejpam-841	109	8	are	be	AUX
ejpam-841	109	9	the	the	DET
ejpam-841	109	10	parameters	parameter	NOUN
ejpam-841	109	11	with	with	ADP
ejpam-841	109	12	proper	proper	ADJ
ejpam-841	109	13	multiplicity	multiplicity	NOUN
ejpam-841	109	14	corresponding	correspond	VERB
ejpam-841	109	15	to	to	ADP
ejpam-841	109	16	the	the	DET
ejpam-841	109	17	point	point	NOUN
ejpam-841	109	18	q	q	NOUN
ejpam-841	109	19	on	on	ADP
ejpam-841	109	20	the	the	DET
ejpam-841	109	21	curve	curve	NOUN
ejpam-841	109	22	f(s	f(s	PROPN
ejpam-841	109	23	,	,	PUNCT
ejpam-841	109	24	t	t	PROPN
ejpam-841	109	25	)	)	PUNCT
ejpam-841	109	26	.	.	PUNCT
ejpam-841	110	1	proposition	proposition	NOUN
ejpam-841	110	2	2	2	NUM
ejpam-841	110	3	.	.	PUNCT
ejpam-841	111	1	[	[	X
ejpam-841	111	2	[	[	X
ejpam-841	111	3	26	26	NUM
ejpam-841	111	4	]	]	X
ejpam-841	111	5	]	]	X
ejpam-841	111	6	suppose	suppose	VERB
ejpam-841	111	7	p(s	p(s	PROPN
ejpam-841	111	8	,	,	PUNCT
ejpam-841	111	9	t	t	PROPN
ejpam-841	111	10	)	)	PUNCT
ejpam-841	111	11	,	,	PUNCT
ejpam-841	111	12	q(s	q(s	PROPN
ejpam-841	111	13	,	,	PUNCT
ejpam-841	111	14	t	t	PROPN
ejpam-841	111	15	)	)	PUNCT
ejpam-841	111	16	and	and	CCONJ
ejpam-841	111	17	r(s	r(s	PROPN
ejpam-841	111	18	,	,	PUNCT
ejpam-841	111	19	t	t	PROPN
ejpam-841	111	20	)	)	PUNCT
ejpam-841	111	21	are	be	AUX
ejpam-841	111	22	a	a	DET
ejpam-841	111	23	µ-basis	µ-basis	NOUN
ejpam-841	111	24	of	of	ADP
ejpam-841	111	25	degrees	degree	NOUN
ejpam-841	111	26	(	(	PUNCT
ejpam-841	111	27	1,1	1,1	NUM
ejpam-841	111	28	,	,	PUNCT
ejpam-841	111	29	d	d	NOUN
ejpam-841	111	30	−	−	PROPN
ejpam-841	111	31	2	2	NUM
ejpam-841	111	32	)	)	PUNCT
ejpam-841	111	33	for	for	ADP
ejpam-841	111	34	the	the	DET
ejpam-841	111	35	rational	rational	ADJ
ejpam-841	111	36	space	space	NOUN
ejpam-841	111	37	curve	curve	NOUN
ejpam-841	111	38	f(s	f(s	PROPN
ejpam-841	111	39	,	,	PUNCT
ejpam-841	111	40	t	t	PROPN
ejpam-841	111	41	)	)	PUNCT
ejpam-841	111	42	.	.	PUNCT
ejpam-841	112	1	then	then	ADV
ejpam-841	112	2	1	1	X
ejpam-841	112	3	.	.	PUNCT
ejpam-841	112	4	f(s	f(s	PROPN
ejpam-841	112	5	,	,	PUNCT
ejpam-841	112	6	t	t	PROPN
ejpam-841	112	7	)	)	PUNCT
ejpam-841	112	8	has	have	VERB
ejpam-841	112	9	no	no	DET
ejpam-841	112	10	singularities	singularity	NOUN
ejpam-841	112	11	if	if	SCONJ
ejpam-841	112	12	and	and	CCONJ
ejpam-841	112	13	only	only	ADV
ejpam-841	112	14	if	if	SCONJ
ejpam-841	112	15	the	the	DET
ejpam-841	112	16	axes	axis	NOUN
ejpam-841	112	17	of	of	ADP
ejpam-841	112	18	p	p	PROPN
ejpam-841	112	19	and	and	CCONJ
ejpam-841	112	20	q	q	NOUN
ejpam-841	112	21	do	do	AUX
ejpam-841	112	22	not	not	PART
ejpam-841	112	23	intersect	intersect	VERB
ejpam-841	112	24	.	.	PUNCT
ejpam-841	113	1	2	2	X
ejpam-841	113	2	.	.	NUM
ejpam-841	113	3	f(s	f(s	PROPN
ejpam-841	113	4	,	,	PUNCT
ejpam-841	113	5	t	t	PROPN
ejpam-841	113	6	)	)	PUNCT
ejpam-841	113	7	has	have	VERB
ejpam-841	113	8	exactly	exactly	ADV
ejpam-841	113	9	one	one	NUM
ejpam-841	113	10	singular	singular	NOUN
ejpam-841	113	11	point	point	NOUN
ejpam-841	113	12	a	a	PRON
ejpam-841	113	13	which	which	PRON
ejpam-841	113	14	is	be	AUX
ejpam-841	113	15	of	of	ADP
ejpam-841	113	16	order	order	NOUN
ejpam-841	114	1	d	d	NOUN
ejpam-841	114	2	−	−	PROPN
ejpam-841	114	3	2	2	NUM
ejpam-841	114	4	if	if	SCONJ
ejpam-841	114	5	and	and	CCONJ
ejpam-841	114	6	only	only	ADV
ejpam-841	114	7	if	if	SCONJ
ejpam-841	114	8	the	the	DET
ejpam-841	114	9	axes	axis	NOUN
ejpam-841	114	10	of	of	ADP
ejpam-841	114	11	p	p	NOUN
ejpam-841	114	12	and	and	CCONJ
ejpam-841	114	13	q	q	NOUN
ejpam-841	114	14	intersect	intersect	NOUN
ejpam-841	114	15	at	at	ADP
ejpam-841	114	16	the	the	DET
ejpam-841	114	17	point	point	NOUN
ejpam-841	114	18	a.	a.	NOUN
ejpam-841	114	19	moreover	moreover	ADV
ejpam-841	114	20	,	,	PUNCT
ejpam-841	114	21	from	from	ADP
ejpam-841	114	22	two	two	NUM
ejpam-841	114	23	µ-basis	µ-basis	NOUN
ejpam-841	114	24	elements	element	NOUN
ejpam-841	114	25	,	,	PUNCT
ejpam-841	114	26	we	we	PRON
ejpam-841	114	27	can	can	AUX
ejpam-841	114	28	generate	generate	VERB
ejpam-841	114	29	a	a	DET
ejpam-841	114	30	quadric	quadric	ADJ
ejpam-841	114	31	surface	surface	NOUN
ejpam-841	114	32	that	that	PRON
ejpam-841	114	33	contains	contain	VERB
ejpam-841	114	34	a	a	DET
ejpam-841	114	35	space	space	NOUN
ejpam-841	114	36	curve	curve	NOUN
ejpam-841	114	37	of	of	ADP
ejpam-841	114	38	type	type	NOUN
ejpam-841	114	39	(	(	PUNCT
ejpam-841	114	40	1,1	1,1	NUM
ejpam-841	114	41	,	,	PUNCT
ejpam-841	114	42	d	d	NOUN
ejpam-841	114	43	−	−	PROPN
ejpam-841	114	44	2	2	NUM
ejpam-841	114	45	)	)	PUNCT
ejpam-841	114	46	.	.	PUNCT
ejpam-841	115	1	lemma	lemma	PROPN
ejpam-841	115	2	1	1	X
ejpam-841	115	3	.	.	PUNCT
ejpam-841	116	1	let	let	VERB
ejpam-841	116	2	p	p	NOUN
ejpam-841	116	3	=	=	NOUN
ejpam-841	116	4	p1s+	p1s+	NOUN
ejpam-841	116	5	p0	p0	NOUN
ejpam-841	116	6	t	t	PROPN
ejpam-841	116	7	and	and	CCONJ
ejpam-841	116	8	q	q	NOUN
ejpam-841	116	9	=	=	SYM
ejpam-841	116	10	q1s+q0	q1s+q0	NOUN
ejpam-841	116	11	t	t	PROPN
ejpam-841	116	12	be	be	VERB
ejpam-841	116	13	two	two	NUM
ejpam-841	116	14	µ-basis	µ-basis	NOUN
ejpam-841	116	15	elements	element	NOUN
ejpam-841	116	16	of	of	ADP
ejpam-841	116	17	degree	degree	NOUN
ejpam-841	116	18	1	1	NUM
ejpam-841	116	19	of	of	ADP
ejpam-841	116	20	a	a	DET
ejpam-841	116	21	rational	rational	ADJ
ejpam-841	116	22	quartic	quartic	ADJ
ejpam-841	116	23	space	space	NOUN
ejpam-841	116	24	curve	curve	NOUN
ejpam-841	116	25	c	c	PROPN
ejpam-841	116	26	.	.	PUNCT
ejpam-841	117	1	then	then	ADV
ejpam-841	117	2	the	the	DET
ejpam-841	117	3	curve	curve	NOUN
ejpam-841	117	4	c	c	PROPN
ejpam-841	117	5	is	be	AUX
ejpam-841	117	6	contained	contain	VERB
ejpam-841	117	7	in	in	ADP
ejpam-841	117	8	the	the	DET
ejpam-841	117	9	irreducible	irreducible	ADJ
ejpam-841	117	10	quadric	quadric	ADJ
ejpam-841	117	11	surface	surface	NOUN
ejpam-841	117	12	defined	define	VERB
ejpam-841	117	13	by	by	ADP
ejpam-841	117	14	sylvs	sylv	NOUN
ejpam-841	117	15	,	,	PUNCT
ejpam-841	117	16	t(p	t(p	PROPN
ejpam-841	117	17	,	,	PUNCT
ejpam-841	117	18	q	q	NOUN
ejpam-841	117	19	)	)	PUNCT
ejpam-841	117	20	=	=	SYM
ejpam-841	117	21	det	det	PROPN
ejpam-841	117	22	�	�	PROPN
ejpam-841	117	23	p1	p1	PROPN
ejpam-841	117	24	p0	p0	PROPN
ejpam-841	117	25	q1	q1	PROPN
ejpam-841	117	26	q0	q0	PROPN
ejpam-841	117	27	�	�	PROPN
ejpam-841	117	28	=	=	SYM
ejpam-841	117	29	0	0	X
ejpam-841	117	30	.	.	PUNCT
ejpam-841	118	1	proof	proof	NOUN
ejpam-841	118	2	.	.	PUNCT
ejpam-841	119	1	since	since	SCONJ
ejpam-841	119	2	deg(p	deg(p	PROPN
ejpam-841	119	3	)	)	PUNCT
ejpam-841	119	4	=	=	SYM
ejpam-841	119	5	deg(q	deg(q	NOUN
ejpam-841	119	6	)	)	PUNCT
ejpam-841	119	7	=	=	SYM
ejpam-841	119	8	(	(	PUNCT
ejpam-841	119	9	1,1	1,1	NUM
ejpam-841	119	10	)	)	PUNCT
ejpam-841	119	11	,	,	PUNCT
ejpam-841	119	12	p	p	NOUN
ejpam-841	119	13	and	and	CCONJ
ejpam-841	119	14	q	q	NOUN
ejpam-841	119	15	are	be	AUX
ejpam-841	119	16	relatively	relatively	ADV
ejpam-841	119	17	prime	prime	ADJ
ejpam-841	119	18	over	over	ADP
ejpam-841	119	19	the	the	DET
ejpam-841	119	20	ring	ring	NOUN
ejpam-841	119	21	c	c	NOUN
ejpam-841	119	22	.	.	PUNCT
ejpam-841	120	1	hence	hence	ADV
ejpam-841	120	2	p	p	NOUN
ejpam-841	120	3	and	and	CCONJ
ejpam-841	120	4	q	q	PROPN
ejpam-841	120	5	form	form	NOUN
ejpam-841	120	6	a	a	DET
ejpam-841	120	7	regular	regular	ADJ
ejpam-841	120	8	sequence	sequence	NOUN
ejpam-841	120	9	over	over	ADP
ejpam-841	120	10	c	c	PROPN
ejpam-841	120	11	,	,	PUNCT
ejpam-841	120	12	and	and	CCONJ
ejpam-841	120	13	therefore	therefore	ADV
ejpam-841	120	14	sylvs	sylvs	PROPN
ejpam-841	120	15	,	,	PUNCT
ejpam-841	120	16	t(p	t(p	PROPN
ejpam-841	120	17	,	,	PUNCT
ejpam-841	120	18	q	q	NOUN
ejpam-841	120	19	)	)	PUNCT
ejpam-841	120	20	6≡	6≡	NUM
ejpam-841	120	21	0	0	NUM
ejpam-841	120	22	.	.	PUNCT
ejpam-841	121	1	moreover	moreover	ADV
ejpam-841	121	2	,	,	PUNCT
ejpam-841	121	3	since	since	SCONJ
ejpam-841	121	4	deg(pi	deg(pi	NOUN
ejpam-841	121	5	)	)	PUNCT
ejpam-841	121	6	=	=	SYM
ejpam-841	121	7	deg(qi	deg(qi	NOUN
ejpam-841	121	8	)	)	PUNCT
ejpam-841	121	9	=	=	SYM
ejpam-841	121	10	1	1	NUM
ejpam-841	121	11	for	for	ADP
ejpam-841	121	12	i	i	PROPN
ejpam-841	121	13	=	=	SYM
ejpam-841	121	14	0,1	0,1	NUM
ejpam-841	121	15	,	,	PUNCT
ejpam-841	121	16	sylvs	sylvs	NOUN
ejpam-841	121	17	,	,	PUNCT
ejpam-841	121	18	t(p	t(p	PROPN
ejpam-841	121	19	,	,	PUNCT
ejpam-841	121	20	q	q	NOUN
ejpam-841	121	21	)	)	PUNCT
ejpam-841	121	22	is	be	AUX
ejpam-841	121	23	of	of	ADP
ejpam-841	121	24	homogeneous	homogeneous	ADJ
ejpam-841	121	25	degree	degree	NOUN
ejpam-841	121	26	2	2	NUM
ejpam-841	121	27	.	.	PUNCT
ejpam-841	122	1	in	in	ADP
ejpam-841	122	2	addition	addition	NOUN
ejpam-841	122	3	,	,	PUNCT
ejpam-841	122	4	sylvs	sylvs	NOUN
ejpam-841	122	5	,	,	PUNCT
ejpam-841	122	6	t(p	t(p	PROPN
ejpam-841	122	7	,	,	PUNCT
ejpam-841	122	8	q	q	NOUN
ejpam-841	122	9	)	)	PUNCT
ejpam-841	122	10	vanishes	vanish	VERB
ejpam-841	122	11	on	on	ADP
ejpam-841	122	12	the	the	DET
ejpam-841	122	13	curve	curve	NOUN
ejpam-841	122	14	c	c	NOUN
ejpam-841	122	15	,	,	PUNCT
ejpam-841	122	16	since	since	SCONJ
ejpam-841	122	17	p	p	PRON
ejpam-841	122	18	,	,	PUNCT
ejpam-841	122	19	q	q	X
ejpam-841	122	20	are	be	AUX
ejpam-841	122	21	two	two	NUM
ejpam-841	122	22	moving	move	VERB
ejpam-841	122	23	planes	plane	NOUN
ejpam-841	122	24	that	that	PRON
ejpam-841	122	25	follow	follow	VERB
ejpam-841	122	26	the	the	DET
ejpam-841	122	27	curve	curve	NOUN
ejpam-841	122	28	.	.	PUNCT
ejpam-841	123	1	finally	finally	ADV
ejpam-841	123	2	,	,	PUNCT
ejpam-841	123	3	this	this	DET
ejpam-841	123	4	quadric	quadric	NOUN
ejpam-841	123	5	is	be	AUX
ejpam-841	123	6	irreducible	irreducible	ADJ
ejpam-841	123	7	,	,	PUNCT
ejpam-841	123	8	otherwise	otherwise	ADV
ejpam-841	123	9	the	the	DET
ejpam-841	123	10	curve	curve	NOUN
ejpam-841	123	11	c	c	PROPN
ejpam-841	123	12	would	would	AUX
ejpam-841	123	13	be	be	AUX
ejpam-841	123	14	contained	contain	VERB
ejpam-841	123	15	in	in	ADP
ejpam-841	123	16	one	one	NUM
ejpam-841	123	17	of	of	ADP
ejpam-841	123	18	the	the	DET
ejpam-841	123	19	linear	linear	ADJ
ejpam-841	123	20	factors	factor	NOUN
ejpam-841	123	21	,	,	PUNCT
ejpam-841	123	22	contradicting	contradict	VERB
ejpam-841	123	23	the	the	DET
ejpam-841	123	24	fact	fact	NOUN
ejpam-841	123	25	that	that	SCONJ
ejpam-841	123	26	the	the	DET
ejpam-841	123	27	curve	curve	NOUN
ejpam-841	123	28	c	c	PROPN
ejpam-841	123	29	is	be	AUX
ejpam-841	123	30	non	non	ADJ
ejpam-841	123	31	-	-	ADJ
ejpam-841	123	32	planar	planar	ADJ
ejpam-841	123	33	.	.	PUNCT
ejpam-841	124	1	j.	j.	PROPN
ejpam-841	124	2	hoffman	hoffman	PROPN
ejpam-841	124	3	,	,	PUNCT
ejpam-841	124	4	h.	h.	PROPN
ejpam-841	124	5	wang	wang	PROPN
ejpam-841	124	6	,	,	PUNCT
ejpam-841	124	7	x.	x.	PROPN
ejpam-841	124	8	jia	jia	PROPN
ejpam-841	124	9	,	,	PUNCT
ejpam-841	124	10	r.	r.	PROPN
ejpam-841	124	11	goldman	goldman	PROPN
ejpam-841	124	12	/	/	SYM
ejpam-841	124	13	eur	eur	PROPN
ejpam-841	124	14	.	.	PUNCT
ejpam-841	125	1	j.	j.	PROPN
ejpam-841	125	2	pure	pure	PROPN
ejpam-841	125	3	appl	appl	PROPN
ejpam-841	125	4	.	.	PROPN
ejpam-841	125	5	math	math	PROPN
ejpam-841	125	6	,	,	PUNCT
ejpam-841	125	7	3	3	NUM
ejpam-841	125	8	(	(	PUNCT
ejpam-841	125	9	2010	2010	NUM
ejpam-841	125	10	)	)	PUNCT
ejpam-841	125	11	,	,	PUNCT
ejpam-841	125	12	602	602	NUM
ejpam-841	125	13	-	-	SYM
ejpam-841	125	14	632	632	NUM
ejpam-841	125	15	607	607	NUM
ejpam-841	125	16	3	3	NUM
ejpam-841	125	17	.	.	NOUN
ejpam-841	125	18	minimal	minimal	ADJ
ejpam-841	125	19	generators	generator	NOUN
ejpam-841	125	20	of	of	ADP
ejpam-841	125	21	the	the	DET
ejpam-841	125	22	rees	rees	PROPN
ejpam-841	125	23	algebra	algebra	NOUN
ejpam-841	125	24	in	in	ADP
ejpam-841	125	25	this	this	DET
ejpam-841	125	26	section	section	NOUN
ejpam-841	125	27	,	,	PUNCT
ejpam-841	125	28	we	we	PRON
ejpam-841	125	29	will	will	AUX
ejpam-841	125	30	investigate	investigate	VERB
ejpam-841	125	31	minimal	minimal	ADJ
ejpam-841	125	32	generators	generator	NOUN
ejpam-841	125	33	for	for	ADP
ejpam-841	125	34	the	the	DET
ejpam-841	125	35	rees	rees	PROPN
ejpam-841	125	36	algebra	algebra	NOUN
ejpam-841	125	37	associated	associate	VERB
ejpam-841	125	38	to	to	ADP
ejpam-841	125	39	rational	rational	ADJ
ejpam-841	125	40	space	space	NOUN
ejpam-841	125	41	curves	curve	NOUN
ejpam-841	125	42	of	of	ADP
ejpam-841	125	43	type	type	NOUN
ejpam-841	125	44	(	(	PUNCT
ejpam-841	125	45	1,1	1,1	NUM
ejpam-841	125	46	,	,	PUNCT
ejpam-841	125	47	d	d	NOUN
ejpam-841	125	48	−	−	PROPN
ejpam-841	125	49	2	2	NUM
ejpam-841	125	50	)	)	PUNCT
ejpam-841	125	51	.	.	PUNCT
ejpam-841	126	1	we	we	PRON
ejpam-841	126	2	will	will	AUX
ejpam-841	126	3	provide	provide	VERB
ejpam-841	126	4	a	a	DET
ejpam-841	126	5	simple	simple	ADJ
ejpam-841	126	6	algorithm	algorithm	NOUN
ejpam-841	126	7	to	to	PART
ejpam-841	126	8	find	find	VERB
ejpam-841	126	9	these	these	DET
ejpam-841	126	10	generators	generator	NOUN
ejpam-841	126	11	based	base	VERB
ejpam-841	126	12	solely	solely	ADV
ejpam-841	126	13	on	on	ADP
ejpam-841	126	14	the	the	DET
ejpam-841	126	15	three	three	NUM
ejpam-841	126	16	µ-basis	µ-basis	NOUN
ejpam-841	126	17	elements	element	NOUN
ejpam-841	126	18	of	of	ADP
ejpam-841	126	19	these	these	DET
ejpam-841	126	20	rational	rational	ADJ
ejpam-841	126	21	space	space	NOUN
ejpam-841	126	22	curves	curve	NOUN
ejpam-841	126	23	.	.	PUNCT
ejpam-841	127	1	our	our	PRON
ejpam-841	127	2	approach	approach	NOUN
ejpam-841	127	3	is	be	AUX
ejpam-841	127	4	to	to	PART
ejpam-841	127	5	study	study	VERB
ejpam-841	127	6	separately	separately	ADV
ejpam-841	127	7	the	the	DET
ejpam-841	127	8	cases	case	NOUN
ejpam-841	127	9	when	when	SCONJ
ejpam-841	127	10	the	the	DET
ejpam-841	127	11	curve	curve	NOUN
ejpam-841	127	12	f(s	f(s	PROPN
ejpam-841	127	13	,	,	PUNCT
ejpam-841	127	14	t	t	PROPN
ejpam-841	127	15	)	)	PUNCT
ejpam-841	127	16	is	be	AUX
ejpam-841	127	17	either	either	CCONJ
ejpam-841	127	18	singular	singular	ADJ
ejpam-841	127	19	or	or	CCONJ
ejpam-841	127	20	nonsingular	nonsingular	ADJ
ejpam-841	127	21	.	.	PUNCT
ejpam-841	128	1	if	if	SCONJ
ejpam-841	128	2	f(s	f(	NOUN
ejpam-841	128	3	,	,	PUNCT
ejpam-841	128	4	t	t	PROPN
ejpam-841	128	5	)	)	PUNCT
ejpam-841	128	6	is	be	AUX
ejpam-841	128	7	singular	singular	ADJ
ejpam-841	128	8	,	,	PUNCT
ejpam-841	128	9	then	then	ADV
ejpam-841	128	10	we	we	PRON
ejpam-841	128	11	also	also	ADV
ejpam-841	128	12	study	study	VERB
ejpam-841	128	13	separately	separately	ADV
ejpam-841	128	14	the	the	DET
ejpam-841	128	15	cases	case	NOUN
ejpam-841	128	16	when	when	SCONJ
ejpam-841	128	17	the	the	DET
ejpam-841	128	18	degree	degree	NOUN
ejpam-841	128	19	of	of	ADP
ejpam-841	128	20	the	the	DET
ejpam-841	128	21	curve	curve	NOUN
ejpam-841	128	22	is	be	AUX
ejpam-841	128	23	either	either	CCONJ
ejpam-841	128	24	even	even	ADV
ejpam-841	128	25	or	or	CCONJ
ejpam-841	128	26	odd	odd	ADJ
ejpam-841	128	27	.	.	PUNCT
ejpam-841	129	1	3.1	3.1	NUM
ejpam-841	129	2	.	.	PUNCT
ejpam-841	129	3	singular	singular	ADJ
ejpam-841	129	4	rational	rational	ADJ
ejpam-841	129	5	space	space	NOUN
ejpam-841	129	6	curves	curve	NOUN
ejpam-841	129	7	of	of	ADP
ejpam-841	129	8	type	type	NOUN
ejpam-841	129	9	(	(	PUNCT
ejpam-841	129	10	1	1	NUM
ejpam-841	129	11	,	,	PUNCT
ejpam-841	129	12	1	1	NUM
ejpam-841	129	13	,	,	PUNCT
ejpam-841	129	14	d	d	NOUN
ejpam-841	130	1	−	−	PROPN
ejpam-841	130	2	2	2	NUM
ejpam-841	130	3	)	)	PUNCT
ejpam-841	130	4	here	here	ADV
ejpam-841	130	5	we	we	PRON
ejpam-841	130	6	will	will	AUX
ejpam-841	130	7	find	find	VERB
ejpam-841	130	8	a	a	DET
ejpam-841	130	9	minimal	minimal	ADJ
ejpam-841	130	10	set	set	NOUN
ejpam-841	130	11	of	of	ADP
ejpam-841	130	12	generators	generator	NOUN
ejpam-841	130	13	for	for	ADP
ejpam-841	130	14	the	the	DET
ejpam-841	130	15	rees	rees	PROPN
ejpam-841	130	16	algebra	algebra	NOUN
ejpam-841	130	17	associated	associate	VERB
ejpam-841	130	18	to	to	ADP
ejpam-841	130	19	singular	singular	ADJ
ejpam-841	130	20	rational	rational	ADJ
ejpam-841	130	21	space	space	NOUN
ejpam-841	130	22	curves	curve	NOUN
ejpam-841	130	23	of	of	ADP
ejpam-841	130	24	type	type	NOUN
ejpam-841	130	25	(	(	PUNCT
ejpam-841	130	26	1,1	1,1	NUM
ejpam-841	130	27	,	,	PUNCT
ejpam-841	130	28	d	d	NOUN
ejpam-841	130	29	−	−	PROPN
ejpam-841	130	30	2	2	NUM
ejpam-841	130	31	)	)	PUNCT
ejpam-841	130	32	.	.	PUNCT
ejpam-841	131	1	remark	remark	PROPN
ejpam-841	131	2	3	3	NUM
ejpam-841	131	3	.	.	PUNCT
ejpam-841	132	1	let	let	VERB
ejpam-841	132	2	p	p	NOUN
ejpam-841	132	3	=	=	NOUN
ejpam-841	132	4	p1s+	p1s+	NOUN
ejpam-841	132	5	p0	p0	NOUN
ejpam-841	132	6	t	t	PROPN
ejpam-841	132	7	and	and	CCONJ
ejpam-841	132	8	q	q	NOUN
ejpam-841	132	9	=	=	NOUN
ejpam-841	133	1	q1s+	q1s+	NOUN
ejpam-841	133	2	q0	q0	PROPN
ejpam-841	133	3	t	t	PROPN
ejpam-841	133	4	be	be	AUX
ejpam-841	133	5	two	two	NUM
ejpam-841	133	6	µ-basis	µ-basis	NOUN
ejpam-841	133	7	elements	element	NOUN
ejpam-841	133	8	of	of	ADP
ejpam-841	133	9	a	a	DET
ejpam-841	133	10	singular	singular	ADJ
ejpam-841	133	11	rational	rational	ADJ
ejpam-841	133	12	space	space	NOUN
ejpam-841	133	13	curve	curve	NOUN
ejpam-841	133	14	f(s	f(s	PROPN
ejpam-841	133	15	,	,	PUNCT
ejpam-841	133	16	t	t	PROPN
ejpam-841	133	17	)	)	PUNCT
ejpam-841	133	18	of	of	ADP
ejpam-841	133	19	type	type	NOUN
ejpam-841	133	20	(	(	PUNCT
ejpam-841	133	21	1,1	1,1	NUM
ejpam-841	133	22	,	,	PUNCT
ejpam-841	133	23	d	d	NOUN
ejpam-841	133	24	−	−	PROPN
ejpam-841	133	25	2	2	NUM
ejpam-841	133	26	)	)	PUNCT
ejpam-841	133	27	,	,	PUNCT
ejpam-841	133	28	where	where	SCONJ
ejpam-841	133	29	p1	p1	NOUN
ejpam-841	133	30	,	,	PUNCT
ejpam-841	133	31	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	133	32	are	be	AUX
ejpam-841	133	33	linear	linear	ADJ
ejpam-841	133	34	forms	form	NOUN
ejpam-841	133	35	in	in	ADP
ejpam-841	133	36	x	x	SYM
ejpam-841	133	37	,	,	PUNCT
ejpam-841	133	38	y	y	PROPN
ejpam-841	133	39	,	,	PUNCT
ejpam-841	133	40	z	z	PROPN
ejpam-841	133	41	,	,	PUNCT
ejpam-841	133	42	w.	w.	PROPN
ejpam-841	133	43	then	then	ADV
ejpam-841	133	44	the	the	DET
ejpam-841	133	45	polynomials	polynomial	NOUN
ejpam-841	133	46	p1	p1	NOUN
ejpam-841	133	47	,	,	PUNCT
ejpam-841	133	48	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	133	49	are	be	AUX
ejpam-841	133	50	linearly	linearly	ADV
ejpam-841	133	51	dependent	dependent	ADJ
ejpam-841	133	52	of	of	ADP
ejpam-841	133	53	rank	rank	NOUN
ejpam-841	133	54	3	3	NUM
ejpam-841	133	55	.	.	PUNCT
ejpam-841	134	1	proof	proof	NOUN
ejpam-841	134	2	.	.	PUNCT
ejpam-841	135	1	by	by	ADP
ejpam-841	135	2	proposition	proposition	NOUN
ejpam-841	135	3	2	2	NUM
ejpam-841	135	4	,	,	PUNCT
ejpam-841	135	5	there	there	PRON
ejpam-841	135	6	is	be	VERB
ejpam-841	135	7	a	a	DET
ejpam-841	135	8	unique	unique	ADJ
ejpam-841	135	9	singular	singular	ADJ
ejpam-841	135	10	point	point	NOUN
ejpam-841	135	11	on	on	ADP
ejpam-841	135	12	the	the	DET
ejpam-841	135	13	curve	curve	NOUN
ejpam-841	135	14	,	,	PUNCT
ejpam-841	135	15	which	which	PRON
ejpam-841	135	16	is	be	AUX
ejpam-841	135	17	the	the	DET
ejpam-841	135	18	intersection	intersection	NOUN
ejpam-841	135	19	of	of	ADP
ejpam-841	135	20	the	the	DET
ejpam-841	135	21	axes	axis	NOUN
ejpam-841	135	22	of	of	ADP
ejpam-841	135	23	p	p	PROPN
ejpam-841	135	24	and	and	CCONJ
ejpam-841	135	25	q.	q.	PROPN
ejpam-841	135	26	thus	thus	ADV
ejpam-841	135	27	,	,	PUNCT
ejpam-841	135	28	v(p1	v(p1	NOUN
ejpam-841	135	29	,	,	PUNCT
ejpam-841	135	30	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	135	31	)	)	PUNCT
ejpam-841	135	32	consists	consist	VERB
ejpam-841	135	33	of	of	ADP
ejpam-841	135	34	just	just	ADV
ejpam-841	135	35	one	one	NUM
ejpam-841	135	36	point	point	NOUN
ejpam-841	135	37	.	.	PUNCT
ejpam-841	136	1	therefore	therefore	ADV
ejpam-841	136	2	,	,	PUNCT
ejpam-841	136	3	the	the	DET
ejpam-841	136	4	polynomials	polynomial	NOUN
ejpam-841	136	5	p1	p1	NOUN
ejpam-841	136	6	,	,	PUNCT
ejpam-841	136	7	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	136	8	are	be	AUX
ejpam-841	136	9	linearly	linearly	ADV
ejpam-841	136	10	dependent	dependent	ADJ
ejpam-841	136	11	of	of	ADP
ejpam-841	136	12	rank	rank	NOUN
ejpam-841	136	13	3	3	NUM
ejpam-841	136	14	.	.	PUNCT
ejpam-841	137	1	lemma	lemma	PROPN
ejpam-841	137	2	2	2	X
ejpam-841	137	3	.	.	PUNCT
ejpam-841	138	1	let	let	VERB
ejpam-841	138	2	p	p	PRON
ejpam-841	138	3	,	,	PUNCT
ejpam-841	138	4	q	q	ADJ
ejpam-841	138	5	,	,	PUNCT
ejpam-841	138	6	r	r	NOUN
ejpam-841	138	7	be	be	AUX
ejpam-841	138	8	a	a	DET
ejpam-841	138	9	µ-basis	µ-basis	NOUN
ejpam-841	138	10	for	for	ADP
ejpam-841	138	11	a	a	DET
ejpam-841	138	12	singular	singular	ADJ
ejpam-841	138	13	rational	rational	ADJ
ejpam-841	138	14	space	space	NOUN
ejpam-841	138	15	curve	curve	NOUN
ejpam-841	138	16	f(s	f(s	PROPN
ejpam-841	138	17	,	,	PUNCT
ejpam-841	138	18	t	t	PROPN
ejpam-841	138	19	)	)	PUNCT
ejpam-841	138	20	of	of	ADP
ejpam-841	138	21	type	type	NOUN
ejpam-841	138	22	(	(	PUNCT
ejpam-841	138	23	1,1	1,1	NUM
ejpam-841	138	24	,	,	PUNCT
ejpam-841	138	25	d	d	NOUN
ejpam-841	138	26	−	−	PROPN
ejpam-841	138	27	2	2	NUM
ejpam-841	138	28	)	)	PUNCT
ejpam-841	138	29	.	.	PUNCT
ejpam-841	139	1	by	by	ADP
ejpam-841	139	2	a	a	DET
ejpam-841	139	3	linear	linear	ADJ
ejpam-841	139	4	transformation	transformation	NOUN
ejpam-841	139	5	on	on	ADP
ejpam-841	139	6	the	the	DET
ejpam-841	139	7	basis	basis	NOUN
ejpam-841	139	8	elements	element	NOUN
ejpam-841	139	9	p	p	X
ejpam-841	139	10	,	,	PUNCT
ejpam-841	139	11	q	q	ADJ
ejpam-841	139	12	,	,	PUNCT
ejpam-841	139	13	and	and	CCONJ
ejpam-841	139	14	by	by	ADP
ejpam-841	139	15	a	a	DET
ejpam-841	139	16	projective	projective	ADJ
ejpam-841	139	17	change	change	NOUN
ejpam-841	139	18	of	of	ADP
ejpam-841	139	19	coordinates	coordinate	NOUN
ejpam-841	139	20	in	in	ADP
ejpam-841	139	21	x	x	SYM
ejpam-841	139	22	,	,	PUNCT
ejpam-841	139	23	y	y	PROPN
ejpam-841	139	24	,	,	PUNCT
ejpam-841	139	25	z	z	PROPN
ejpam-841	139	26	,	,	PUNCT
ejpam-841	139	27	w	w	PROPN
ejpam-841	139	28	,	,	PUNCT
ejpam-841	139	29	we	we	PRON
ejpam-841	139	30	can	can	AUX
ejpam-841	139	31	adjust	adjust	VERB
ejpam-841	139	32	the	the	DET
ejpam-841	139	33	elements	element	NOUN
ejpam-841	139	34	of	of	ADP
ejpam-841	139	35	the	the	DET
ejpam-841	139	36	µ-basis	µ-basis	NOUN
ejpam-841	139	37	so	so	SCONJ
ejpam-841	139	38	that	that	SCONJ
ejpam-841	139	39	p	p	X
ejpam-841	139	40	=	=	X
ejpam-841	139	41	ys	ys	NOUN
ejpam-841	139	42	−	−	PROPN
ejpam-841	139	43	x	x	SYM
ejpam-841	139	44	t	t	PROPN
ejpam-841	139	45	,	,	PUNCT
ejpam-841	139	46	q	q	NOUN
ejpam-841	140	1	=	=	PUNCT
ejpam-841	140	2	zs	zs	PROPN
ejpam-841	140	3	−	−	PROPN
ejpam-841	140	4	y	y	PROPN
ejpam-841	140	5	t	t	PROPN
ejpam-841	140	6	,	,	PUNCT
ejpam-841	140	7	and	and	CCONJ
ejpam-841	140	8	transform	transform	VERB
ejpam-841	140	9	the	the	DET
ejpam-841	140	10	singular	singular	ADJ
ejpam-841	140	11	point	point	NOUN
ejpam-841	140	12	to	to	ADP
ejpam-841	140	13	(	(	PUNCT
ejpam-841	140	14	0,0,0,1	0,0,0,1	NUM
ejpam-841	140	15	)	)	PUNCT
ejpam-841	140	16	.	.	PUNCT
ejpam-841	141	1	proof	proof	NOUN
ejpam-841	141	2	.	.	PUNCT
ejpam-841	142	1	let	let	VERB
ejpam-841	142	2	p	p	NOUN
ejpam-841	142	3	=	=	NOUN
ejpam-841	142	4	p1s+	p1s+	NOUN
ejpam-841	142	5	p0	p0	NOUN
ejpam-841	142	6	t	t	PROPN
ejpam-841	142	7	and	and	CCONJ
ejpam-841	142	8	q	q	NOUN
ejpam-841	142	9	=	=	SYM
ejpam-841	142	10	q1s+q0	q1s+q0	PROPN
ejpam-841	142	11	t	t	PROPN
ejpam-841	142	12	where	where	SCONJ
ejpam-841	142	13	p1	p1	PROPN
ejpam-841	142	14	,	,	PUNCT
ejpam-841	142	15	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	142	16	are	be	AUX
ejpam-841	142	17	linear	linear	ADJ
ejpam-841	142	18	forms	form	NOUN
ejpam-841	142	19	in	in	ADP
ejpam-841	142	20	x	x	SYM
ejpam-841	142	21	,	,	PUNCT
ejpam-841	142	22	y	y	PROPN
ejpam-841	142	23	,	,	PUNCT
ejpam-841	142	24	z	z	PROPN
ejpam-841	142	25	,	,	PUNCT
ejpam-841	142	26	w.	w.	NOUN
ejpam-841	142	27	since	since	SCONJ
ejpam-841	142	28	p	p	PRON
ejpam-841	142	29	,	,	PUNCT
ejpam-841	142	30	q	q	PROPN
ejpam-841	142	31	are	be	AUX
ejpam-841	142	32	µ-basis	µ-basis	NOUN
ejpam-841	142	33	elements	element	NOUN
ejpam-841	142	34	,	,	PUNCT
ejpam-841	142	35	gcd(p1	gcd(p1	NOUN
ejpam-841	142	36	,	,	PUNCT
ejpam-841	142	37	p0	p0	NOUN
ejpam-841	142	38	)	)	PUNCT
ejpam-841	143	1	=	=	SYM
ejpam-841	143	2	gcd(q1,q0	gcd(q1,q0	NOUN
ejpam-841	143	3	)	)	PUNCT
ejpam-841	144	1	=	=	SYM
ejpam-841	144	2	1	1	NUM
ejpam-841	144	3	;	;	PUNCT
ejpam-841	144	4	otherwise	otherwise	ADV
ejpam-841	144	5	the	the	DET
ejpam-841	144	6	curve	curve	NOUN
ejpam-841	144	7	would	would	AUX
ejpam-841	144	8	be	be	AUX
ejpam-841	144	9	contained	contain	VERB
ejpam-841	144	10	in	in	ADP
ejpam-841	144	11	the	the	DET
ejpam-841	144	12	plane	plane	NOUN
ejpam-841	144	13	p1	p1	NOUN
ejpam-841	144	14	=	=	SYM
ejpam-841	144	15	p0	p0	NOUN
ejpam-841	144	16	=	=	SYM
ejpam-841	144	17	0	0	NUM
ejpam-841	144	18	or	or	CCONJ
ejpam-841	144	19	q1	q1	PROPN
ejpam-841	145	1	=	=	SYM
ejpam-841	145	2	q0	q0	PROPN
ejpam-841	145	3	=	=	NOUN
ejpam-841	145	4	0	0	X
ejpam-841	145	5	.	.	PUNCT
ejpam-841	146	1	also	also	ADV
ejpam-841	146	2	,	,	PUNCT
ejpam-841	146	3	note	note	VERB
ejpam-841	146	4	that	that	SCONJ
ejpam-841	146	5	gcd(p1,q1	gcd(p1,q1	NOUN
ejpam-841	146	6	)	)	PUNCT
ejpam-841	146	7	=	=	SYM
ejpam-841	146	8	gcd(p0,q0	gcd(p0,q0	PROPN
ejpam-841	146	9	)	)	PUNCT
ejpam-841	146	10	=	=	SYM
ejpam-841	147	1	1	1	X
ejpam-841	147	2	.	.	PUNCT
ejpam-841	148	1	otherwise	otherwise	ADV
ejpam-841	148	2	,	,	PUNCT
ejpam-841	148	3	if	if	SCONJ
ejpam-841	148	4	q1	q1	PROPN
ejpam-841	148	5	=	=	SYM
ejpam-841	148	6	ap1	ap1	PROPN
ejpam-841	148	7	for	for	ADP
ejpam-841	148	8	some	some	DET
ejpam-841	148	9	non	non	ADJ
ejpam-841	148	10	-	-	ADJ
ejpam-841	148	11	zero	zero	ADJ
ejpam-841	148	12	constant	constant	ADJ
ejpam-841	148	13	a	a	PRON
ejpam-841	148	14	,	,	PUNCT
ejpam-841	148	15	then	then	ADV
ejpam-841	148	16	q	q	X
ejpam-841	148	17	=	=	SYM
ejpam-841	148	18	q1s	q1s	NOUN
ejpam-841	148	19	+	+	CCONJ
ejpam-841	148	20	q0	q0	PROPN
ejpam-841	148	21	t	t	NOUN
ejpam-841	148	22	=	=	PUNCT
ejpam-841	149	1	ap1s	ap1s	PROPN
ejpam-841	149	2	+	+	CCONJ
ejpam-841	149	3	q0	q0	PROPN
ejpam-841	149	4	t	t	PROPN
ejpam-841	149	5	,	,	PUNCT
ejpam-841	149	6	and	and	CCONJ
ejpam-841	149	7	q−	q−	PROPN
ejpam-841	149	8	ap	ap	PROPN
ejpam-841	150	1	=	=	PUNCT
ejpam-841	150	2	(	(	PUNCT
ejpam-841	150	3	ap1s+	ap1s+	PROPN
ejpam-841	150	4	q0t)−	q0t)−	PROPN
ejpam-841	150	5	a(p1s+	a(p1s+	PROPN
ejpam-841	150	6	p0	p0	PROPN
ejpam-841	150	7	t	t	PROPN
ejpam-841	150	8	)	)	PUNCT
ejpam-841	150	9	=	=	PUNCT
ejpam-841	151	1	(	(	PUNCT
ejpam-841	151	2	q0	q0	PROPN
ejpam-841	151	3	−	−	PROPN
ejpam-841	151	4	ap0)t	ap0)t	PROPN
ejpam-841	151	5	.	.	PUNCT
ejpam-841	152	1	but	but	CCONJ
ejpam-841	152	2	this	this	PRON
ejpam-841	152	3	is	be	AUX
ejpam-841	152	4	impossible	impossible	ADJ
ejpam-841	152	5	,	,	PUNCT
ejpam-841	152	6	since	since	SCONJ
ejpam-841	152	7	q−	q−	PROPN
ejpam-841	152	8	ap	ap	PROPN
ejpam-841	152	9	is	be	AUX
ejpam-841	152	10	a	a	DET
ejpam-841	152	11	moving	move	VERB
ejpam-841	152	12	plane	plane	NOUN
ejpam-841	152	13	that	that	PRON
ejpam-841	152	14	follows	follow	VERB
ejpam-841	152	15	the	the	DET
ejpam-841	152	16	space	space	NOUN
ejpam-841	152	17	curve	curve	NOUN
ejpam-841	152	18	f(s	f(s	PROPN
ejpam-841	152	19	,	,	PUNCT
ejpam-841	152	20	t	t	PROPN
ejpam-841	152	21	)	)	PUNCT
ejpam-841	152	22	,	,	PUNCT
ejpam-841	152	23	so	so	CCONJ
ejpam-841	152	24	the	the	DET
ejpam-841	152	25	space	space	NOUN
ejpam-841	152	26	curve	curve	NOUN
ejpam-841	152	27	f(s	f(s	PROPN
ejpam-841	152	28	,	,	PUNCT
ejpam-841	152	29	t	t	PROPN
ejpam-841	152	30	)	)	PUNCT
ejpam-841	152	31	would	would	AUX
ejpam-841	152	32	be	be	AUX
ejpam-841	152	33	contained	contain	VERB
ejpam-841	152	34	in	in	ADP
ejpam-841	152	35	the	the	DET
ejpam-841	152	36	plane	plane	NOUN
ejpam-841	152	37	q0	q0	NOUN
ejpam-841	152	38	−	−	PROPN
ejpam-841	152	39	ap0	ap0	PROPN
ejpam-841	152	40	.	.	PUNCT
ejpam-841	153	1	thus	thus	ADV
ejpam-841	153	2	gcd(p1,q1	gcd(p1,q1	VERB
ejpam-841	153	3	)	)	PUNCT
ejpam-841	153	4	=	=	SYM
ejpam-841	153	5	1	1	X
ejpam-841	153	6	.	.	X
ejpam-841	153	7	similarly	similarly	ADV
ejpam-841	153	8	,	,	PUNCT
ejpam-841	153	9	gcd(p0,q0	gcd(p0,q0	PROPN
ejpam-841	153	10	)	)	PUNCT
ejpam-841	153	11	=	=	PUNCT
ejpam-841	154	1	1	1	X
ejpam-841	154	2	.	.	PUNCT
ejpam-841	154	3	since	since	SCONJ
ejpam-841	154	4	gcd(q1,q0	gcd(q1,q0	PROPN
ejpam-841	154	5	)	)	PUNCT
ejpam-841	154	6	=	=	SYM
ejpam-841	154	7	1	1	X
ejpam-841	154	8	,	,	PUNCT
ejpam-841	154	9	by	by	ADP
ejpam-841	154	10	a	a	DET
ejpam-841	154	11	change	change	NOUN
ejpam-841	154	12	of	of	ADP
ejpam-841	154	13	coordinates	coordinate	NOUN
ejpam-841	154	14	we	we	PRON
ejpam-841	154	15	can	can	AUX
ejpam-841	154	16	let	let	VERB
ejpam-841	154	17	y	y	PRON
ejpam-841	154	18	′	′	NUM
ejpam-841	154	19	=	=	PUNCT
ejpam-841	154	20	q1	q1	PROPN
ejpam-841	154	21	and	and	CCONJ
ejpam-841	154	22	x	x	NOUN
ejpam-841	154	23	′	′	NUM
ejpam-841	154	24	=	=	SYM
ejpam-841	154	25	−q0	−q0	NOUN
ejpam-841	154	26	.	.	PUNCT
ejpam-841	155	1	hence	hence	ADV
ejpam-841	155	2	q	q	NOUN
ejpam-841	156	1	=	=	PUNCT
ejpam-841	156	2	y	y	PROPN
ejpam-841	156	3	′s	′s	PROPN
ejpam-841	156	4	−	−	PROPN
ejpam-841	156	5	x	x	SYM
ejpam-841	156	6	′t	′t	NOUN
ejpam-841	156	7	.	.	PUNCT
ejpam-841	157	1	since	since	SCONJ
ejpam-841	157	2	gcd(p1,q1	gcd(p1,q1	PROPN
ejpam-841	157	3	)	)	PUNCT
ejpam-841	157	4	=	=	SYM
ejpam-841	157	5	1	1	NUM
ejpam-841	157	6	,	,	PUNCT
ejpam-841	157	7	it	it	PRON
ejpam-841	157	8	follows	follow	VERB
ejpam-841	157	9	that	that	SCONJ
ejpam-841	157	10	p1	p1	PROPN
ejpam-841	157	11	6=	6=	PROPN
ejpam-841	157	12	a	a	DET
ejpam-841	157	13	y	y	NOUN
ejpam-841	157	14	′	′	NUM
ejpam-841	157	15	for	for	ADP
ejpam-841	157	16	any	any	DET
ejpam-841	157	17	non	non	ADJ
ejpam-841	157	18	-	-	ADJ
ejpam-841	157	19	zero	zero	NUM
ejpam-841	157	20	constant	constant	ADJ
ejpam-841	157	21	.	.	PUNCT
ejpam-841	158	1	we	we	PRON
ejpam-841	158	2	will	will	AUX
ejpam-841	158	3	now	now	ADV
ejpam-841	158	4	prove	prove	VERB
ejpam-841	158	5	the	the	DET
ejpam-841	158	6	lemma	lemma	PROPN
ejpam-841	158	7	by	by	ADP
ejpam-841	158	8	establishing	establish	VERB
ejpam-841	158	9	the	the	DET
ejpam-841	158	10	following	follow	VERB
ejpam-841	158	11	two	two	NUM
ejpam-841	158	12	cases	case	NOUN
ejpam-841	158	13	.	.	PUNCT
ejpam-841	159	1	case	case	NOUN
ejpam-841	159	2	1	1	NUM
ejpam-841	159	3	:	:	PUNCT
ejpam-841	159	4	p1	p1	NOUN
ejpam-841	159	5	=	=	NOUN
ejpam-841	159	6	ax	ax	NOUN
ejpam-841	159	7	′+	′+	PUNCT
ejpam-841	159	8	b	b	SYM
ejpam-841	159	9	y	y	NOUN
ejpam-841	159	10	′	′	VERB
ejpam-841	159	11	for	for	ADP
ejpam-841	159	12	some	some	PRON
ejpam-841	159	13	a	a	PRON
ejpam-841	159	14	,	,	PUNCT
ejpam-841	159	15	b	b	NOUN
ejpam-841	159	16	∈k	∈k	NOUN
ejpam-841	159	17	and	and	CCONJ
ejpam-841	159	18	a	a	DET
ejpam-841	159	19	6=	6=	NUM
ejpam-841	159	20	0	0	NUM
ejpam-841	159	21	.	.	PUNCT
ejpam-841	160	1	then	then	ADV
ejpam-841	160	2	p	p	NOUN
ejpam-841	160	3	=	=	PUNCT
ejpam-841	160	4	(	(	PUNCT
ejpam-841	160	5	ax	ax	INTJ
ejpam-841	160	6	′+	′+	PUNCT
ejpam-841	160	7	b	b	NOUN
ejpam-841	160	8	y	y	PROPN
ejpam-841	160	9	′)s+p0	′)s+p0	PROPN
ejpam-841	160	10	t.	t.	NOUN
ejpam-841	160	11	therefore	therefore	ADV
ejpam-841	160	12	,	,	PUNCT
ejpam-841	160	13	the	the	DET
ejpam-841	160	14	moving	move	VERB
ejpam-841	160	15	plane	plane	NOUN
ejpam-841	160	16	1	1	NUM
ejpam-841	160	17	a	a	DET
ejpam-841	160	18	p−	p−	NOUN
ejpam-841	160	19	b	b	NOUN
ejpam-841	160	20	a	a	DET
ejpam-841	160	21	q	q	NOUN
ejpam-841	160	22	=	=	PUNCT
ejpam-841	160	23	x	x	NOUN
ejpam-841	160	24	′s+	′s+	X
ejpam-841	160	25	p0+bx	p0+bx	NOUN
ejpam-841	160	26	′	′	NUM
ejpam-841	160	27	a	a	DET
ejpam-841	160	28	t	t	NOUN
ejpam-841	160	29	is	be	AUX
ejpam-841	160	30	linearly	linearly	ADV
ejpam-841	160	31	independent	independent	ADJ
ejpam-841	160	32	from	from	ADP
ejpam-841	160	33	the	the	DET
ejpam-841	160	34	moving	move	VERB
ejpam-841	160	35	plane	plane	NOUN
ejpam-841	160	36	q	q	NOUN
ejpam-841	160	37	,	,	PUNCT
ejpam-841	160	38	and	and	CCONJ
ejpam-841	160	39	follows	follow	VERB
ejpam-841	160	40	the	the	DET
ejpam-841	160	41	space	space	NOUN
ejpam-841	160	42	curve	curve	NOUN
ejpam-841	160	43	f(s	f(s	PROPN
ejpam-841	160	44	,	,	PUNCT
ejpam-841	160	45	t	t	PROPN
ejpam-841	160	46	)	)	PUNCT
ejpam-841	160	47	.	.	PUNCT
ejpam-841	161	1	thus	thus	ADV
ejpam-841	161	2	,	,	PUNCT
ejpam-841	161	3	q	q	X
ejpam-841	161	4	,	,	PUNCT
ejpam-841	161	5	1	1	NUM
ejpam-841	161	6	a	a	DET
ejpam-841	161	7	p	p	NOUN
ejpam-841	162	1	−	−	PROPN
ejpam-841	162	2	b	b	PROPN
ejpam-841	162	3	a	a	DET
ejpam-841	162	4	q	q	NOUN
ejpam-841	162	5	and	and	CCONJ
ejpam-841	162	6	r	r	NOUN
ejpam-841	162	7	also	also	ADV
ejpam-841	162	8	form	form	VERB
ejpam-841	162	9	a	a	DET
ejpam-841	162	10	µ-basis	µ-basis	NOUN
ejpam-841	162	11	for	for	ADP
ejpam-841	162	12	the	the	DET
ejpam-841	162	13	space	space	NOUN
ejpam-841	162	14	curve	curve	NOUN
ejpam-841	162	15	f(s	f(s	PROPN
ejpam-841	162	16	,	,	PUNCT
ejpam-841	162	17	t	t	PROPN
ejpam-841	162	18	)	)	PUNCT
ejpam-841	162	19	.	.	PUNCT
ejpam-841	163	1	since	since	SCONJ
ejpam-841	163	2	the	the	DET
ejpam-841	163	3	space	space	NOUN
ejpam-841	163	4	curve	curve	NOUN
ejpam-841	163	5	f(s	f(s	PROPN
ejpam-841	163	6	,	,	PUNCT
ejpam-841	163	7	t	t	PROPN
ejpam-841	163	8	)	)	PUNCT
ejpam-841	163	9	is	be	AUX
ejpam-841	163	10	singular	singular	ADJ
ejpam-841	163	11	,	,	PUNCT
ejpam-841	163	12	the	the	DET
ejpam-841	163	13	axes	axis	NOUN
ejpam-841	163	14	of	of	ADP
ejpam-841	163	15	q	q	PROPN
ejpam-841	163	16	and	and	CCONJ
ejpam-841	163	17	1	1	NUM
ejpam-841	163	18	a	a	DET
ejpam-841	163	19	p−	p−	NOUN
ejpam-841	163	20	b	b	NOUN
ejpam-841	163	21	a	a	DET
ejpam-841	163	22	q	q	NOUN
ejpam-841	163	23	intersect	intersect	NOUN
ejpam-841	163	24	at	at	ADP
ejpam-841	163	25	one	one	NUM
ejpam-841	163	26	point	point	NOUN
ejpam-841	163	27	.	.	PUNCT
ejpam-841	164	1	thus	thus	ADV
ejpam-841	164	2	,	,	PUNCT
ejpam-841	164	3	by	by	ADP
ejpam-841	164	4	remark	remark	NOUN
ejpam-841	164	5	3	3	NUM
ejpam-841	164	6	,	,	PUNCT
ejpam-841	164	7	x	x	PROPN
ejpam-841	164	8	′	′	NUM
ejpam-841	164	9	,	,	PUNCT
ejpam-841	164	10	y	y	PROPN
ejpam-841	164	11	′	′	NOUN
ejpam-841	164	12	,	,	PUNCT
ejpam-841	164	13	p0+bx	p0+bx	X
ejpam-841	164	14	′	′	NUM
ejpam-841	164	15	a	a	PRON
ejpam-841	164	16	must	must	AUX
ejpam-841	164	17	have	have	AUX
ejpam-841	164	18	rank	rank	NOUN
ejpam-841	164	19	3	3	NUM
ejpam-841	164	20	.	.	PUNCT
ejpam-841	165	1	hence	hence	ADV
ejpam-841	165	2	,	,	PUNCT
ejpam-841	165	3	by	by	ADP
ejpam-841	165	4	the	the	DET
ejpam-841	165	5	projective	projective	ADJ
ejpam-841	165	6	change	change	NOUN
ejpam-841	165	7	of	of	ADP
ejpam-841	165	8	coordinates	coordinate	NOUN
ejpam-841	165	9	,	,	PUNCT
ejpam-841	165	10	x	x	PUNCT
ejpam-841	165	11	=	=	PUNCT
ejpam-841	165	12	−	−	PROPN
ejpam-841	165	13	p0+bx	p0+bx	NOUN
ejpam-841	165	14	′	′	NUM
ejpam-841	165	15	a	a	PRON
ejpam-841	165	16	,	,	PUNCT
ejpam-841	165	17	y	y	PROPN
ejpam-841	165	18	=	=	PUNCT
ejpam-841	165	19	x	x	SYM
ejpam-841	165	20	′	′	NUM
ejpam-841	165	21	,	,	PUNCT
ejpam-841	165	22	z	z	PROPN
ejpam-841	165	23	=	=	SYM
ejpam-841	165	24	y	y	PROPN
ejpam-841	165	25	′	′	NOUN
ejpam-841	165	26	,	,	PUNCT
ejpam-841	165	27	we	we	PRON
ejpam-841	165	28	obtain	obtain	VERB
ejpam-841	165	29	a	a	DET
ejpam-841	165	30	µ-basis	µ-basis	NOUN
ejpam-841	165	31	with	with	ADP
ejpam-841	165	32	elements	element	NOUN
ejpam-841	165	33	p	p	X
ejpam-841	165	34	=	=	PUNCT
ejpam-841	165	35	ys−	ys−	PUNCT
ejpam-841	165	36	x	x	SYM
ejpam-841	165	37	t	t	NOUN
ejpam-841	165	38	and	and	CCONJ
ejpam-841	165	39	q	q	NOUN
ejpam-841	165	40	=	=	SYM
ejpam-841	165	41	zs−	zs−	NUM
ejpam-841	165	42	y	y	PROPN
ejpam-841	165	43	t.	t.	PROPN
ejpam-841	165	44	j.	j.	PROPN
ejpam-841	165	45	hoffman	hoffman	PROPN
ejpam-841	165	46	,	,	PUNCT
ejpam-841	165	47	h.	h.	PROPN
ejpam-841	165	48	wang	wang	PROPN
ejpam-841	165	49	,	,	PUNCT
ejpam-841	165	50	x.	x.	PROPN
ejpam-841	165	51	jia	jia	PROPN
ejpam-841	165	52	,	,	PUNCT
ejpam-841	165	53	r.	r.	PROPN
ejpam-841	165	54	goldman	goldman	PROPN
ejpam-841	165	55	/	/	SYM
ejpam-841	165	56	eur	eur	PROPN
ejpam-841	165	57	.	.	PUNCT
ejpam-841	166	1	j.	j.	PROPN
ejpam-841	166	2	pure	pure	PROPN
ejpam-841	166	3	appl	appl	PROPN
ejpam-841	166	4	.	.	PROPN
ejpam-841	166	5	math	math	PROPN
ejpam-841	166	6	,	,	PUNCT
ejpam-841	166	7	3	3	NUM
ejpam-841	166	8	(	(	PUNCT
ejpam-841	166	9	2010	2010	NUM
ejpam-841	166	10	)	)	PUNCT
ejpam-841	166	11	,	,	PUNCT
ejpam-841	166	12	602	602	NUM
ejpam-841	166	13	-	-	SYM
ejpam-841	166	14	632	632	NUM
ejpam-841	166	15	608	608	NUM
ejpam-841	166	16	case	case	NOUN
ejpam-841	166	17	2	2	NUM
ejpam-841	166	18	:	:	PUNCT
ejpam-841	166	19	p1	p1	NOUN
ejpam-841	166	20	6=	6=	NOUN
ejpam-841	166	21	ax	ax	NOUN
ejpam-841	166	22	′	′	NUM
ejpam-841	167	1	+	+	CCONJ
ejpam-841	167	2	b	b	X
ejpam-841	167	3	y	y	NOUN
ejpam-841	167	4	′	′	VERB
ejpam-841	167	5	for	for	ADP
ejpam-841	167	6	all	all	PRON
ejpam-841	167	7	possible	possible	ADJ
ejpam-841	167	8	a	a	DET
ejpam-841	167	9	,	,	PUNCT
ejpam-841	167	10	b	b	PROPN
ejpam-841	167	11	∈	∈	PROPN
ejpam-841	167	12	k.	k.	PROPN
ejpam-841	168	1	then	then	ADV
ejpam-841	168	2	p1	p1	PROPN
ejpam-841	168	3	is	be	AUX
ejpam-841	168	4	linearly	linearly	ADV
ejpam-841	168	5	independent	independent	ADJ
ejpam-841	168	6	of	of	ADP
ejpam-841	168	7	x	x	PROPN
ejpam-841	168	8	′	′	NUM
ejpam-841	168	9	,	,	PUNCT
ejpam-841	168	10	y	y	PROPN
ejpam-841	168	11	′.	′.	NOUN
ejpam-841	168	12	without	without	ADP
ejpam-841	168	13	loss	loss	NOUN
ejpam-841	168	14	of	of	ADP
ejpam-841	168	15	generality	generality	NOUN
ejpam-841	168	16	,	,	PUNCT
ejpam-841	168	17	we	we	PRON
ejpam-841	168	18	can	can	AUX
ejpam-841	168	19	let	let	VERB
ejpam-841	168	20	z′	z′	NUM
ejpam-841	168	21	=	=	SYM
ejpam-841	168	22	p1	p1	PROPN
ejpam-841	168	23	.	.	PUNCT
ejpam-841	169	1	then	then	ADV
ejpam-841	169	2	p	p	X
ejpam-841	169	3	=	=	PUNCT
ejpam-841	169	4	z′s+	z′s+	NOUN
ejpam-841	169	5	p0	p0	NOUN
ejpam-841	169	6	t.	t.	PROPN
ejpam-841	169	7	since	since	SCONJ
ejpam-841	169	8	the	the	DET
ejpam-841	169	9	space	space	NOUN
ejpam-841	169	10	curve	curve	NOUN
ejpam-841	169	11	f(s	f(s	PROPN
ejpam-841	169	12	,	,	PUNCT
ejpam-841	169	13	t	t	PROPN
ejpam-841	169	14	)	)	PUNCT
ejpam-841	169	15	is	be	AUX
ejpam-841	169	16	singular	singular	ADJ
ejpam-841	169	17	,	,	PUNCT
ejpam-841	169	18	by	by	ADP
ejpam-841	169	19	remark	remark	NOUN
ejpam-841	169	20	3	3	NUM
ejpam-841	169	21	x	x	SYM
ejpam-841	169	22	′	′	NUM
ejpam-841	169	23	,	,	PUNCT
ejpam-841	169	24	y	y	PROPN
ejpam-841	169	25	′	′	PROPN
ejpam-841	169	26	,	,	PUNCT
ejpam-841	169	27	z′	z′	PROPN
ejpam-841	169	28	,	,	PUNCT
ejpam-841	169	29	p0	p0	NOUN
ejpam-841	169	30	has	have	AUX
ejpam-841	169	31	rank	rank	VERB
ejpam-841	169	32	3	3	NUM
ejpam-841	169	33	,	,	PUNCT
ejpam-841	169	34	so	so	ADV
ejpam-841	169	35	p0	p0	NOUN
ejpam-841	169	36	=	=	PUNCT
ejpam-841	169	37	ax	ax	NOUN
ejpam-841	169	38	′+	′+	PUNCT
ejpam-841	169	39	b	b	NOUN
ejpam-841	169	40	y	y	PROPN
ejpam-841	169	41	′+	′+	PUNCT
ejpam-841	169	42	cz′	cz′	NOUN
ejpam-841	169	43	for	for	ADP
ejpam-841	169	44	some	some	DET
ejpam-841	169	45	non	non	ADJ
ejpam-841	169	46	-	-	ADJ
ejpam-841	169	47	zero	zero	ADJ
ejpam-841	169	48	constant	constant	ADJ
ejpam-841	169	49	a	a	DET
ejpam-841	169	50	,	,	PUNCT
ejpam-841	169	51	b	b	NOUN
ejpam-841	169	52	,	,	PUNCT
ejpam-841	169	53	c.	c.	PROPN
ejpam-841	169	54	moreover	moreover	ADV
ejpam-841	169	55	,	,	PUNCT
ejpam-841	169	56	gcd(p0,q0	gcd(p0,q0	PROPN
ejpam-841	169	57	)	)	PUNCT
ejpam-841	169	58	=	=	SYM
ejpam-841	170	1	1	1	NUM
ejpam-841	170	2	implies	imply	VERB
ejpam-841	170	3	that	that	PRON
ejpam-841	170	4	b	b	NOUN
ejpam-841	170	5	,	,	PUNCT
ejpam-841	170	6	c	c	PROPN
ejpam-841	170	7	can	can	AUX
ejpam-841	170	8	not	not	PART
ejpam-841	170	9	both	both	PRON
ejpam-841	170	10	be	be	AUX
ejpam-841	170	11	zero	zero	NUM
ejpam-841	170	12	.	.	PUNCT
ejpam-841	171	1	now	now	ADV
ejpam-841	171	2	consider	consider	VERB
ejpam-841	171	3	the	the	DET
ejpam-841	171	4	two	two	NUM
ejpam-841	171	5	moving	move	VERB
ejpam-841	171	6	planes	plane	NOUN
ejpam-841	171	7	bq+	bq+	NOUN
ejpam-841	171	8	cp	cp	PROPN
ejpam-841	171	9	=	=	SYM
ejpam-841	171	10	b(y	b(y	PROPN
ejpam-841	171	11	′s−	′s−	NOUN
ejpam-841	171	12	x	x	SYM
ejpam-841	171	13	′	′	NUM
ejpam-841	171	14	t	t	PROPN
ejpam-841	171	15	)	)	PUNCT
ejpam-841	172	1	+	+	CCONJ
ejpam-841	173	1	c[z′s+	c[z′s+	NUM
ejpam-841	173	2	(	(	PUNCT
ejpam-841	173	3	ax	ax	NOUN
ejpam-841	173	4	′+	′+	PUNCT
ejpam-841	173	5	b	b	SYM
ejpam-841	173	6	y	y	PROPN
ejpam-841	173	7	′	′	VERB
ejpam-841	174	1	+	+	CCONJ
ejpam-841	174	2	cz′)t	cz′)t	NOUN
ejpam-841	174	3	]	]	X
ejpam-841	175	1	=	=	PUNCT
ejpam-841	175	2	(	(	PUNCT
ejpam-841	175	3	b	b	X
ejpam-841	175	4	y	y	NOUN
ejpam-841	175	5	′	′	NOUN
ejpam-841	176	1	+	+	CCONJ
ejpam-841	176	2	cz′)s+	cz′)s+	ADJ
ejpam-841	177	1	[	[	X
ejpam-841	177	2	(	(	PUNCT
ejpam-841	177	3	ac	ac	PROPN
ejpam-841	177	4	−	−	PROPN
ejpam-841	177	5	b)x	b)x	PUNCT
ejpam-841	177	6	′+	′+	PUNCT
ejpam-841	177	7	bc	bc	PROPN
ejpam-841	177	8	y	y	PROPN
ejpam-841	178	1	′	′	NUM
ejpam-841	179	1	+	+	CCONJ
ejpam-841	179	2	c2z′]t	c2z′]t	NUM
ejpam-841	179	3	,	,	PUNCT
ejpam-841	179	4	−aq−	−aq−	X
ejpam-841	180	1	p	p	X
ejpam-841	180	2	=	=	NOUN
ejpam-841	180	3	−a(y	−a(y	NOUN
ejpam-841	180	4	′s−	′s−	NOUN
ejpam-841	180	5	x	x	NOUN
ejpam-841	180	6	′	′	NUM
ejpam-841	181	1	t)−	t)−	PROPN
ejpam-841	182	1	[	[	X
ejpam-841	182	2	z′s+	z′s+	X
ejpam-841	182	3	(	(	PUNCT
ejpam-841	182	4	ax	ax	NOUN
ejpam-841	182	5	′+	′+	PUNCT
ejpam-841	182	6	b	b	SYM
ejpam-841	182	7	y	y	PROPN
ejpam-841	182	8	′	′	VERB
ejpam-841	182	9	+	+	CCONJ
ejpam-841	182	10	cz′)t	cz′)t	NOUN
ejpam-841	182	11	]	]	X
ejpam-841	183	1	=	=	PUNCT
ejpam-841	184	1	(	(	PUNCT
ejpam-841	184	2	−a	−a	NOUN
ejpam-841	184	3	y	y	NOUN
ejpam-841	184	4	′	′	NUM
ejpam-841	185	1	−	−	PROPN
ejpam-841	185	2	z′)s−	z′)s−	NUM
ejpam-841	185	3	(	(	PUNCT
ejpam-841	185	4	b	b	X
ejpam-841	185	5	y	y	NOUN
ejpam-841	185	6	′	′	VERB
ejpam-841	186	1	+	+	CCONJ
ejpam-841	186	2	cz′)t	cz′)t	NOUN
ejpam-841	186	3	.	.	PUNCT
ejpam-841	187	1	notice	notice	VERB
ejpam-841	187	2	that	that	SCONJ
ejpam-841	187	3	ac	ac	PROPN
ejpam-841	187	4	−	−	PROPN
ejpam-841	187	5	b	b	PROPN
ejpam-841	187	6	6=	6=	PROPN
ejpam-841	187	7	0	0	NUM
ejpam-841	187	8	,	,	PUNCT
ejpam-841	187	9	otherwise	otherwise	ADV
ejpam-841	187	10	the	the	DET
ejpam-841	187	11	moving	move	VERB
ejpam-841	187	12	plane	plane	NOUN
ejpam-841	187	13	bq	bq	NOUN
ejpam-841	188	1	+	+	CCONJ
ejpam-841	188	2	cp	cp	NOUN
ejpam-841	188	3	=	=	SYM
ejpam-841	188	4	(	(	PUNCT
ejpam-841	188	5	b	b	X
ejpam-841	188	6	y	y	NOUN
ejpam-841	188	7	′	′	NUM
ejpam-841	189	1	+	+	CCONJ
ejpam-841	189	2	cz′)s	cz′)s	X
ejpam-841	190	1	+	+	CCONJ
ejpam-841	191	1	[	[	X
ejpam-841	191	2	bc	bc	X
ejpam-841	191	3	y	y	PROPN
ejpam-841	191	4	′	′	NUM
ejpam-841	192	1	+	+	CCONJ
ejpam-841	192	2	c2z′]t	c2z′]t	ADJ
ejpam-841	192	3	=	=	SYM
ejpam-841	192	4	(	(	PUNCT
ejpam-841	192	5	b	b	PROPN
ejpam-841	192	6	y	y	PROPN
ejpam-841	192	7	′+	′+	PUNCT
ejpam-841	192	8	cz′)(s+	cz′)(s+	PROPN
ejpam-841	192	9	ct	ct	PROPN
ejpam-841	192	10	)	)	PUNCT
ejpam-841	192	11	,	,	PUNCT
ejpam-841	192	12	which	which	PRON
ejpam-841	192	13	is	be	AUX
ejpam-841	192	14	not	not	PART
ejpam-841	192	15	possible	possible	ADJ
ejpam-841	192	16	because	because	SCONJ
ejpam-841	192	17	the	the	DET
ejpam-841	192	18	curve	curve	NOUN
ejpam-841	192	19	f(s	f(s	PROPN
ejpam-841	192	20	,	,	PUNCT
ejpam-841	192	21	t	t	PROPN
ejpam-841	192	22	)	)	PUNCT
ejpam-841	192	23	is	be	AUX
ejpam-841	192	24	not	not	PART
ejpam-841	192	25	planar	planar	ADJ
ejpam-841	192	26	.	.	PUNCT
ejpam-841	193	1	thus	thus	ADV
ejpam-841	193	2	bq+	bq+	ADJ
ejpam-841	193	3	cp	cp	INTJ
ejpam-841	193	4	and	and	CCONJ
ejpam-841	193	5	−aq−	−aq−	PROPN
ejpam-841	193	6	p	p	NOUN
ejpam-841	193	7	are	be	AUX
ejpam-841	193	8	two	two	NUM
ejpam-841	193	9	linearly	linearly	ADV
ejpam-841	193	10	independent	independent	ADJ
ejpam-841	193	11	moving	move	VERB
ejpam-841	193	12	planes	plane	NOUN
ejpam-841	193	13	that	that	PRON
ejpam-841	193	14	follow	follow	VERB
ejpam-841	193	15	the	the	DET
ejpam-841	193	16	space	space	NOUN
ejpam-841	193	17	curve	curve	NOUN
ejpam-841	193	18	f(s	f(s	PROPN
ejpam-841	193	19	,	,	PUNCT
ejpam-841	193	20	t	t	PROPN
ejpam-841	193	21	)	)	PUNCT
ejpam-841	193	22	.	.	PUNCT
ejpam-841	194	1	therefore	therefore	ADV
ejpam-841	194	2	,	,	PUNCT
ejpam-841	194	3	by	by	ADP
ejpam-841	194	4	a	a	DET
ejpam-841	194	5	projective	projective	ADJ
ejpam-841	194	6	change	change	NOUN
ejpam-841	194	7	of	of	ADP
ejpam-841	194	8	coordinates	coordinate	NOUN
ejpam-841	194	9	,	,	PUNCT
ejpam-841	194	10	if	if	SCONJ
ejpam-841	194	11	we	we	PRON
ejpam-841	194	12	let	let	VERB
ejpam-841	194	13	x	x	PUNCT
ejpam-841	194	14	=	=	PUNCT
ejpam-841	194	15	−[(ac	−[(ac	PROPN
ejpam-841	194	16	−	−	NOUN
ejpam-841	194	17	b)x	b)x	PRON
ejpam-841	194	18	′	′	NUM
ejpam-841	195	1	+	+	CCONJ
ejpam-841	196	1	bc	bc	PROPN
ejpam-841	196	2	y	y	PROPN
ejpam-841	196	3	′	′	NUM
ejpam-841	197	1	+	+	CCONJ
ejpam-841	197	2	c2z′	c2z′	PROPN
ejpam-841	197	3	]	]	X
ejpam-841	197	4	,	,	PUNCT
ejpam-841	198	1	y	y	PROPN
ejpam-841	198	2	=	=	SYM
ejpam-841	198	3	b	b	PROPN
ejpam-841	198	4	y	y	PROPN
ejpam-841	198	5	′	′	NUM
ejpam-841	199	1	+	+	CCONJ
ejpam-841	199	2	cz′	cz′	NOUN
ejpam-841	199	3	,	,	PUNCT
ejpam-841	199	4	z	z	NOUN
ejpam-841	199	5	=	=	PUNCT
ejpam-841	199	6	−a	−a	NOUN
ejpam-841	199	7	y	y	NOUN
ejpam-841	200	1	′	′	NUM
ejpam-841	201	1	−	−	PROPN
ejpam-841	202	1	z′	z′	NOUN
ejpam-841	202	2	,	,	PUNCT
ejpam-841	202	3	then	then	ADV
ejpam-841	202	4	we	we	PRON
ejpam-841	202	5	obtain	obtain	VERB
ejpam-841	202	6	a	a	DET
ejpam-841	202	7	µ-basis	µ-basis	NOUN
ejpam-841	202	8	in	in	ADP
ejpam-841	202	9	the	the	DET
ejpam-841	202	10	desired	desire	VERB
ejpam-841	202	11	form	form	NOUN
ejpam-841	202	12	:	:	PUNCT
ejpam-841	202	13	p	p	X
ejpam-841	202	14	=	=	PUNCT
ejpam-841	202	15	ys−	ys−	PUNCT
ejpam-841	202	16	x	x	SYM
ejpam-841	202	17	t	t	NOUN
ejpam-841	202	18	and	and	CCONJ
ejpam-841	202	19	q	q	NOUN
ejpam-841	202	20	=	=	SYM
ejpam-841	202	21	zs−	zs−	NUM
ejpam-841	202	22	y	y	PROPN
ejpam-841	202	23	t.	t.	PROPN
ejpam-841	202	24	finally	finally	ADV
ejpam-841	202	25	,	,	PUNCT
ejpam-841	202	26	by	by	ADP
ejpam-841	202	27	proposition	proposition	NOUN
ejpam-841	202	28	2	2	NUM
ejpam-841	202	29	,	,	PUNCT
ejpam-841	202	30	the	the	DET
ejpam-841	202	31	singular	singular	ADJ
ejpam-841	202	32	point	point	NOUN
ejpam-841	202	33	is	be	AUX
ejpam-841	202	34	(	(	PUNCT
ejpam-841	202	35	0,0,0,1	0,0,0,1	NOUN
ejpam-841	202	36	)	)	PUNCT
ejpam-841	202	37	.	.	PUNCT
ejpam-841	203	1	by	by	ADP
ejpam-841	203	2	lemma	lemma	PROPN
ejpam-841	203	3	2	2	NUM
ejpam-841	203	4	,	,	PUNCT
ejpam-841	203	5	we	we	PRON
ejpam-841	203	6	can	can	AUX
ejpam-841	203	7	choose	choose	VERB
ejpam-841	203	8	the	the	DET
ejpam-841	203	9	µ-basis	µ-basis	NOUN
ejpam-841	203	10	elements	element	NOUN
ejpam-841	203	11	so	so	SCONJ
ejpam-841	203	12	that	that	SCONJ
ejpam-841	203	13	tp	tp	ADP
ejpam-841	203	14	+	+	CCONJ
ejpam-841	203	15	sq	sq	PROPN
ejpam-841	203	16	=	=	NOUN
ejpam-841	203	17	zs2	zs2	PROPN
ejpam-841	204	1	−	−	NOUN
ejpam-841	204	2	x	x	SYM
ejpam-841	204	3	t2	t2	PROPN
ejpam-841	204	4	.	.	PUNCT
ejpam-841	205	1	therefore	therefore	ADV
ejpam-841	205	2	,	,	PUNCT
ejpam-841	205	3	since	since	SCONJ
ejpam-841	205	4	the	the	DET
ejpam-841	205	5	µ-basis	µ-basis	NOUN
ejpam-841	205	6	vanishes	vanish	VERB
ejpam-841	205	7	on	on	ADP
ejpam-841	205	8	the	the	DET
ejpam-841	205	9	curve	curve	NOUN
ejpam-841	205	10	,	,	PUNCT
ejpam-841	205	11	we	we	PRON
ejpam-841	205	12	have	have	VERB
ejpam-841	205	13	the	the	DET
ejpam-841	205	14	following	follow	VERB
ejpam-841	205	15	relations	relation	NOUN
ejpam-841	205	16	for	for	ADP
ejpam-841	205	17	any	any	DET
ejpam-841	205	18	point	point	NOUN
ejpam-841	205	19	on	on	ADP
ejpam-841	205	20	the	the	DET
ejpam-841	205	21	space	space	NOUN
ejpam-841	205	22	curve	curve	NOUN
ejpam-841	205	23	f(s	f(s	PROPN
ejpam-841	205	24	,	,	PUNCT
ejpam-841	205	25	t	t	PROPN
ejpam-841	205	26	)	)	PUNCT
ejpam-841	205	27	t	t	PROPN
ejpam-841	205	28	s	s	PART
ejpam-841	205	29	=	=	SYM
ejpam-841	205	30	y	y	PROPN
ejpam-841	205	31	x	x	PROPN
ejpam-841	205	32	,	,	PUNCT
ejpam-841	205	33	t2	t2	NOUN
ejpam-841	205	34	s2	s2	NOUN
ejpam-841	205	35	=	=	PUNCT
ejpam-841	206	1	z	z	NOUN
ejpam-841	206	2	x	x	X
ejpam-841	206	3	,	,	PUNCT
ejpam-841	206	4	s	s	PROPN
ejpam-841	206	5	t	t	NOUN
ejpam-841	206	6	=	=	SYM
ejpam-841	206	7	y	y	PROPN
ejpam-841	206	8	z	z	PROPN
ejpam-841	206	9	,	,	PUNCT
ejpam-841	206	10	s2	s2	NOUN
ejpam-841	206	11	t2	t2	NOUN
ejpam-841	206	12	=	=	PUNCT
ejpam-841	206	13	x	x	SYM
ejpam-841	206	14	z	z	NOUN
ejpam-841	206	15	.	.	PUNCT
ejpam-841	207	1	(	(	PUNCT
ejpam-841	207	2	2	2	X
ejpam-841	207	3	)	)	PUNCT
ejpam-841	207	4	we	we	PRON
ejpam-841	207	5	will	will	AUX
ejpam-841	207	6	find	find	VERB
ejpam-841	207	7	the	the	DET
ejpam-841	207	8	implicit	implicit	ADJ
ejpam-841	207	9	equations	equation	NOUN
ejpam-841	207	10	of	of	ADP
ejpam-841	207	11	these	these	DET
ejpam-841	207	12	space	space	NOUN
ejpam-841	207	13	curves	curve	NOUN
ejpam-841	207	14	,	,	PUNCT
ejpam-841	207	15	and	and	CCONJ
ejpam-841	207	16	also	also	ADV
ejpam-841	207	17	the	the	DET
ejpam-841	207	18	defining	define	VERB
ejpam-841	207	19	equations	equation	NOUN
ejpam-841	207	20	for	for	ADP
ejpam-841	207	21	the	the	DET
ejpam-841	207	22	rees	rees	PROPN
ejpam-841	207	23	algebras	algebra	NOUN
ejpam-841	207	24	of	of	ADP
ejpam-841	207	25	these	these	DET
ejpam-841	207	26	space	space	NOUN
ejpam-841	207	27	curve	curve	NOUN
ejpam-841	207	28	by	by	ADP
ejpam-841	207	29	studying	study	VERB
ejpam-841	207	30	separately	separately	ADV
ejpam-841	207	31	the	the	DET
ejpam-841	207	32	case	case	NOUN
ejpam-841	207	33	when	when	SCONJ
ejpam-841	207	34	d	d	NOUN
ejpam-841	207	35	is	be	AUX
ejpam-841	207	36	even	even	ADV
ejpam-841	207	37	and	and	CCONJ
ejpam-841	207	38	the	the	DET
ejpam-841	207	39	case	case	NOUN
ejpam-841	207	40	when	when	SCONJ
ejpam-841	207	41	d	d	NOUN
ejpam-841	207	42	is	be	AUX
ejpam-841	207	43	odd	odd	ADJ
ejpam-841	207	44	.	.	PUNCT
ejpam-841	208	1	3.1.1	3.1.1	NUM
ejpam-841	208	2	.	.	PUNCT
ejpam-841	208	3	degree	degree	NOUN
ejpam-841	208	4	d	d	NOUN
ejpam-841	208	5	=	=	SYM
ejpam-841	208	6	2k	2k	NUM
ejpam-841	208	7	,	,	PUNCT
ejpam-841	208	8	k	k	X
ejpam-841	208	9	≥	≥	NUM
ejpam-841	208	10	2	2	NUM
ejpam-841	208	11	first	first	ADV
ejpam-841	208	12	,	,	PUNCT
ejpam-841	208	13	observe	observe	VERB
ejpam-841	208	14	that	that	SCONJ
ejpam-841	208	15	when	when	SCONJ
ejpam-841	208	16	d	d	PROPN
ejpam-841	208	17	=	=	SYM
ejpam-841	208	18	2k	2k	NUM
ejpam-841	208	19	,	,	PUNCT
ejpam-841	208	20	the	the	DET
ejpam-841	208	21	µ-basis	µ-basis	NOUN
ejpam-841	208	22	element	element	NOUN
ejpam-841	208	23	r	r	NOUN
ejpam-841	208	24	can	can	AUX
ejpam-841	208	25	be	be	AUX
ejpam-841	208	26	written	write	VERB
ejpam-841	208	27	as	as	ADP
ejpam-841	208	28	r	r	NOUN
ejpam-841	208	29	=	=	PUNCT
ejpam-841	208	30	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	208	31	+	+	NUM
ejpam-841	208	32	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	208	33	t	t	NOUN
ejpam-841	208	34	+	+	CCONJ
ejpam-841	208	35	·	·	PUNCT
ejpam-841	208	36	·	·	PUNCT
ejpam-841	208	37	·	·	PUNCT
ejpam-841	208	38	+	+	NUM
ejpam-841	208	39	r0	r0	NOUN
ejpam-841	208	40	td−2	td−2	NOUN
ejpam-841	208	41	=	=	SYM
ejpam-841	208	42	d−2	d−2	PROPN
ejpam-841	208	43	∑	∑	PUNCT
ejpam-841	208	44	j=0	j=0	PROPN
ejpam-841	208	45	r	r	NOUN
ejpam-841	208	46	js	js	PROPN
ejpam-841	208	47	j	j	PROPN
ejpam-841	208	48	td−2−	td−2−	NUM
ejpam-841	208	49	j	j	PROPN
ejpam-841	208	50	=	=	SYM
ejpam-841	208	51	s2i	s2i	PROPN
ejpam-841	208	52	[	[	PUNCT
ejpam-841	208	53	d−2	d−2	PROPN
ejpam-841	208	54	∑	∑	PUNCT
ejpam-841	208	55	j=0	j=0	PROPN
ejpam-841	208	56	r	r	PROPN
ejpam-841	208	57	j	j	PROPN
ejpam-841	208	58	s	s	PROPN
ejpam-841	208	59	j	j	PROPN
ejpam-841	208	60	td−2−	td−2−	NUM
ejpam-841	208	61	j	j	PROPN
ejpam-841	208	62	s2i	s2i	VERB
ejpam-841	208	63	]	]	PUNCT
ejpam-841	208	64	=	=	SYM
ejpam-841	208	65	s2i	s2i	PROPN
ejpam-841	208	66	[	[	PUNCT
ejpam-841	208	67	2i−1	2i−1	NUM
ejpam-841	208	68	∑	∑	PUNCT
ejpam-841	208	69	j=0	j=0	PROPN
ejpam-841	208	70	r	r	PROPN
ejpam-841	208	71	j	j	PROPN
ejpam-841	208	72	t	t	PROPN
ejpam-841	208	73	d−2−2i	d−2−2i	PROPN
ejpam-841	208	74	(	(	PUNCT
ejpam-841	208	75	t	t	PROPN
ejpam-841	208	76	s	s	PART
ejpam-841	208	77	)	)	PUNCT
ejpam-841	208	78	2i−	2i−	NUM
ejpam-841	208	79	j	j	NOUN
ejpam-841	209	1	+	+	CCONJ
ejpam-841	209	2	d−2	d−2	PROPN
ejpam-841	209	3	∑	∑	PROPN
ejpam-841	210	1	j=2i	j=2i	PROPN
ejpam-841	210	2	r	r	AUX
ejpam-841	210	3	js	js	PROPN
ejpam-841	210	4	j−2i	j−2i	PROPN
ejpam-841	210	5	td−2−	td−2−	PROPN
ejpam-841	210	6	j	j	X
ejpam-841	210	7	]	]	X
ejpam-841	210	8	=	=	SYM
ejpam-841	210	9	s2i	s2i	PROPN
ejpam-841	210	10	{	{	PUNCT
ejpam-841	210	11	i−1	i−1	PROPN
ejpam-841	210	12	∑	∑	PUNCT
ejpam-841	210	13	j=0	j=0	PROPN
ejpam-841	211	1	[	[	X
ejpam-841	211	2	r2	r2	PROPN
ejpam-841	211	3	j	j	PROPN
ejpam-841	211	4	t	t	PROPN
ejpam-841	211	5	d−2−2i	d−2−2i	PROPN
ejpam-841	211	6	(	(	PUNCT
ejpam-841	211	7	t2	t2	PROPN
ejpam-841	211	8	s2	s2	PROPN
ejpam-841	211	9	)	)	PUNCT
ejpam-841	211	10	i−	i−	PROPN
ejpam-841	211	11	j	j	PROPN
ejpam-841	211	12	+	+	CCONJ
ejpam-841	211	13	r2	r2	PROPN
ejpam-841	211	14	j+1	j+1	PUNCT
ejpam-841	211	15	td−2−2i	td−2−2i	VERB
ejpam-841	211	16	(	(	PUNCT
ejpam-841	211	17	t2	t2	PROPN
ejpam-841	211	18	s2	s2	PROPN
ejpam-841	211	19	)	)	PUNCT
ejpam-841	211	20	i−	i−	PROPN
ejpam-841	211	21	j−1	j−1	PROPN
ejpam-841	211	22	(	(	PUNCT
ejpam-841	211	23	t	t	PROPN
ejpam-841	211	24	s	s	PART
ejpam-841	211	25	)	)	PUNCT
ejpam-841	211	26	]	]	PUNCT
ejpam-841	212	1	+	+	CCONJ
ejpam-841	212	2	d−2−2i	d−2−2i	PROPN
ejpam-841	212	3	∑	∑	PUNCT
ejpam-841	212	4	j=0	j=0	PROPN
ejpam-841	212	5	r2i+	r2i+	PROPN
ejpam-841	212	6	js	js	ADP
ejpam-841	212	7	j	j	PROPN
ejpam-841	212	8	td−2−2i−	td−2−2i−	PROPN
ejpam-841	212	9	j	j	PROPN
ejpam-841	212	10	}	}	PUNCT
ejpam-841	212	11	,	,	PUNCT
ejpam-841	212	12	for	for	ADP
ejpam-841	212	13	all	all	DET
ejpam-841	212	14	i	i	PRON
ejpam-841	212	15	=	=	NOUN
ejpam-841	212	16	1	1	NUM
ejpam-841	212	17	,	,	PUNCT
ejpam-841	212	18	.	.	PUNCT
ejpam-841	212	19	.	.	PUNCT
ejpam-841	213	1	.	.	PUNCT
ejpam-841	214	1	,	,	PUNCT
ejpam-841	214	2	k−	k−	PROPN
ejpam-841	214	3	1	1	NUM
ejpam-841	214	4	,	,	PUNCT
ejpam-841	214	5	let	let	VERB
ejpam-841	214	6	r	r	NOUN
ejpam-841	214	7	′i	′i	NOUN
ejpam-841	214	8	(	(	PUNCT
ejpam-841	214	9	a	a	DET
ejpam-841	214	10	,	,	PUNCT
ejpam-841	214	11	b	b	NOUN
ejpam-841	214	12	)	)	PUNCT
ejpam-841	214	13	=	=	SYM
ejpam-841	214	14	i−1	i−1	PROPN
ejpam-841	214	15	∑	∑	PUNCT
ejpam-841	214	16	j=0	j=0	PROPN
ejpam-841	215	1	[	[	X
ejpam-841	215	2	r2	r2	PROPN
ejpam-841	215	3	j	j	PROPN
ejpam-841	215	4	t	t	PROPN
ejpam-841	215	5	d−2−2i(b)i−	d−2−2i(b)i−	PROPN
ejpam-841	215	6	j	j	PROPN
ejpam-841	215	7	+	+	CCONJ
ejpam-841	215	8	r2	r2	PROPN
ejpam-841	215	9	j+1	j+1	PUNCT
ejpam-841	215	10	td−2−2i(b)i−	td−2−2i(b)i−	PROPN
ejpam-841	215	11	j−1(a)]+	j−1(a)]+	PROPN
ejpam-841	215	12	d−2−2i	d−2−2i	PROPN
ejpam-841	215	13	∑	∑	PUNCT
ejpam-841	215	14	j=0	j=0	PROPN
ejpam-841	215	15	r2i+	r2i+	PROPN
ejpam-841	215	16	js	js	ADP
ejpam-841	215	17	j	j	PROPN
ejpam-841	215	18	td−2−2i−	td−2−2i−	PROPN
ejpam-841	215	19	j	j	PROPN
ejpam-841	215	20	.	.	PUNCT
ejpam-841	216	1	j.	j.	PROPN
ejpam-841	216	2	hoffman	hoffman	PROPN
ejpam-841	216	3	,	,	PUNCT
ejpam-841	216	4	h.	h.	PROPN
ejpam-841	216	5	wang	wang	PROPN
ejpam-841	216	6	,	,	PUNCT
ejpam-841	216	7	x.	x.	PROPN
ejpam-841	216	8	jia	jia	PROPN
ejpam-841	216	9	,	,	PUNCT
ejpam-841	216	10	r.	r.	PROPN
ejpam-841	216	11	goldman	goldman	PROPN
ejpam-841	216	12	/	/	SYM
ejpam-841	216	13	eur	eur	PROPN
ejpam-841	216	14	.	.	PUNCT
ejpam-841	217	1	j.	j.	PROPN
ejpam-841	217	2	pure	pure	PROPN
ejpam-841	217	3	appl	appl	PROPN
ejpam-841	217	4	.	.	PROPN
ejpam-841	217	5	math	math	PROPN
ejpam-841	217	6	,	,	PUNCT
ejpam-841	217	7	3	3	NUM
ejpam-841	217	8	(	(	PUNCT
ejpam-841	217	9	2010	2010	NUM
ejpam-841	217	10	)	)	PUNCT
ejpam-841	217	11	,	,	PUNCT
ejpam-841	217	12	602	602	NUM
ejpam-841	217	13	-	-	SYM
ejpam-841	217	14	632	632	NUM
ejpam-841	217	15	609	609	NUM
ejpam-841	217	16	then	then	ADV
ejpam-841	217	17	r	r	NOUN
ejpam-841	217	18	′i	′i	NOUN
ejpam-841	217	19	(	(	PUNCT
ejpam-841	217	20	y	y	NOUN
ejpam-841	217	21	x	x	INTJ
ejpam-841	217	22	,	,	PUNCT
ejpam-841	217	23	z	z	NOUN
ejpam-841	217	24	x	x	SYM
ejpam-841	217	25	)	)	PUNCT
ejpam-841	217	26	=	=	NOUN
ejpam-841	217	27	td−2−2i	td−2−2i	PUNCT
ejpam-841	217	28	i−1	i−1	PROPN
ejpam-841	217	29	∑	∑	PUNCT
ejpam-841	217	30	j=0	j=0	PROPN
ejpam-841	218	1	[	[	PUNCT
ejpam-841	218	2	r2	r2	PROPN
ejpam-841	218	3	j	j	PROPN
ejpam-841	218	4	(	(	PUNCT
ejpam-841	218	5	z	z	NOUN
ejpam-841	218	6	x	x	SYM
ejpam-841	218	7	)	)	PUNCT
ejpam-841	218	8	i−	i−	PROPN
ejpam-841	218	9	j	j	PROPN
ejpam-841	218	10	+	+	CCONJ
ejpam-841	218	11	r2	r2	PROPN
ejpam-841	218	12	j+1	j+1	PROPN
ejpam-841	218	13	(	(	PUNCT
ejpam-841	218	14	z	z	NOUN
ejpam-841	218	15	x	x	SYM
ejpam-841	218	16	)	)	PUNCT
ejpam-841	218	17	i−	i−	PROPN
ejpam-841	218	18	j−1	j−1	PROPN
ejpam-841	218	19	(	(	PUNCT
ejpam-841	218	20	y	y	NOUN
ejpam-841	218	21	x	x	PROPN
ejpam-841	218	22	)	)	PUNCT
ejpam-841	218	23	]	]	PUNCT
ejpam-841	219	1	+	+	CCONJ
ejpam-841	219	2	d−2−2i	d−2−2i	PROPN
ejpam-841	219	3	∑	∑	PUNCT
ejpam-841	219	4	j=0	j=0	PROPN
ejpam-841	219	5	r2i+	r2i+	PROPN
ejpam-841	219	6	js	js	ADP
ejpam-841	219	7	j	j	PROPN
ejpam-841	219	8	td−2−2i−	td−2−2i−	PROPN
ejpam-841	219	9	j	j	PROPN
ejpam-841	219	10	.	.	PUNCT
ejpam-841	220	1	(	(	PUNCT
ejpam-841	220	2	3	3	X
ejpam-841	220	3	)	)	PUNCT
ejpam-841	220	4	theorem	theorem	NOUN
ejpam-841	220	5	1	1	NUM
ejpam-841	220	6	.	.	PUNCT
ejpam-841	221	1	a	a	DET
ejpam-841	221	2	minimal	minimal	ADJ
ejpam-841	221	3	set	set	NOUN
ejpam-841	221	4	of	of	ADP
ejpam-841	221	5	generators	generator	NOUN
ejpam-841	221	6	for	for	ADP
ejpam-841	221	7	the	the	DET
ejpam-841	221	8	defining	define	VERB
ejpam-841	221	9	equation	equation	NOUN
ejpam-841	221	10	of	of	ADP
ejpam-841	221	11	the	the	DET
ejpam-841	221	12	rees	rees	PROPN
ejpam-841	221	13	algebra	algebra	NOUN
ejpam-841	221	14	associated	associate	VERB
ejpam-841	221	15	to	to	ADP
ejpam-841	221	16	a	a	DET
ejpam-841	221	17	singular	singular	ADJ
ejpam-841	221	18	rational	rational	ADJ
ejpam-841	221	19	space	space	NOUN
ejpam-841	221	20	curve	curve	NOUN
ejpam-841	221	21	of	of	ADP
ejpam-841	221	22	type	type	NOUN
ejpam-841	221	23	(	(	PUNCT
ejpam-841	221	24	1,1	1,1	NUM
ejpam-841	221	25	,	,	PUNCT
ejpam-841	221	26	d	d	NOUN
ejpam-841	221	27	−	−	PROPN
ejpam-841	221	28	2	2	NUM
ejpam-841	221	29	)	)	PUNCT
ejpam-841	221	30	where	where	SCONJ
ejpam-841	221	31	d	d	NOUN
ejpam-841	221	32	=	=	SYM
ejpam-841	221	33	2k	2k	NUM
ejpam-841	221	34	are	be	AUX
ejpam-841	221	35	given	give	VERB
ejpam-841	221	36	by	by	ADP
ejpam-841	221	37	the	the	DET
ejpam-841	221	38	following	follow	VERB
ejpam-841	221	39	k+	k+	NOUN
ejpam-841	221	40	3	3	NUM
ejpam-841	221	41	polynomials	polynomial	NOUN
ejpam-841	221	42	:	:	PUNCT
ejpam-841	221	43	1	1	NUM
ejpam-841	221	44	.	.	X
ejpam-841	222	1	three	three	NUM
ejpam-841	222	2	µ-basis	µ-basis	NOUN
ejpam-841	222	3	elements	element	NOUN
ejpam-841	222	4	:	:	PUNCT
ejpam-841	222	5	p	p	X
ejpam-841	222	6	,	,	PUNCT
ejpam-841	222	7	q	q	ADJ
ejpam-841	222	8	,	,	PUNCT
ejpam-841	222	9	r	r	NOUN
ejpam-841	222	10	,	,	PUNCT
ejpam-841	222	11	where	where	SCONJ
ejpam-841	222	12	deg(p	deg(p	PROPN
ejpam-841	222	13	)	)	PUNCT
ejpam-841	222	14	=	=	SYM
ejpam-841	222	15	deg(q	deg(q	NOUN
ejpam-841	222	16	)	)	PUNCT
ejpam-841	222	17	=	=	SYM
ejpam-841	222	18	(	(	PUNCT
ejpam-841	222	19	1,1	1,1	NUM
ejpam-841	222	20	)	)	PUNCT
ejpam-841	222	21	,	,	PUNCT
ejpam-841	222	22	and	and	CCONJ
ejpam-841	222	23	deg(r	deg(r	PROPN
ejpam-841	222	24	)	)	PUNCT
ejpam-841	222	25	=	=	PUNCT
ejpam-841	222	26	(	(	PUNCT
ejpam-841	222	27	d	d	NOUN
ejpam-841	222	28	−	−	PROPN
ejpam-841	222	29	2,1	2,1	NUM
ejpam-841	222	30	)	)	PUNCT
ejpam-841	222	31	;	;	PUNCT
ejpam-841	222	32	2	2	X
ejpam-841	222	33	.	.	X
ejpam-841	222	34	two	two	NUM
ejpam-841	222	35	implicit	implicit	ADJ
ejpam-841	222	36	equations	equation	NOUN
ejpam-841	222	37	:	:	PUNCT
ejpam-841	222	38	sylvs	sylv	NOUN
ejpam-841	222	39	,	,	PUNCT
ejpam-841	222	40	t(p	t(p	PROPN
ejpam-841	222	41	,	,	PUNCT
ejpam-841	222	42	q	q	NOUN
ejpam-841	222	43	)	)	PUNCT
ejpam-841	222	44	of	of	ADP
ejpam-841	222	45	degree	degree	NOUN
ejpam-841	222	46	(	(	PUNCT
ejpam-841	222	47	0,2	0,2	NUM
ejpam-841	222	48	)	)	PUNCT
ejpam-841	222	49	,	,	PUNCT
ejpam-841	222	50	and	and	CCONJ
ejpam-841	222	51	x	x	X
ejpam-841	222	52	k−1r	k−1r	PROPN
ejpam-841	222	53	′	′	NUM
ejpam-841	222	54	k−1	k−1	PROPN
ejpam-841	222	55	(	(	PUNCT
ejpam-841	222	56	y	y	NOUN
ejpam-841	222	57	x	x	INTJ
ejpam-841	222	58	,	,	PUNCT
ejpam-841	222	59	z	z	NOUN
ejpam-841	222	60	x	x	SYM
ejpam-841	222	61	)	)	PUNCT
ejpam-841	222	62	of	of	ADP
ejpam-841	222	63	degree	degree	NOUN
ejpam-841	222	64	(	(	PUNCT
ejpam-841	222	65	0	0	NUM
ejpam-841	222	66	,	,	PUNCT
ejpam-841	222	67	k	k	NOUN
ejpam-841	222	68	)	)	PUNCT
ejpam-841	222	69	;	;	PUNCT
ejpam-841	222	70	3	3	X
ejpam-841	222	71	.	.	X
ejpam-841	222	72	k−	k−	NOUN
ejpam-841	222	73	2	2	NUM
ejpam-841	222	74	moving	move	VERB
ejpam-841	222	75	surfaces	surface	NOUN
ejpam-841	222	76	:	:	PUNCT
ejpam-841	222	77	x	x	X
ejpam-841	223	1	i	i	NOUN
ejpam-841	223	2	r	r	VERB
ejpam-841	223	3	′	′	NUM
ejpam-841	224	1	i	i	PRON
ejpam-841	224	2	(	(	PUNCT
ejpam-841	224	3	y	y	NOUN
ejpam-841	224	4	x	x	INTJ
ejpam-841	224	5	,	,	PUNCT
ejpam-841	224	6	z	z	NOUN
ejpam-841	224	7	x	x	SYM
ejpam-841	224	8	)	)	PUNCT
ejpam-841	224	9	of	of	ADP
ejpam-841	224	10	degree	degree	NOUN
ejpam-841	224	11	(	(	PUNCT
ejpam-841	224	12	d	d	NOUN
ejpam-841	224	13	−	−	PROPN
ejpam-841	224	14	2−	2−	NUM
ejpam-841	224	15	2i	2i	NOUN
ejpam-841	224	16	,	,	PUNCT
ejpam-841	224	17	i	i	PRON
ejpam-841	224	18	+	+	NOUN
ejpam-841	224	19	1	1	X
ejpam-841	224	20	)	)	PUNCT
ejpam-841	224	21	for	for	ADP
ejpam-841	224	22	i	i	PROPN
ejpam-841	224	23	=	=	NOUN
ejpam-841	224	24	1	1	NUM
ejpam-841	224	25	,	,	PUNCT
ejpam-841	224	26	.	.	PUNCT
ejpam-841	224	27	.	.	PUNCT
ejpam-841	224	28	.	.	PUNCT
ejpam-841	225	1	,	,	PUNCT
ejpam-841	225	2	k−	k−	PROPN
ejpam-841	225	3	2	2	NUM
ejpam-841	225	4	;	;	PUNCT
ejpam-841	225	5	where	where	SCONJ
ejpam-841	225	6	r	r	NOUN
ejpam-841	225	7	′i	′i	NOUN
ejpam-841	225	8	(	(	PUNCT
ejpam-841	225	9	y	y	NOUN
ejpam-841	225	10	x	x	INTJ
ejpam-841	225	11	,	,	PUNCT
ejpam-841	225	12	z	z	NOUN
ejpam-841	225	13	x	x	PUNCT
ejpam-841	225	14	)	)	PUNCT
ejpam-841	225	15	is	be	AUX
ejpam-841	225	16	defined	define	VERB
ejpam-841	225	17	as	as	ADP
ejpam-841	225	18	in	in	ADP
ejpam-841	225	19	equation	equation	NOUN
ejpam-841	225	20	(	(	PUNCT
ejpam-841	225	21	3	3	NUM
ejpam-841	225	22	)	)	PUNCT
ejpam-841	225	23	for	for	ADP
ejpam-841	225	24	i	i	PROPN
ejpam-841	225	25	=	=	NOUN
ejpam-841	225	26	1	1	NUM
ejpam-841	225	27	,	,	PUNCT
ejpam-841	225	28	.	.	PUNCT
ejpam-841	225	29	.	.	PUNCT
ejpam-841	225	30	.	.	PUNCT
ejpam-841	226	1	k−	k−	PROPN
ejpam-841	226	2	1	1	NUM
ejpam-841	226	3	.	.	PUNCT
ejpam-841	227	1	proof	proof	NOUN
ejpam-841	227	2	.	.	PUNCT
ejpam-841	228	1	we	we	PRON
ejpam-841	228	2	will	will	AUX
ejpam-841	228	3	apply	apply	VERB
ejpam-841	228	4	the	the	DET
ejpam-841	228	5	results	result	NOUN
ejpam-841	228	6	of	of	ADP
ejpam-841	228	7	kustin	kustin	NOUN
ejpam-841	228	8	,	,	PUNCT
ejpam-841	228	9	polini	polini	NOUN
ejpam-841	228	10	and	and	CCONJ
ejpam-841	228	11	ulrich	ulrich	PROPN
ejpam-841	228	12	[	[	X
ejpam-841	228	13	18	18	NUM
ejpam-841	228	14	]	]	PUNCT
ejpam-841	228	15	,	,	PUNCT
ejpam-841	228	16	by	by	ADP
ejpam-841	228	17	listing	list	VERB
ejpam-841	228	18	the	the	DET
ejpam-841	228	19	minimal	minimal	ADJ
ejpam-841	228	20	set	set	NOUN
ejpam-841	228	21	of	of	ADP
ejpam-841	228	22	generators	generator	NOUN
ejpam-841	228	23	in	in	ADP
ejpam-841	228	24	their	their	PRON
ejpam-841	228	25	paper	paper	NOUN
ejpam-841	228	26	,	,	PUNCT
ejpam-841	228	27	and	and	CCONJ
ejpam-841	228	28	comparing	compare	VERB
ejpam-841	228	29	these	these	DET
ejpam-841	228	30	generators	generator	NOUN
ejpam-841	228	31	with	with	ADP
ejpam-841	228	32	the	the	DET
ejpam-841	228	33	generators	generator	NOUN
ejpam-841	228	34	listed	list	VERB
ejpam-841	228	35	in	in	ADP
ejpam-841	228	36	our	our	PRON
ejpam-841	228	37	theorem	theorem	NOUN
ejpam-841	228	38	.	.	PUNCT
ejpam-841	229	1	first	first	ADV
ejpam-841	229	2	,	,	PUNCT
ejpam-841	229	3	we	we	PRON
ejpam-841	229	4	note	note	VERB
ejpam-841	229	5	that	that	SCONJ
ejpam-841	229	6	the	the	DET
ejpam-841	229	7	notation	notation	NOUN
ejpam-841	229	8	x	x	PUNCT
ejpam-841	229	9	,	,	PUNCT
ejpam-841	229	10	y	y	PROPN
ejpam-841	229	11	,	,	PUNCT
ejpam-841	229	12	z	z	PROPN
ejpam-841	229	13	,	,	PUNCT
ejpam-841	229	14	w	w	PROPN
ejpam-841	229	15	,	,	PUNCT
ejpam-841	229	16	t	t	PROPN
ejpam-841	229	17	,	,	PUNCT
ejpam-841	229	18	s	s	PROPN
ejpam-841	229	19	,	,	PUNCT
ejpam-841	229	20	ri	ri	PROPN
ejpam-841	229	21	,	,	PUNCT
ejpam-841	229	22	i	i	PRON
ejpam-841	229	23	=	=	NOUN
ejpam-841	229	24	0	0	NUM
ejpam-841	229	25	,	,	PUNCT
ejpam-841	229	26	.	.	PUNCT
ejpam-841	229	27	.	.	PUNCT
ejpam-841	229	28	.	.	PUNCT
ejpam-841	230	1	,	,	PUNCT
ejpam-841	231	1	d	d	X
ejpam-841	231	2	−	−	PROPN
ejpam-841	231	3	2	2	NUM
ejpam-841	231	4	in	in	ADP
ejpam-841	231	5	this	this	DET
ejpam-841	231	6	paper	paper	NOUN
ejpam-841	231	7	is	be	AUX
ejpam-841	231	8	the	the	DET
ejpam-841	231	9	same	same	ADJ
ejpam-841	231	10	as	as	ADP
ejpam-841	231	11	t1	t1	NOUN
ejpam-841	231	12	,	,	PUNCT
ejpam-841	231	13	t2	t2	NOUN
ejpam-841	231	14	,	,	PUNCT
ejpam-841	231	15	t3	t3	PROPN
ejpam-841	231	16	,	,	PUNCT
ejpam-841	231	17	t4	t4	PROPN
ejpam-841	231	18	,	,	PUNCT
ejpam-841	231	19	x	x	INTJ
ejpam-841	231	20	,	,	PUNCT
ejpam-841	231	21	y	y	PROPN
ejpam-841	231	22	,	,	PUNCT
ejpam-841	231	23	ci	ci	PROPN
ejpam-841	231	24	,	,	PUNCT
ejpam-841	231	25	i	i	PRON
ejpam-841	231	26	=	=	NOUN
ejpam-841	231	27	0	0	NUM
ejpam-841	231	28	,	,	PUNCT
ejpam-841	231	29	.	.	PUNCT
ejpam-841	231	30	.	.	PUNCT
ejpam-841	232	1	.	.	PUNCT
ejpam-841	233	1	,	,	PUNCT
ejpam-841	233	2	d−2	d−2	PROPN
ejpam-841	233	3	in	in	ADP
ejpam-841	233	4	their	their	PRON
ejpam-841	233	5	notation	notation	NOUN
ejpam-841	233	6	.	.	PUNCT
ejpam-841	234	1	moreover	moreover	ADV
ejpam-841	234	2	,	,	PUNCT
ejpam-841	234	3	in	in	ADP
ejpam-841	234	4	our	our	PRON
ejpam-841	234	5	setting	setting	NOUN
ejpam-841	234	6	,	,	PUNCT
ejpam-841	234	7	we	we	PRON
ejpam-841	234	8	identify	identify	VERB
ejpam-841	234	9	the	the	DET
ejpam-841	234	10	following	follow	VERB
ejpam-841	234	11	items	item	NOUN
ejpam-841	234	12	in	in	ADP
ejpam-841	234	13	their	their	PRON
ejpam-841	234	14	paper	paper	NOUN
ejpam-841	234	15	for	for	ADP
ejpam-841	234	16	singular	singular	ADJ
ejpam-841	234	17	curves	curve	NOUN
ejpam-841	234	18	of	of	ADP
ejpam-841	234	19	even	even	ADJ
ejpam-841	234	20	degrees	degree	NOUN
ejpam-841	234	21	:	:	PUNCT
ejpam-841	234	22	ρ	ρ	PROPN
ejpam-841	234	23	=	=	SYM
ejpam-841	234	24	1	1	NUM
ejpam-841	234	25	,	,	PUNCT
ejpam-841	234	26	ℓ	ℓ	NOUN
ejpam-841	234	27	=	=	SYM
ejpam-841	234	28	2	2	NUM
ejpam-841	234	29	,	,	PUNCT
ejpam-841	234	30	σ1	σ1	NOUN
ejpam-841	234	31	=	=	SYM
ejpam-841	234	32	2	2	NUM
ejpam-841	234	33	,	,	PUNCT
ejpam-841	234	34	σ2	σ2	NOUN
ejpam-841	234	35	=	=	SYM
ejpam-841	234	36	1	1	NUM
ejpam-841	234	37	,	,	PUNCT
ejpam-841	234	38	t2,1	t2,1	PROPN
ejpam-841	234	39	=	=	SYM
ejpam-841	234	40	s	s	PROPN
ejpam-841	234	41	,	,	PUNCT
ejpam-841	234	42	t2,2	t2,2	PROPN
ejpam-841	234	43	=	=	SYM
ejpam-841	234	44	t	t	PROPN
ejpam-841	234	45	,	,	PUNCT
ejpam-841	234	46	t1,1	t1,1	NOUN
ejpam-841	234	47	=	=	PUNCT
ejpam-841	234	48	x	x	SYM
ejpam-841	234	49	,	,	PUNCT
ejpam-841	234	50	t1,2	t1,2	PROPN
ejpam-841	234	51	=	=	SYM
ejpam-841	234	52	y	y	PROPN
ejpam-841	234	53	,	,	PUNCT
ejpam-841	234	54	t1,3	t1,3	PROPN
ejpam-841	234	55	=	=	PUNCT
ejpam-841	234	56	z.	z.	PROPN
ejpam-841	234	57	by	by	ADP
ejpam-841	234	58	theorem	theorem	NOUN
ejpam-841	234	59	3.2	3.2	NUM
ejpam-841	234	60	in	in	ADP
ejpam-841	234	61	[	[	X
ejpam-841	234	62	18	18	NUM
ejpam-841	234	63	]	]	PUNCT
ejpam-841	234	64	,	,	PUNCT
ejpam-841	234	65	we	we	PRON
ejpam-841	234	66	have	have	VERB
ejpam-841	234	67	a=	a=	VERB
ejpam-841	234	68	(	(	PUNCT
ejpam-841	234	69	a1	a1	NOUN
ejpam-841	234	70	)	)	PUNCT
ejpam-841	234	71	=	=	SYM
ejpam-841	234	72	0,1	0,1	NUM
ejpam-841	234	73	,	,	PUNCT
ejpam-841	234	74	.	.	PUNCT
ejpam-841	234	75	.	.	PUNCT
ejpam-841	235	1	.	.	PUNCT
ejpam-841	236	1	,	,	PUNCT
ejpam-841	236	2	k−	k−	PROPN
ejpam-841	236	3	2	2	NUM
ejpam-841	236	4	;	;	PUNCT
ejpam-841	236	5	f	f	X
ejpam-841	236	6	(	(	PUNCT
ejpam-841	236	7	a	a	X
ejpam-841	236	8	)	)	PUNCT
ejpam-841	236	9	=	=	SYM
ejpam-841	236	10	f	f	PROPN
ejpam-841	236	11	(	(	PUNCT
ejpam-841	236	12	a1	a1	PROPN
ejpam-841	236	13	)	)	PUNCT
ejpam-841	236	14	=	=	SYM
ejpam-841	237	1	d	d	ADP
ejpam-841	237	2	−	−	PROPN
ejpam-841	237	3	3−	3−	NUM
ejpam-841	237	4	2a1	2a1	NUM
ejpam-841	237	5	;	;	PUNCT
ejpam-841	237	6	r(a	r(a	NUM
ejpam-841	237	7	)	)	PUNCT
ejpam-841	237	8	=	=	SYM
ejpam-841	237	9	r(a1	r(a1	NOUN
ejpam-841	237	10	)	)	PUNCT
ejpam-841	237	11	=	=	SYM
ejpam-841	237	12	1	1	NUM
ejpam-841	237	13	;	;	PUNCT
ejpam-841	237	14	f	f	PROPN
ejpam-841	237	15	(	(	PUNCT
ejpam-841	237	16	;)	;)	PUNCT
ejpam-841	237	17	=	=	PUNCT
ejpam-841	237	18	k−	k−	PROPN
ejpam-841	237	19	2	2	NUM
ejpam-841	237	20	;	;	PUNCT
ejpam-841	237	21	r	r	NOUN
ejpam-841	237	22	(;	(;	X
ejpam-841	237	23	)	)	PUNCT
ejpam-841	237	24	=	=	SYM
ejpam-841	237	25	1	1	NUM
ejpam-841	237	26	;	;	PUNCT
ejpam-841	237	27	t	t	NOUN
ejpam-841	237	28	;	;	PUNCT
ejpam-841	237	29	=	=	SYM
ejpam-841	237	30	1	1	X
ejpam-841	237	31	.	.	PUNCT
ejpam-841	237	32	by	by	ADP
ejpam-841	237	33	definition	definition	NOUN
ejpam-841	237	34	3.5	3.5	NUM
ejpam-841	237	35	and	and	CCONJ
ejpam-841	237	36	the	the	DET
ejpam-841	237	37	description	description	NOUN
ejpam-841	237	38	in	in	ADP
ejpam-841	237	39	[	[	X
ejpam-841	237	40	18	18	NUM
ejpam-841	237	41	]	]	PUNCT
ejpam-841	237	42	,	,	PUNCT
ejpam-841	237	43	the	the	DET
ejpam-841	237	44	generators	generator	NOUN
ejpam-841	237	45	for	for	ADP
ejpam-841	237	46	the	the	DET
ejpam-841	237	47	rees	rees	PROPN
ejpam-841	237	48	algebra	algebra	NOUN
ejpam-841	237	49	are	be	AUX
ejpam-841	237	50	p	p	NOUN
ejpam-841	237	51	,	,	PUNCT
ejpam-841	237	52	q	q	NOUN
ejpam-841	237	53	,	,	PUNCT
ejpam-841	237	54	sylvs	sylvs	NOUN
ejpam-841	237	55	,	,	PUNCT
ejpam-841	237	56	t(p	t(p	PROPN
ejpam-841	237	57	,	,	PUNCT
ejpam-841	237	58	q	q	NOUN
ejpam-841	237	59	)	)	PUNCT
ejpam-841	237	60	,	,	PUNCT
ejpam-841	237	61	f1	f1	NOUN
ejpam-841	237	62	and	and	CCONJ
ejpam-841	237	63	ga1,1	ga1,1	NOUN
ejpam-841	237	64	.	.	PUNCT
ejpam-841	238	1	we	we	PRON
ejpam-841	238	2	shall	shall	AUX
ejpam-841	238	3	now	now	ADV
ejpam-841	238	4	write	write	VERB
ejpam-841	238	5	f1	f1	NOUN
ejpam-841	238	6	and	and	CCONJ
ejpam-841	238	7	ga1,1	ga1,1	NOUN
ejpam-841	238	8	explicitly	explicitly	ADV
ejpam-841	238	9	and	and	CCONJ
ejpam-841	238	10	compare	compare	VERB
ejpam-841	238	11	these	these	DET
ejpam-841	238	12	expressions	expression	NOUN
ejpam-841	238	13	with	with	ADP
ejpam-841	238	14	the	the	DET
ejpam-841	238	15	generators	generator	NOUN
ejpam-841	238	16	listed	list	VERB
ejpam-841	238	17	in	in	ADP
ejpam-841	238	18	the	the	DET
ejpam-841	238	19	statement	statement	NOUN
ejpam-841	238	20	of	of	ADP
ejpam-841	238	21	our	our	PRON
ejpam-841	238	22	theorem	theorem	NOUN
ejpam-841	238	23	.	.	PUNCT
ejpam-841	239	1	we	we	PRON
ejpam-841	239	2	have	have	VERB
ejpam-841	239	3	f1	f1	NOUN
ejpam-841	239	4	=	=	SYM
ejpam-841	239	5	z	z	PROPN
ejpam-841	239	6	∑	∑	PROPN
ejpam-841	239	7	i+	i+	PROPN
ejpam-841	239	8	j	j	PROPN
ejpam-841	239	9	=	=	PROPN
ejpam-841	239	10	k−3	k−3	PROPN
ejpam-841	239	11	x	x	INTJ
ejpam-841	239	12	iz	iz	INTJ
ejpam-841	239	13	j(r2iz	j(r2iz	NOUN
ejpam-841	240	1	+	+	CCONJ
ejpam-841	240	2	r2i+1	r2i+1	PROPN
ejpam-841	240	3	y	y	NOUN
ejpam-841	240	4	)	)	PUNCT
ejpam-841	241	1	+	+	CCONJ
ejpam-841	241	2	x	x	SYM
ejpam-841	241	3	k−2(rd−2	k−2(rd−2	NOUN
ejpam-841	241	4	x	x	PUNCT
ejpam-841	242	1	+	+	CCONJ
ejpam-841	242	2	rd−3	rd−3	PROPN
ejpam-841	242	3	y	y	PROPN
ejpam-841	242	4	+	+	CCONJ
ejpam-841	242	5	rd−4z	rd−4z	NOUN
ejpam-841	242	6	)	)	PUNCT
ejpam-841	242	7	=	=	SYM
ejpam-841	243	1	∑	∑	PROPN
ejpam-841	243	2	i+	i+	X
ejpam-841	243	3	j	j	PROPN
ejpam-841	243	4	=	=	PROPN
ejpam-841	243	5	k−3	k−3	PROPN
ejpam-841	243	6	[	[	X
ejpam-841	243	7	r2i	r2i	ADP
ejpam-841	243	8	x	x	INTJ
ejpam-841	243	9	iz	iz	INTJ
ejpam-841	243	10	j+2	j+2	PROPN
ejpam-841	244	1	+	+	CCONJ
ejpam-841	244	2	r2i+1	r2i+1	NOUN
ejpam-841	244	3	x	x	VERB
ejpam-841	244	4	i	i	NOUN
ejpam-841	244	5	yz	yz	PROPN
ejpam-841	244	6	j+1	j+1	X
ejpam-841	244	7	]	]	X
ejpam-841	245	1	+	+	CCONJ
ejpam-841	245	2	(	(	PUNCT
ejpam-841	245	3	rd−2	rd−2	PROPN
ejpam-841	245	4	x	x	SYM
ejpam-841	245	5	k−1	k−1	PROPN
ejpam-841	245	6	+	+	X
ejpam-841	245	7	rd−3	rd−3	PROPN
ejpam-841	245	8	x	x	VERB
ejpam-841	245	9	k−2	k−2	PROPN
ejpam-841	245	10	y	y	PROPN
ejpam-841	245	11	+	+	CCONJ
ejpam-841	245	12	rd−4	rd−4	PROPN
ejpam-841	245	13	x	x	SYM
ejpam-841	245	14	k−2z	k−2z	PROPN
ejpam-841	245	15	)	)	PUNCT
ejpam-841	245	16	=	=	PUNCT
ejpam-841	246	1	x	x	X
ejpam-841	246	2	k−1{rd−2	k−1{rd−2	NOUN
ejpam-841	246	3	+	+	CCONJ
ejpam-841	246	4	rd−3	rd−3	NOUN
ejpam-841	246	5	(	(	PUNCT
ejpam-841	246	6	y	y	NOUN
ejpam-841	246	7	x	x	PROPN
ejpam-841	246	8	)	)	PUNCT
ejpam-841	247	1	+	+	CCONJ
ejpam-841	247	2	rd−4	rd−4	NOUN
ejpam-841	247	3	(	(	PUNCT
ejpam-841	247	4	z	z	NOUN
ejpam-841	247	5	x	x	PUNCT
ejpam-841	247	6	)	)	PUNCT
ejpam-841	248	1	+	+	CCONJ
ejpam-841	248	2	k−3	k−3	PROPN
ejpam-841	248	3	∑	∑	PROPN
ejpam-841	248	4	i=0	i=0	PROPN
ejpam-841	248	5	[	[	PUNCT
ejpam-841	248	6	r2i	r2i	ADP
ejpam-841	248	7	(	(	PUNCT
ejpam-841	248	8	z	z	NOUN
ejpam-841	248	9	x	x	X
ejpam-841	248	10	)	)	PUNCT
ejpam-841	248	11	k−1−i	k−1−i	PROPN
ejpam-841	248	12	+	+	NUM
ejpam-841	248	13	r2i+1	r2i+1	PROPN
ejpam-841	248	14	(	(	PUNCT
ejpam-841	248	15	z	z	NOUN
ejpam-841	248	16	x	x	SYM
ejpam-841	248	17	)	)	PUNCT
ejpam-841	248	18	k−2−i	k−2−i	PROPN
ejpam-841	248	19	(	(	PUNCT
ejpam-841	248	20	y	y	NOUN
ejpam-841	248	21	x	x	PROPN
ejpam-841	248	22	)	)	PUNCT
ejpam-841	248	23	]	]	PUNCT
ejpam-841	248	24	}	}	PUNCT
ejpam-841	248	25	=	=	PUNCT
ejpam-841	248	26	x	x	SYM
ejpam-841	248	27	k−1{rd−2	k−1{rd−2	X
ejpam-841	248	28	+	+	CCONJ
ejpam-841	248	29	k−2	k−2	PROPN
ejpam-841	248	30	∑	∑	PROPN
ejpam-841	248	31	i=0	i=0	PROPN
ejpam-841	248	32	[	[	X
ejpam-841	248	33	r2i	r2i	ADP
ejpam-841	248	34	(	(	PUNCT
ejpam-841	248	35	z	z	NOUN
ejpam-841	248	36	x	x	X
ejpam-841	248	37	)	)	PUNCT
ejpam-841	248	38	k−1−i	k−1−i	PROPN
ejpam-841	248	39	+	+	NUM
ejpam-841	248	40	r2i+1	r2i+1	PROPN
ejpam-841	248	41	(	(	PUNCT
ejpam-841	248	42	z	z	NOUN
ejpam-841	248	43	x	x	SYM
ejpam-841	248	44	)	)	PUNCT
ejpam-841	248	45	k−2−i	k−2−i	PROPN
ejpam-841	248	46	(	(	PUNCT
ejpam-841	248	47	y	y	NOUN
ejpam-841	248	48	x	x	PROPN
ejpam-841	248	49	)	)	PUNCT
ejpam-841	248	50	]	]	PUNCT
ejpam-841	248	51	}	}	PUNCT
ejpam-841	248	52	j.	j.	PROPN
ejpam-841	248	53	hoffman	hoffman	PROPN
ejpam-841	248	54	,	,	PUNCT
ejpam-841	248	55	h.	h.	PROPN
ejpam-841	248	56	wang	wang	PROPN
ejpam-841	248	57	,	,	PUNCT
ejpam-841	248	58	x.	x.	PROPN
ejpam-841	248	59	jia	jia	PROPN
ejpam-841	248	60	,	,	PUNCT
ejpam-841	248	61	r.	r.	PROPN
ejpam-841	248	62	goldman	goldman	PROPN
ejpam-841	248	63	/	/	SYM
ejpam-841	248	64	eur	eur	PROPN
ejpam-841	248	65	.	.	PUNCT
ejpam-841	249	1	j.	j.	PROPN
ejpam-841	249	2	pure	pure	PROPN
ejpam-841	249	3	appl	appl	PROPN
ejpam-841	249	4	.	.	PROPN
ejpam-841	249	5	math	math	PROPN
ejpam-841	249	6	,	,	PUNCT
ejpam-841	249	7	3	3	NUM
ejpam-841	249	8	(	(	PUNCT
ejpam-841	249	9	2010	2010	NUM
ejpam-841	249	10	)	)	PUNCT
ejpam-841	249	11	,	,	PUNCT
ejpam-841	249	12	602	602	NUM
ejpam-841	249	13	-	-	SYM
ejpam-841	249	14	632	632	NUM
ejpam-841	250	1	610	610	NUM
ejpam-841	250	2	=	=	SYM
ejpam-841	250	3	x	x	PRON
ejpam-841	250	4	k−1r	k−1r	PROPN
ejpam-841	250	5	′k−1	′k−1	PROPN
ejpam-841	250	6	(	(	PUNCT
ejpam-841	250	7	y	y	NOUN
ejpam-841	250	8	x	x	PROPN
ejpam-841	250	9	,	,	PUNCT
ejpam-841	250	10	z	z	NOUN
ejpam-841	250	11	x	x	NOUN
ejpam-841	250	12	)	)	PUNCT
ejpam-841	250	13	.	.	PUNCT
ejpam-841	251	1	hence	hence	ADV
ejpam-841	251	2	f1	f1	PROPN
ejpam-841	251	3	is	be	AUX
ejpam-841	251	4	one	one	NUM
ejpam-841	251	5	of	of	ADP
ejpam-841	251	6	our	our	PRON
ejpam-841	251	7	implicit	implicit	ADJ
ejpam-841	251	8	equations	equation	NOUN
ejpam-841	251	9	and	and	CCONJ
ejpam-841	251	10	deg	deg	PROPN
ejpam-841	251	11	(	(	PUNCT
ejpam-841	251	12	f1	f1	NOUN
ejpam-841	251	13	)	)	PUNCT
ejpam-841	251	14	=	=	PUNCT
ejpam-841	251	15	(	(	PUNCT
ejpam-841	251	16	0	0	NUM
ejpam-841	251	17	,	,	PUNCT
ejpam-841	251	18	k	k	NOUN
ejpam-841	251	19	)	)	PUNCT
ejpam-841	251	20	.	.	PUNCT
ejpam-841	252	1	ga1,1	ga1,1	NOUN
ejpam-841	252	2	=	=	PROPN
ejpam-841	252	3	td−2−2a1	td−2−2a1	PROPN
ejpam-841	252	4	∑	∑	PUNCT
ejpam-841	252	5	i+	i+	PROPN
ejpam-841	252	6	j	j	X
ejpam-841	252	7	=	=	NOUN
ejpam-841	252	8	a1−1	a1−1	NOUN
ejpam-841	252	9	x	x	INTJ
ejpam-841	253	1	iz	iz	INTJ
ejpam-841	253	2	j(r2iz	j(r2iz	NOUN
ejpam-841	254	1	+	+	CCONJ
ejpam-841	254	2	r2i+1	r2i+1	PROPN
ejpam-841	254	3	y	y	NOUN
ejpam-841	254	4	)	)	PUNCT
ejpam-841	255	1	+	+	CCONJ
ejpam-841	255	2	x	x	SYM
ejpam-841	255	3	a1	a1	NOUN
ejpam-841	255	4	t	t	PROPN
ejpam-841	255	5	×	×	NOUN
ejpam-841	255	6	∑	∑	PUNCT
ejpam-841	255	7	i+	i+	PROPN
ejpam-841	255	8	j	j	PROPN
ejpam-841	255	9	=	=	PROPN
ejpam-841	255	10	d−4−2a1	d−4−2a1	NOUN
ejpam-841	255	11	si	si	PROPN
ejpam-841	255	12	t	t	PROPN
ejpam-841	255	13	j	j	PROPN
ejpam-841	255	14	r2a1+i	r2a1+i	PROPN
ejpam-841	255	15	t	t	PROPN
ejpam-841	255	16	+	+	CCONJ
ejpam-841	255	17	x	x	PRON
ejpam-841	255	18	a1sd−3−2a1(rd−2s+	a1sd−3−2a1(rd−2s+	VERB
ejpam-841	255	19	rd−3	rd−3	PROPN
ejpam-841	255	20	t	t	PROPN
ejpam-841	255	21	)	)	PUNCT
ejpam-841	255	22	=	=	PUNCT
ejpam-841	256	1	x	x	X
ejpam-841	256	2	a1{td−2−2a1	a1{td−2−2a1	NOUN
ejpam-841	256	3	a1−1	a1−1	PRON
ejpam-841	256	4	∑	∑	PUNCT
ejpam-841	256	5	i=0	i=0	PROPN
ejpam-841	256	6	[	[	PUNCT
ejpam-841	256	7	r2i	r2i	ADP
ejpam-841	256	8	(	(	PUNCT
ejpam-841	256	9	z	z	NOUN
ejpam-841	256	10	x	x	NOUN
ejpam-841	256	11	)	)	PUNCT
ejpam-841	256	12	a1−i	a1−i	PROPN
ejpam-841	256	13	+	+	SYM
ejpam-841	256	14	r2i+1	r2i+1	PROPN
ejpam-841	256	15	(	(	PUNCT
ejpam-841	256	16	z	z	NOUN
ejpam-841	256	17	x	x	SYM
ejpam-841	256	18	)	)	PUNCT
ejpam-841	256	19	a1−i−1	a1−i−1	PROPN
ejpam-841	256	20	(	(	PUNCT
ejpam-841	256	21	y	y	PROPN
ejpam-841	256	22	x	x	PROPN
ejpam-841	256	23	)	)	PUNCT
ejpam-841	256	24	]	]	PUNCT
ejpam-841	257	1	+	+	CCONJ
ejpam-841	257	2	d−2−2a1	d−2−2a1	PUNCT
ejpam-841	257	3	∑	∑	ADP
ejpam-841	257	4	i=0	i=0	PROPN
ejpam-841	257	5	r2a1+is	r2a1+is	PROPN
ejpam-841	257	6	i	i	PRON
ejpam-841	257	7	td−2−2a1−i	td−2−2a1−i	PROPN
ejpam-841	257	8	}	}	PUNCT
ejpam-841	257	9	=	=	SYM
ejpam-841	257	10	(	(	PUNCT
ejpam-841	257	11	x	x	SYM
ejpam-841	257	12	a1	a1	NOUN
ejpam-841	257	13	r	r	NOUN
ejpam-841	257	14	′a1	′a1	NOUN
ejpam-841	257	15	(	(	PUNCT
ejpam-841	257	16	y	y	NOUN
ejpam-841	257	17	x	x	PROPN
ejpam-841	257	18	,	,	PUNCT
ejpam-841	257	19	z	z	NOUN
ejpam-841	257	20	x	x	PUNCT
ejpam-841	257	21	)	)	PUNCT
ejpam-841	257	22	for	for	ADP
ejpam-841	257	23	a1	a1	NOUN
ejpam-841	257	24	=	=	SYM
ejpam-841	257	25	1	1	NUM
ejpam-841	257	26	,	,	PUNCT
ejpam-841	257	27	·	·	PUNCT
ejpam-841	257	28	·	·	PUNCT
ejpam-841	257	29	·	·	PUNCT
ejpam-841	257	30	,	,	PUNCT
ejpam-841	257	31	k−	k−	NOUN
ejpam-841	257	32	2	2	NUM
ejpam-841	257	33	,	,	PUNCT
ejpam-841	257	34	r	r	NOUN
ejpam-841	257	35	for	for	ADP
ejpam-841	257	36	a1	a1	NOUN
ejpam-841	257	37	=	=	SYM
ejpam-841	257	38	0	0	X
ejpam-841	257	39	.	.	PUNCT
ejpam-841	258	1	hence	hence	ADV
ejpam-841	258	2	ga1,1	ga1,1	NOUN
ejpam-841	258	3	are	be	AUX
ejpam-841	258	4	our	our	PRON
ejpam-841	258	5	moving	move	VERB
ejpam-841	258	6	surfaces	surface	NOUN
ejpam-841	258	7	,	,	PUNCT
ejpam-841	258	8	and	and	CCONJ
ejpam-841	258	9	deg(ga1,1	deg(ga1,1	NOUN
ejpam-841	258	10	)	)	PUNCT
ejpam-841	258	11	=	=	PUNCT
ejpam-841	259	1	(	(	PUNCT
ejpam-841	259	2	d	d	NOUN
ejpam-841	259	3	−	−	PROPN
ejpam-841	259	4	2−	2−	NUM
ejpam-841	259	5	2a1	2a1	NUM
ejpam-841	259	6	,	,	PUNCT
ejpam-841	259	7	a1	a1	NOUN
ejpam-841	259	8	+	+	CCONJ
ejpam-841	259	9	1	1	NUM
ejpam-841	259	10	)	)	PUNCT
ejpam-841	259	11	.	.	PUNCT
ejpam-841	260	1	therefore	therefore	ADV
ejpam-841	260	2	,	,	PUNCT
ejpam-841	260	3	the	the	DET
ejpam-841	260	4	generators	generator	NOUN
ejpam-841	260	5	provided	provide	VERB
ejpam-841	260	6	by	by	ADP
ejpam-841	260	7	our	our	PRON
ejpam-841	260	8	theorem	theorem	NOUN
ejpam-841	260	9	are	be	AUX
ejpam-841	260	10	the	the	DET
ejpam-841	260	11	same	same	ADJ
ejpam-841	260	12	as	as	SCONJ
ejpam-841	260	13	the	the	DET
ejpam-841	260	14	generators	generator	NOUN
ejpam-841	260	15	described	describe	VERB
ejpam-841	260	16	in	in	ADP
ejpam-841	260	17	[	[	X
ejpam-841	260	18	18	18	NUM
ejpam-841	260	19	]	]	PUNCT
ejpam-841	260	20	.	.	PUNCT
ejpam-841	261	1	thus	thus	ADV
ejpam-841	261	2	,	,	PUNCT
ejpam-841	261	3	we	we	PRON
ejpam-841	261	4	have	have	AUX
ejpam-841	261	5	proved	prove	VERB
ejpam-841	261	6	our	our	PRON
ejpam-841	261	7	claim	claim	NOUN
ejpam-841	261	8	.	.	PUNCT
ejpam-841	262	1	3.1.2	3.1.2	X
ejpam-841	262	2	.	.	PUNCT
ejpam-841	262	3	degree	degree	NOUN
ejpam-841	263	1	d	d	NOUN
ejpam-841	263	2	=	=	SYM
ejpam-841	263	3	2k+	2k+	NUM
ejpam-841	263	4	1	1	NUM
ejpam-841	263	5	,	,	PUNCT
ejpam-841	263	6	k	k	X
ejpam-841	263	7	≥	≥	NUM
ejpam-841	263	8	2	2	NUM
ejpam-841	263	9	first	first	ADV
ejpam-841	263	10	,	,	PUNCT
ejpam-841	263	11	observe	observe	VERB
ejpam-841	263	12	that	that	SCONJ
ejpam-841	263	13	when	when	SCONJ
ejpam-841	263	14	d	d	PROPN
ejpam-841	263	15	=	=	SYM
ejpam-841	263	16	2k+	2k+	NUM
ejpam-841	263	17	1	1	NUM
ejpam-841	263	18	,	,	PUNCT
ejpam-841	263	19	the	the	DET
ejpam-841	263	20	µ-basis	µ-basis	NOUN
ejpam-841	263	21	element	element	NOUN
ejpam-841	263	22	r	r	NOUN
ejpam-841	263	23	can	can	AUX
ejpam-841	263	24	be	be	AUX
ejpam-841	263	25	written	write	VERB
ejpam-841	263	26	as	as	ADP
ejpam-841	263	27	r	r	NOUN
ejpam-841	263	28	=	=	PUNCT
ejpam-841	263	29	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	263	30	+	+	NUM
ejpam-841	263	31	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	263	32	t	t	NOUN
ejpam-841	263	33	+	+	CCONJ
ejpam-841	263	34	·	·	PUNCT
ejpam-841	263	35	·	·	PUNCT
ejpam-841	263	36	·	·	PUNCT
ejpam-841	263	37	+	+	NUM
ejpam-841	263	38	r0	r0	NOUN
ejpam-841	263	39	td−2	td−2	NOUN
ejpam-841	263	40	=	=	SYM
ejpam-841	263	41	d−2	d−2	PROPN
ejpam-841	263	42	∑	∑	PUNCT
ejpam-841	263	43	j=0	j=0	PROPN
ejpam-841	263	44	r	r	NOUN
ejpam-841	263	45	js	js	PROPN
ejpam-841	263	46	j	j	PROPN
ejpam-841	263	47	td−2−	td−2−	NUM
ejpam-841	263	48	j	j	NOUN
ejpam-841	263	49	=	=	SYM
ejpam-841	263	50	sd−2	sd−2	NOUN
ejpam-841	263	51	d−2	d−2	PROPN
ejpam-841	263	52	∑	∑	PUNCT
ejpam-841	263	53	j=0	j=0	PROPN
ejpam-841	263	54	r	r	NOUN
ejpam-841	263	55	j	j	PROPN
ejpam-841	263	56	td−2−	td−2−	NUM
ejpam-841	263	57	j	j	PROPN
ejpam-841	264	1	sd−2−	sd−2−	X
ejpam-841	264	2	j	j	NOUN
ejpam-841	264	3	=	=	SYM
ejpam-841	264	4	sd−2	sd−2	PROPN
ejpam-841	264	5	k−1	k−1	PROPN
ejpam-841	264	6	∑	∑	PUNCT
ejpam-841	264	7	j=0	j=0	PROPN
ejpam-841	265	1	[	[	X
ejpam-841	265	2	r2	r2	PROPN
ejpam-841	265	3	j	j	PROPN
ejpam-841	265	4	(	(	PUNCT
ejpam-841	265	5	t2	t2	PROPN
ejpam-841	265	6	s2	s2	PROPN
ejpam-841	265	7	)	)	PUNCT
ejpam-841	266	1	k−1−	k−1−	PROPN
ejpam-841	266	2	j	j	PROPN
ejpam-841	266	3	(	(	PUNCT
ejpam-841	266	4	t	t	PROPN
ejpam-841	266	5	s	s	PART
ejpam-841	266	6	)	)	PUNCT
ejpam-841	266	7	+	+	NUM
ejpam-841	266	8	r2	r2	PROPN
ejpam-841	266	9	j+1	j+1	PROPN
ejpam-841	266	10	(	(	PUNCT
ejpam-841	266	11	t2	t2	NOUN
ejpam-841	266	12	s2	s2	PROPN
ejpam-841	266	13	)	)	PUNCT
ejpam-841	267	1	k−1−	k−1−	PROPN
ejpam-841	267	2	j	j	X
ejpam-841	267	3	]	]	X
ejpam-841	267	4	.	.	PUNCT
ejpam-841	268	1	let	let	VERB
ejpam-841	268	2	r	r	NOUN
ejpam-841	268	3	′′(a	′′(a	PROPN
ejpam-841	268	4	,	,	PUNCT
ejpam-841	268	5	b	b	NOUN
ejpam-841	268	6	)	)	PUNCT
ejpam-841	268	7	=	=	SYM
ejpam-841	268	8	k−1	k−1	PROPN
ejpam-841	268	9	∑	∑	PUNCT
ejpam-841	268	10	j=0	j=0	PROPN
ejpam-841	269	1	[	[	PUNCT
ejpam-841	269	2	r2	r2	PROPN
ejpam-841	269	3	j(b	j(b	PROPN
ejpam-841	269	4	)	)	PUNCT
ejpam-841	270	1	k−1−	k−1−	PROPN
ejpam-841	270	2	j(a	j(a	PROPN
ejpam-841	270	3	)	)	PUNCT
ejpam-841	271	1	+	+	CCONJ
ejpam-841	271	2	r2	r2	PROPN
ejpam-841	271	3	j+1(b	j+1(b	PROPN
ejpam-841	271	4	)	)	PUNCT
ejpam-841	272	1	k−1−	k−1−	PROPN
ejpam-841	272	2	j	j	X
ejpam-841	272	3	]	]	X
ejpam-841	272	4	.	.	PUNCT
ejpam-841	273	1	then	then	ADV
ejpam-841	273	2	r	r	PROPN
ejpam-841	273	3	′′	′′	PROPN
ejpam-841	273	4	(	(	PUNCT
ejpam-841	273	5	y	y	PROPN
ejpam-841	273	6	x	x	PROPN
ejpam-841	273	7	,	,	PUNCT
ejpam-841	273	8	z	z	NOUN
ejpam-841	273	9	x	x	SYM
ejpam-841	273	10	)	)	PUNCT
ejpam-841	274	1	=	=	SYM
ejpam-841	274	2	k−1	k−1	PROPN
ejpam-841	274	3	∑	∑	PUNCT
ejpam-841	274	4	j=0	j=0	PROPN
ejpam-841	275	1	[	[	X
ejpam-841	275	2	r2	r2	PROPN
ejpam-841	275	3	j	j	PROPN
ejpam-841	275	4	(	(	PUNCT
ejpam-841	275	5	z	z	NOUN
ejpam-841	275	6	x	x	SYM
ejpam-841	275	7	)	)	PUNCT
ejpam-841	276	1	k−1−	k−1−	PROPN
ejpam-841	276	2	j	j	PROPN
ejpam-841	276	3	(	(	PUNCT
ejpam-841	276	4	y	y	NOUN
ejpam-841	276	5	x	x	PROPN
ejpam-841	276	6	)	)	PUNCT
ejpam-841	277	1	+	+	CCONJ
ejpam-841	277	2	r2	r2	PROPN
ejpam-841	277	3	j+1	j+1	PROPN
ejpam-841	277	4	(	(	PUNCT
ejpam-841	277	5	z	z	NOUN
ejpam-841	277	6	x	x	SYM
ejpam-841	277	7	)	)	PUNCT
ejpam-841	277	8	k−1−	k−1−	PROPN
ejpam-841	277	9	j	j	X
ejpam-841	277	10	]	]	X
ejpam-841	277	11	.	.	PUNCT
ejpam-841	278	1	(	(	PUNCT
ejpam-841	278	2	4	4	X
ejpam-841	278	3	)	)	PUNCT
ejpam-841	278	4	on	on	ADP
ejpam-841	278	5	the	the	DET
ejpam-841	278	6	other	other	ADJ
ejpam-841	278	7	hand	hand	NOUN
ejpam-841	278	8	,	,	PUNCT
ejpam-841	278	9	the	the	DET
ejpam-841	278	10	µ-basis	µ-basis	NOUN
ejpam-841	278	11	element	element	NOUN
ejpam-841	278	12	r	r	NOUN
ejpam-841	278	13	also	also	ADV
ejpam-841	278	14	can	can	AUX
ejpam-841	278	15	be	be	AUX
ejpam-841	278	16	written	write	VERB
ejpam-841	278	17	as	as	ADP
ejpam-841	278	18	r	r	NOUN
ejpam-841	278	19	=	=	PUNCT
ejpam-841	278	20	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	278	21	+	+	NUM
ejpam-841	278	22	rd−3sd−3	rd−3sd−3	PART
ejpam-841	278	23	t	t	NOUN
ejpam-841	278	24	+	+	X
ejpam-841	278	25	·	·	PUNCT
ejpam-841	278	26	·	·	PUNCT
ejpam-841	278	27	·	·	PUNCT
ejpam-841	279	1	+	+	NUM
ejpam-841	279	2	r0	r0	NOUN
ejpam-841	279	3	td−2	td−2	NOUN
ejpam-841	279	4	=	=	SYM
ejpam-841	279	5	d−2	d−2	PROPN
ejpam-841	279	6	∑	∑	PUNCT
ejpam-841	279	7	j=0	j=0	PROPN
ejpam-841	279	8	r	r	NOUN
ejpam-841	279	9	js	js	PROPN
ejpam-841	279	10	j	j	PROPN
ejpam-841	279	11	td−2−	td−2−	NUM
ejpam-841	279	12	j	j	PROPN
ejpam-841	279	13	=	=	SYM
ejpam-841	279	14	td−2	td−2	PROPN
ejpam-841	279	15	d−2	d−2	PROPN
ejpam-841	279	16	∑	∑	PUNCT
ejpam-841	279	17	j=0	j=0	PROPN
ejpam-841	279	18	r	r	PROPN
ejpam-841	279	19	j	j	PROPN
ejpam-841	279	20	s	s	PROPN
ejpam-841	279	21	j	j	PROPN
ejpam-841	279	22	t	t	PROPN
ejpam-841	279	23	j	j	PROPN
ejpam-841	279	24	=	=	PROPN
ejpam-841	279	25	td−2	td−2	PROPN
ejpam-841	279	26	{	{	PUNCT
ejpam-841	279	27	k−1	k−1	PROPN
ejpam-841	279	28	∑	∑	PUNCT
ejpam-841	279	29	j=0	j=0	PROPN
ejpam-841	280	1	[	[	X
ejpam-841	280	2	r2	r2	PROPN
ejpam-841	280	3	j	j	PROPN
ejpam-841	280	4	(	(	PUNCT
ejpam-841	280	5	s2	s2	PROPN
ejpam-841	280	6	t2	t2	PROPN
ejpam-841	280	7	)	)	PUNCT
ejpam-841	280	8	j	j	PROPN
ejpam-841	281	1	+	+	CCONJ
ejpam-841	281	2	r2	r2	PROPN
ejpam-841	281	3	j+1	j+1	PROPN
ejpam-841	281	4	(	(	PUNCT
ejpam-841	281	5	s2	s2	NOUN
ejpam-841	281	6	t2	t2	PROPN
ejpam-841	281	7	)	)	PUNCT
ejpam-841	281	8	j	j	PROPN
ejpam-841	281	9	(	(	PUNCT
ejpam-841	281	10	s	s	PROPN
ejpam-841	281	11	t	t	PROPN
ejpam-841	281	12	)	)	PUNCT
ejpam-841	281	13	]	]	PUNCT
ejpam-841	281	14	}	}	PUNCT
ejpam-841	281	15	.	.	PUNCT
ejpam-841	282	1	j.	j.	PROPN
ejpam-841	282	2	hoffman	hoffman	PROPN
ejpam-841	282	3	,	,	PUNCT
ejpam-841	282	4	h.	h.	PROPN
ejpam-841	282	5	wang	wang	PROPN
ejpam-841	282	6	,	,	PUNCT
ejpam-841	282	7	x.	x.	PROPN
ejpam-841	282	8	jia	jia	PROPN
ejpam-841	282	9	,	,	PUNCT
ejpam-841	282	10	r.	r.	PROPN
ejpam-841	282	11	goldman	goldman	PROPN
ejpam-841	282	12	/	/	SYM
ejpam-841	282	13	eur	eur	PROPN
ejpam-841	282	14	.	.	PUNCT
ejpam-841	283	1	j.	j.	PROPN
ejpam-841	283	2	pure	pure	PROPN
ejpam-841	283	3	appl	appl	PROPN
ejpam-841	283	4	.	.	PROPN
ejpam-841	283	5	math	math	PROPN
ejpam-841	283	6	,	,	PUNCT
ejpam-841	283	7	3	3	NUM
ejpam-841	283	8	(	(	PUNCT
ejpam-841	283	9	2010	2010	NUM
ejpam-841	283	10	)	)	PUNCT
ejpam-841	283	11	,	,	PUNCT
ejpam-841	283	12	602	602	NUM
ejpam-841	283	13	-	-	SYM
ejpam-841	283	14	632	632	NUM
ejpam-841	283	15	611	611	NUM
ejpam-841	283	16	let	let	VERB
ejpam-841	283	17	r	r	PROPN
ejpam-841	283	18	′′′(a	′′′(a	PROPN
ejpam-841	283	19	,	,	PUNCT
ejpam-841	283	20	b	b	NOUN
ejpam-841	283	21	)	)	PUNCT
ejpam-841	283	22	=	=	SYM
ejpam-841	283	23	k−1	k−1	PROPN
ejpam-841	283	24	∑	∑	PUNCT
ejpam-841	283	25	j=0	j=0	PROPN
ejpam-841	283	26	[	[	PUNCT
ejpam-841	283	27	r2	r2	PROPN
ejpam-841	283	28	j(b	j(b	PROPN
ejpam-841	283	29	)	)	PUNCT
ejpam-841	283	30	j	j	PROPN
ejpam-841	283	31	+	+	NUM
ejpam-841	283	32	r2	r2	PROPN
ejpam-841	283	33	j+1(b	j+1(b	PROPN
ejpam-841	283	34	)	)	PUNCT
ejpam-841	283	35	j(a	j(a	PROPN
ejpam-841	283	36	)	)	PUNCT
ejpam-841	283	37	]	]	X
ejpam-841	283	38	;	;	PUNCT
ejpam-841	283	39	then	then	ADV
ejpam-841	283	40	r	r	PROPN
ejpam-841	283	41	′′′	′′′	PROPN
ejpam-841	283	42	(	(	PUNCT
ejpam-841	283	43	y	y	PROPN
ejpam-841	283	44	z	z	PROPN
ejpam-841	283	45	,	,	PUNCT
ejpam-841	283	46	x	x	X
ejpam-841	283	47	z	z	NOUN
ejpam-841	283	48	)	)	PUNCT
ejpam-841	284	1	=	=	SYM
ejpam-841	284	2	k−1	k−1	PROPN
ejpam-841	284	3	∑	∑	PUNCT
ejpam-841	284	4	j=0	j=0	PROPN
ejpam-841	284	5	[	[	X
ejpam-841	284	6	r2	r2	PROPN
ejpam-841	284	7	j	j	PROPN
ejpam-841	284	8	(	(	PUNCT
ejpam-841	284	9	x	x	PROPN
ejpam-841	284	10	z	z	NOUN
ejpam-841	284	11	)	)	PUNCT
ejpam-841	284	12	j	j	PROPN
ejpam-841	284	13	+	+	CCONJ
ejpam-841	284	14	r2	r2	PROPN
ejpam-841	284	15	j+1	j+1	PROPN
ejpam-841	284	16	(	(	PUNCT
ejpam-841	284	17	x	x	NOUN
ejpam-841	284	18	z	z	NOUN
ejpam-841	284	19	)	)	PUNCT
ejpam-841	284	20	j	j	PROPN
ejpam-841	284	21	(	(	PUNCT
ejpam-841	284	22	y	y	PROPN
ejpam-841	284	23	z	z	PROPN
ejpam-841	284	24	)	)	PUNCT
ejpam-841	284	25	]	]	PUNCT
ejpam-841	284	26	.	.	PUNCT
ejpam-841	285	1	(	(	PUNCT
ejpam-841	285	2	5	5	NUM
ejpam-841	285	3	)	)	PUNCT
ejpam-841	285	4	moreover	moreover	ADV
ejpam-841	285	5	,	,	PUNCT
ejpam-841	285	6	for	for	ADP
ejpam-841	285	7	all	all	DET
ejpam-841	285	8	i	i	PRON
ejpam-841	285	9	=	=	NOUN
ejpam-841	285	10	1	1	NUM
ejpam-841	285	11	,	,	PUNCT
ejpam-841	285	12	.	.	PUNCT
ejpam-841	285	13	.	.	PUNCT
ejpam-841	285	14	.	.	PUNCT
ejpam-841	286	1	,	,	PUNCT
ejpam-841	286	2	k−	k−	PROPN
ejpam-841	286	3	1	1	NUM
ejpam-841	286	4	,	,	PUNCT
ejpam-841	286	5	r	r	NOUN
ejpam-841	286	6	=	=	SYM
ejpam-841	286	7	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	286	8	+	+	NUM
ejpam-841	286	9	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	286	10	t	t	NOUN
ejpam-841	286	11	+	+	CCONJ
ejpam-841	286	12	·	·	PUNCT
ejpam-841	286	13	·	·	PUNCT
ejpam-841	286	14	·	·	PUNCT
ejpam-841	286	15	+	+	NUM
ejpam-841	286	16	r0	r0	NOUN
ejpam-841	286	17	td−2	td−2	NOUN
ejpam-841	286	18	=	=	SYM
ejpam-841	286	19	d−2	d−2	PROPN
ejpam-841	286	20	∑	∑	PUNCT
ejpam-841	286	21	j=0	j=0	PROPN
ejpam-841	286	22	r	r	NOUN
ejpam-841	286	23	js	js	PROPN
ejpam-841	286	24	j	j	PROPN
ejpam-841	286	25	td−2−	td−2−	NUM
ejpam-841	286	26	j	j	PROPN
ejpam-841	286	27	=	=	SYM
ejpam-841	286	28	s2i	s2i	PROPN
ejpam-841	286	29	[	[	PUNCT
ejpam-841	286	30	d−2	d−2	PROPN
ejpam-841	286	31	∑	∑	PUNCT
ejpam-841	286	32	j=0	j=0	PROPN
ejpam-841	286	33	r	r	PROPN
ejpam-841	286	34	j	j	PROPN
ejpam-841	286	35	s	s	PROPN
ejpam-841	286	36	j	j	PROPN
ejpam-841	286	37	td−2−	td−2−	NUM
ejpam-841	286	38	j	j	PROPN
ejpam-841	286	39	s2i	s2i	VERB
ejpam-841	286	40	]	]	PUNCT
ejpam-841	286	41	=	=	SYM
ejpam-841	286	42	s2i	s2i	PROPN
ejpam-841	286	43	[	[	PUNCT
ejpam-841	286	44	2i−1	2i−1	NUM
ejpam-841	286	45	∑	∑	PUNCT
ejpam-841	286	46	j=0	j=0	PROPN
ejpam-841	286	47	r	r	PROPN
ejpam-841	286	48	j	j	PROPN
ejpam-841	286	49	t	t	PROPN
ejpam-841	286	50	d−2−2i	d−2−2i	PROPN
ejpam-841	286	51	(	(	PUNCT
ejpam-841	286	52	t	t	PROPN
ejpam-841	286	53	s	s	PART
ejpam-841	286	54	)	)	PUNCT
ejpam-841	287	1	2i−	2i−	NUM
ejpam-841	287	2	j	j	NOUN
ejpam-841	287	3	+	+	CCONJ
ejpam-841	287	4	d−2	d−2	PROPN
ejpam-841	287	5	∑	∑	PROPN
ejpam-841	287	6	j=2i	j=2i	PROPN
ejpam-841	287	7	r	r	NOUN
ejpam-841	287	8	js	js	PROPN
ejpam-841	287	9	j−2i	j−2i	PROPN
ejpam-841	287	10	td−2−	td−2−	PROPN
ejpam-841	287	11	j	j	X
ejpam-841	287	12	]	]	X
ejpam-841	287	13	=	=	SYM
ejpam-841	287	14	s2i	s2i	PROPN
ejpam-841	287	15	{	{	PUNCT
ejpam-841	287	16	i−1	i−1	PROPN
ejpam-841	287	17	∑	∑	PUNCT
ejpam-841	287	18	j=0	j=0	PROPN
ejpam-841	288	1	[	[	X
ejpam-841	288	2	r2	r2	PROPN
ejpam-841	288	3	j	j	PROPN
ejpam-841	288	4	t	t	PROPN
ejpam-841	288	5	d−2−2i	d−2−2i	PROPN
ejpam-841	288	6	(	(	PUNCT
ejpam-841	288	7	t2	t2	PROPN
ejpam-841	288	8	s2	s2	PROPN
ejpam-841	288	9	)	)	PUNCT
ejpam-841	288	10	i−	i−	PROPN
ejpam-841	288	11	j	j	PROPN
ejpam-841	288	12	+	+	CCONJ
ejpam-841	288	13	r2	r2	PROPN
ejpam-841	288	14	j+1	j+1	PUNCT
ejpam-841	288	15	td−2−2i	td−2−2i	VERB
ejpam-841	288	16	(	(	PUNCT
ejpam-841	288	17	t2	t2	PROPN
ejpam-841	288	18	s2	s2	PROPN
ejpam-841	288	19	)	)	PUNCT
ejpam-841	288	20	i−	i−	PROPN
ejpam-841	288	21	j−1	j−1	PROPN
ejpam-841	288	22	(	(	PUNCT
ejpam-841	288	23	t	t	PROPN
ejpam-841	288	24	s	s	PART
ejpam-841	288	25	)	)	PUNCT
ejpam-841	288	26	]	]	PUNCT
ejpam-841	289	1	+	+	CCONJ
ejpam-841	289	2	d−2−2i	d−2−2i	PROPN
ejpam-841	289	3	∑	∑	PUNCT
ejpam-841	289	4	j=0	j=0	PROPN
ejpam-841	289	5	r2i+	r2i+	PROPN
ejpam-841	289	6	js	js	ADP
ejpam-841	289	7	j	j	PROPN
ejpam-841	289	8	td−2−2i−	td−2−2i−	PROPN
ejpam-841	289	9	j	j	PROPN
ejpam-841	289	10	}	}	PUNCT
ejpam-841	289	11	.	.	PUNCT
ejpam-841	290	1	for	for	ADP
ejpam-841	290	2	all	all	DET
ejpam-841	290	3	i	i	PRON
ejpam-841	290	4	=	=	NOUN
ejpam-841	290	5	1	1	NUM
ejpam-841	290	6	,	,	PUNCT
ejpam-841	290	7	.	.	PUNCT
ejpam-841	290	8	.	.	PUNCT
ejpam-841	290	9	.	.	PUNCT
ejpam-841	291	1	,	,	PUNCT
ejpam-841	291	2	k−	k−	PROPN
ejpam-841	291	3	1	1	NUM
ejpam-841	291	4	,	,	PUNCT
ejpam-841	291	5	let	let	VERB
ejpam-841	291	6	r	r	NOUN
ejpam-841	291	7	′′i	′′i	NOUN
ejpam-841	291	8	(	(	PUNCT
ejpam-841	291	9	a	a	DET
ejpam-841	291	10	,	,	PUNCT
ejpam-841	291	11	b	b	NOUN
ejpam-841	291	12	)	)	PUNCT
ejpam-841	291	13	=	=	NOUN
ejpam-841	292	1	td−2−2i	td−2−2i	PUNCT
ejpam-841	292	2	i−1	i−1	PROPN
ejpam-841	292	3	∑	∑	PUNCT
ejpam-841	292	4	j=0	j=0	PROPN
ejpam-841	292	5	[	[	PUNCT
ejpam-841	292	6	r2	r2	PROPN
ejpam-841	292	7	j(b	j(b	PROPN
ejpam-841	292	8	)	)	PUNCT
ejpam-841	292	9	i−	i−	PROPN
ejpam-841	292	10	j	j	PROPN
ejpam-841	292	11	+	+	CCONJ
ejpam-841	292	12	r2	r2	PROPN
ejpam-841	292	13	j+1(b	j+1(b	PROPN
ejpam-841	292	14	)	)	PUNCT
ejpam-841	292	15	i−	i−	PROPN
ejpam-841	292	16	j−1(a)]+	j−1(a)]+	PROPN
ejpam-841	292	17	d−2−2i	d−2−2i	PROPN
ejpam-841	292	18	∑	∑	PUNCT
ejpam-841	292	19	j=0	j=0	PROPN
ejpam-841	292	20	r2i+	r2i+	PROPN
ejpam-841	292	21	js	js	ADP
ejpam-841	292	22	j	j	PROPN
ejpam-841	292	23	td−2−2i−	td−2−2i−	PROPN
ejpam-841	292	24	j	j	PROPN
ejpam-841	292	25	;	;	PUNCT
ejpam-841	292	26	then	then	ADV
ejpam-841	292	27	r	r	NOUN
ejpam-841	292	28	′′i	′′i	NOUN
ejpam-841	292	29	(	(	PUNCT
ejpam-841	292	30	y	y	NOUN
ejpam-841	292	31	x	x	INTJ
ejpam-841	292	32	,	,	PUNCT
ejpam-841	292	33	z	z	NOUN
ejpam-841	292	34	x	x	SYM
ejpam-841	292	35	)	)	PUNCT
ejpam-841	293	1	=	=	NOUN
ejpam-841	293	2	td−2−2i	td−2−2i	PUNCT
ejpam-841	293	3	i−1	i−1	PROPN
ejpam-841	293	4	∑	∑	PUNCT
ejpam-841	293	5	j=0	j=0	PROPN
ejpam-841	294	1	[	[	PUNCT
ejpam-841	294	2	r2	r2	PROPN
ejpam-841	294	3	j	j	PROPN
ejpam-841	294	4	(	(	PUNCT
ejpam-841	294	5	z	z	NOUN
ejpam-841	294	6	x	x	SYM
ejpam-841	294	7	)	)	PUNCT
ejpam-841	294	8	i−	i−	PROPN
ejpam-841	294	9	j	j	PROPN
ejpam-841	294	10	+	+	CCONJ
ejpam-841	294	11	r2	r2	PROPN
ejpam-841	294	12	j+1	j+1	PROPN
ejpam-841	294	13	(	(	PUNCT
ejpam-841	294	14	z	z	NOUN
ejpam-841	294	15	x	x	SYM
ejpam-841	294	16	)	)	PUNCT
ejpam-841	294	17	i−	i−	PROPN
ejpam-841	294	18	j−1	j−1	PROPN
ejpam-841	294	19	(	(	PUNCT
ejpam-841	294	20	y	y	NOUN
ejpam-841	294	21	x	x	PROPN
ejpam-841	294	22	)	)	PUNCT
ejpam-841	294	23	]	]	PUNCT
ejpam-841	295	1	+	+	CCONJ
ejpam-841	295	2	d−2−2i	d−2−2i	PROPN
ejpam-841	295	3	∑	∑	PUNCT
ejpam-841	295	4	j=0	j=0	PROPN
ejpam-841	295	5	r2i+	r2i+	PROPN
ejpam-841	295	6	js	js	ADP
ejpam-841	295	7	j	j	PROPN
ejpam-841	295	8	td−2−2i−	td−2−2i−	PROPN
ejpam-841	295	9	j	j	PROPN
ejpam-841	295	10	.	.	PUNCT
ejpam-841	296	1	(	(	PUNCT
ejpam-841	296	2	6	6	NUM
ejpam-841	296	3	)	)	PUNCT
ejpam-841	296	4	theorem	theorem	NOUN
ejpam-841	296	5	2	2	NUM
ejpam-841	296	6	.	.	PUNCT
ejpam-841	296	7	a	a	DET
ejpam-841	296	8	minimal	minimal	ADJ
ejpam-841	296	9	set	set	NOUN
ejpam-841	296	10	of	of	ADP
ejpam-841	296	11	generators	generator	NOUN
ejpam-841	296	12	for	for	ADP
ejpam-841	296	13	the	the	DET
ejpam-841	296	14	rees	rees	PROPN
ejpam-841	296	15	algebra	algebra	NOUN
ejpam-841	296	16	associated	associate	VERB
ejpam-841	296	17	to	to	ADP
ejpam-841	296	18	a	a	DET
ejpam-841	296	19	singular	singular	ADJ
ejpam-841	296	20	rational	rational	ADJ
ejpam-841	296	21	space	space	NOUN
ejpam-841	296	22	curve	curve	NOUN
ejpam-841	296	23	of	of	ADP
ejpam-841	296	24	type	type	NOUN
ejpam-841	296	25	(	(	PUNCT
ejpam-841	296	26	1,1	1,1	NUM
ejpam-841	296	27	,	,	PUNCT
ejpam-841	296	28	d	d	NOUN
ejpam-841	296	29	−	−	PROPN
ejpam-841	296	30	2	2	NUM
ejpam-841	296	31	)	)	PUNCT
ejpam-841	296	32	where	where	SCONJ
ejpam-841	296	33	d	d	NOUN
ejpam-841	296	34	=	=	SYM
ejpam-841	296	35	2k+	2k+	NUM
ejpam-841	296	36	1	1	NUM
ejpam-841	296	37	are	be	AUX
ejpam-841	296	38	given	give	VERB
ejpam-841	296	39	by	by	ADP
ejpam-841	296	40	the	the	DET
ejpam-841	296	41	following	follow	VERB
ejpam-841	296	42	k+	k+	NOUN
ejpam-841	296	43	5	5	NUM
ejpam-841	296	44	polynomials	polynomial	NOUN
ejpam-841	296	45	:	:	PUNCT
ejpam-841	297	1	1	1	NUM
ejpam-841	297	2	.	.	X
ejpam-841	298	1	three	three	NUM
ejpam-841	298	2	µ-basis	µ-basis	NOUN
ejpam-841	298	3	elements	element	NOUN
ejpam-841	298	4	:	:	PUNCT
ejpam-841	298	5	p	p	X
ejpam-841	298	6	,	,	PUNCT
ejpam-841	298	7	q	q	ADJ
ejpam-841	298	8	,	,	PUNCT
ejpam-841	298	9	r	r	NOUN
ejpam-841	298	10	,	,	PUNCT
ejpam-841	298	11	where	where	SCONJ
ejpam-841	298	12	deg(p	deg(p	PROPN
ejpam-841	298	13	)	)	PUNCT
ejpam-841	298	14	=	=	SYM
ejpam-841	298	15	deg(q	deg(q	NOUN
ejpam-841	298	16	)	)	PUNCT
ejpam-841	298	17	=	=	SYM
ejpam-841	298	18	(	(	PUNCT
ejpam-841	298	19	1,1	1,1	NUM
ejpam-841	298	20	)	)	PUNCT
ejpam-841	298	21	and	and	CCONJ
ejpam-841	298	22	deg(r	deg(r	PROPN
ejpam-841	298	23	)	)	PUNCT
ejpam-841	298	24	=	=	PUNCT
ejpam-841	298	25	(	(	PUNCT
ejpam-841	298	26	d	d	NOUN
ejpam-841	298	27	−	−	PROPN
ejpam-841	298	28	2,1	2,1	NUM
ejpam-841	298	29	)	)	PUNCT
ejpam-841	298	30	;	;	PUNCT
ejpam-841	298	31	2	2	X
ejpam-841	298	32	.	.	X
ejpam-841	298	33	three	three	NUM
ejpam-841	298	34	implicit	implicit	ADJ
ejpam-841	298	35	equations	equation	NOUN
ejpam-841	298	36	:	:	PUNCT
ejpam-841	298	37	sylvs	sylv	NOUN
ejpam-841	298	38	,	,	PUNCT
ejpam-841	298	39	t(p	t(p	PROPN
ejpam-841	298	40	,	,	PUNCT
ejpam-841	298	41	q	q	NOUN
ejpam-841	298	42	)	)	PUNCT
ejpam-841	298	43	of	of	ADP
ejpam-841	298	44	degree	degree	NOUN
ejpam-841	298	45	(	(	PUNCT
ejpam-841	298	46	0,2	0,2	NUM
ejpam-841	298	47	)	)	PUNCT
ejpam-841	298	48	,	,	PUNCT
ejpam-841	298	49	x	x	PROPN
ejpam-841	298	50	kr	kr	PROPN
ejpam-841	298	51	′′	′′	PROPN
ejpam-841	298	52	(	(	PUNCT
ejpam-841	298	53	y	y	PROPN
ejpam-841	298	54	x	x	PROPN
ejpam-841	298	55	,	,	PUNCT
ejpam-841	298	56	z	z	NOUN
ejpam-841	298	57	x	x	PUNCT
ejpam-841	298	58	)	)	PUNCT
ejpam-841	298	59	and	and	CCONJ
ejpam-841	298	60	zkr	zkr	PROPN
ejpam-841	298	61	′′′	′′′	PROPN
ejpam-841	298	62	(	(	PUNCT
ejpam-841	298	63	y	y	PROPN
ejpam-841	298	64	z	z	PROPN
ejpam-841	298	65	,	,	PUNCT
ejpam-841	298	66	x	x	X
ejpam-841	298	67	z	z	NOUN
ejpam-841	298	68	)	)	PUNCT
ejpam-841	298	69	of	of	ADP
ejpam-841	298	70	degree	degree	NOUN
ejpam-841	298	71	(	(	PUNCT
ejpam-841	298	72	0	0	NUM
ejpam-841	298	73	,	,	PUNCT
ejpam-841	298	74	k+	k+	NOUN
ejpam-841	298	75	1	1	NUM
ejpam-841	298	76	)	)	PUNCT
ejpam-841	298	77	;	;	PUNCT
ejpam-841	298	78	3	3	X
ejpam-841	298	79	.	.	PUNCT
ejpam-841	298	80	k−	k−	NOUN
ejpam-841	298	81	1	1	NUM
ejpam-841	298	82	moving	move	VERB
ejpam-841	298	83	surfaces	surface	NOUN
ejpam-841	298	84	:	:	PUNCT
ejpam-841	298	85	x	x	PUNCT
ejpam-841	298	86	i	i	NOUN
ejpam-841	298	87	r	r	NOUN
ejpam-841	298	88	′′	′′	PROPN
ejpam-841	299	1	i	i	PRON
ejpam-841	299	2	(	(	PUNCT
ejpam-841	299	3	y	y	NOUN
ejpam-841	299	4	x	x	INTJ
ejpam-841	299	5	,	,	PUNCT
ejpam-841	299	6	z	z	NOUN
ejpam-841	299	7	x	x	SYM
ejpam-841	299	8	)	)	PUNCT
ejpam-841	299	9	of	of	ADP
ejpam-841	299	10	degree	degree	NOUN
ejpam-841	299	11	(	(	PUNCT
ejpam-841	299	12	d	d	NOUN
ejpam-841	299	13	−	−	PROPN
ejpam-841	299	14	2−	2−	NUM
ejpam-841	299	15	2i	2i	NOUN
ejpam-841	299	16	,	,	PUNCT
ejpam-841	299	17	i	i	PRON
ejpam-841	299	18	+	+	NOUN
ejpam-841	299	19	1	1	X
ejpam-841	299	20	)	)	PUNCT
ejpam-841	299	21	for	for	ADP
ejpam-841	299	22	i	i	PROPN
ejpam-841	299	23	=	=	NOUN
ejpam-841	299	24	1	1	NUM
ejpam-841	299	25	,	,	PUNCT
ejpam-841	299	26	.	.	PUNCT
ejpam-841	299	27	.	.	PUNCT
ejpam-841	299	28	.	.	PUNCT
ejpam-841	300	1	,	,	PUNCT
ejpam-841	300	2	k−	k−	PROPN
ejpam-841	300	3	1	1	NUM
ejpam-841	300	4	;	;	PUNCT
ejpam-841	300	5	where	where	SCONJ
ejpam-841	300	6	r	r	NOUN
ejpam-841	300	7	′′	′′	PROPN
ejpam-841	300	8	(	(	PUNCT
ejpam-841	300	9	y	y	PROPN
ejpam-841	300	10	x	x	PROPN
ejpam-841	300	11	,	,	PUNCT
ejpam-841	300	12	z	z	NOUN
ejpam-841	300	13	x	x	PUNCT
ejpam-841	300	14	)	)	PUNCT
ejpam-841	300	15	,	,	PUNCT
ejpam-841	300	16	r	r	NOUN
ejpam-841	300	17	′′′	′′′	PROPN
ejpam-841	300	18	(	(	PUNCT
ejpam-841	300	19	y	y	PROPN
ejpam-841	300	20	z	z	PROPN
ejpam-841	300	21	,	,	PUNCT
ejpam-841	300	22	x	x	X
ejpam-841	300	23	z	z	NOUN
ejpam-841	300	24	)	)	PUNCT
ejpam-841	300	25	and	and	CCONJ
ejpam-841	300	26	r	r	NOUN
ejpam-841	300	27	′′i	′′i	NOUN
ejpam-841	300	28	(	(	PUNCT
ejpam-841	300	29	y	y	NOUN
ejpam-841	300	30	x	x	INTJ
ejpam-841	300	31	,	,	PUNCT
ejpam-841	300	32	z	z	NOUN
ejpam-841	300	33	x	x	PUNCT
ejpam-841	300	34	)	)	PUNCT
ejpam-841	300	35	are	be	AUX
ejpam-841	300	36	defined	define	VERB
ejpam-841	300	37	in	in	ADP
ejpam-841	300	38	equations	equation	NOUN
ejpam-841	300	39	(	(	PUNCT
ejpam-841	300	40	4	4	NUM
ejpam-841	300	41	)	)	PUNCT
ejpam-841	300	42	,	,	PUNCT
ejpam-841	300	43	(	(	PUNCT
ejpam-841	300	44	5	5	NUM
ejpam-841	300	45	)	)	PUNCT
ejpam-841	300	46	and	and	CCONJ
ejpam-841	300	47	(	(	PUNCT
ejpam-841	300	48	6	6	NUM
ejpam-841	300	49	)	)	PUNCT
ejpam-841	300	50	.	.	PUNCT
ejpam-841	301	1	proof	proof	NOUN
ejpam-841	301	2	.	.	PUNCT
ejpam-841	302	1	we	we	PRON
ejpam-841	302	2	will	will	AUX
ejpam-841	302	3	apply	apply	VERB
ejpam-841	302	4	the	the	DET
ejpam-841	302	5	results	result	NOUN
ejpam-841	302	6	of	of	ADP
ejpam-841	302	7	kustin	kustin	NOUN
ejpam-841	302	8	,	,	PUNCT
ejpam-841	302	9	polini	polini	NOUN
ejpam-841	302	10	and	and	CCONJ
ejpam-841	302	11	ulrich	ulrich	PROPN
ejpam-841	302	12	[	[	X
ejpam-841	302	13	18	18	NUM
ejpam-841	302	14	]	]	PUNCT
ejpam-841	302	15	,	,	PUNCT
ejpam-841	302	16	by	by	ADP
ejpam-841	302	17	listing	list	VERB
ejpam-841	302	18	the	the	DET
ejpam-841	302	19	set	set	NOUN
ejpam-841	302	20	of	of	ADP
ejpam-841	302	21	minimal	minimal	ADJ
ejpam-841	302	22	generators	generator	NOUN
ejpam-841	302	23	in	in	ADP
ejpam-841	302	24	their	their	PRON
ejpam-841	302	25	paper	paper	NOUN
ejpam-841	302	26	,	,	PUNCT
ejpam-841	302	27	and	and	CCONJ
ejpam-841	302	28	comparing	compare	VERB
ejpam-841	302	29	these	these	DET
ejpam-841	302	30	generators	generator	NOUN
ejpam-841	302	31	with	with	ADP
ejpam-841	302	32	the	the	DET
ejpam-841	302	33	generators	generator	NOUN
ejpam-841	302	34	listed	list	VERB
ejpam-841	302	35	in	in	ADP
ejpam-841	302	36	the	the	DET
ejpam-841	302	37	statement	statement	NOUN
ejpam-841	302	38	of	of	ADP
ejpam-841	302	39	our	our	PRON
ejpam-841	302	40	theorem	theorem	PROPN
ejpam-841	302	41	.	.	PUNCT
ejpam-841	303	1	j.	j.	PROPN
ejpam-841	303	2	hoffman	hoffman	PROPN
ejpam-841	303	3	,	,	PUNCT
ejpam-841	303	4	h.	h.	PROPN
ejpam-841	303	5	wang	wang	PROPN
ejpam-841	303	6	,	,	PUNCT
ejpam-841	303	7	x.	x.	PROPN
ejpam-841	303	8	jia	jia	PROPN
ejpam-841	303	9	,	,	PUNCT
ejpam-841	303	10	r.	r.	PROPN
ejpam-841	303	11	goldman	goldman	PROPN
ejpam-841	303	12	/	/	SYM
ejpam-841	303	13	eur	eur	PROPN
ejpam-841	303	14	.	.	PUNCT
ejpam-841	304	1	j.	j.	PROPN
ejpam-841	304	2	pure	pure	PROPN
ejpam-841	304	3	appl	appl	PROPN
ejpam-841	304	4	.	.	PROPN
ejpam-841	304	5	math	math	PROPN
ejpam-841	304	6	,	,	PUNCT
ejpam-841	304	7	3	3	NUM
ejpam-841	304	8	(	(	PUNCT
ejpam-841	304	9	2010	2010	NUM
ejpam-841	304	10	)	)	PUNCT
ejpam-841	304	11	,	,	PUNCT
ejpam-841	304	12	602	602	NUM
ejpam-841	304	13	-	-	SYM
ejpam-841	304	14	632	632	NUM
ejpam-841	304	15	612	612	NUM
ejpam-841	304	16	first	first	ADV
ejpam-841	304	17	,	,	PUNCT
ejpam-841	304	18	we	we	PRON
ejpam-841	304	19	note	note	VERB
ejpam-841	304	20	that	that	SCONJ
ejpam-841	304	21	the	the	DET
ejpam-841	304	22	notation	notation	NOUN
ejpam-841	304	23	x	x	PUNCT
ejpam-841	304	24	,	,	PUNCT
ejpam-841	304	25	y	y	PROPN
ejpam-841	304	26	,	,	PUNCT
ejpam-841	304	27	z	z	PROPN
ejpam-841	304	28	,	,	PUNCT
ejpam-841	304	29	w	w	PROPN
ejpam-841	304	30	,	,	PUNCT
ejpam-841	304	31	t	t	PROPN
ejpam-841	304	32	,	,	PUNCT
ejpam-841	304	33	s	s	PROPN
ejpam-841	304	34	,	,	PUNCT
ejpam-841	304	35	ri	ri	PROPN
ejpam-841	304	36	,	,	PUNCT
ejpam-841	304	37	i	i	PRON
ejpam-841	304	38	=	=	NOUN
ejpam-841	304	39	0	0	NUM
ejpam-841	304	40	,	,	PUNCT
ejpam-841	304	41	.	.	PUNCT
ejpam-841	304	42	.	.	PUNCT
ejpam-841	304	43	.	.	PUNCT
ejpam-841	305	1	,	,	PUNCT
ejpam-841	306	1	d	d	X
ejpam-841	306	2	−	−	PROPN
ejpam-841	306	3	2	2	NUM
ejpam-841	306	4	in	in	ADP
ejpam-841	306	5	this	this	DET
ejpam-841	306	6	paper	paper	NOUN
ejpam-841	306	7	is	be	AUX
ejpam-841	306	8	the	the	DET
ejpam-841	306	9	same	same	ADJ
ejpam-841	306	10	as	as	ADP
ejpam-841	306	11	t1	t1	NOUN
ejpam-841	306	12	,	,	PUNCT
ejpam-841	306	13	t2	t2	NOUN
ejpam-841	306	14	,	,	PUNCT
ejpam-841	306	15	t3	t3	PROPN
ejpam-841	306	16	,	,	PUNCT
ejpam-841	306	17	t4	t4	PROPN
ejpam-841	306	18	,	,	PUNCT
ejpam-841	306	19	x	x	INTJ
ejpam-841	306	20	,	,	PUNCT
ejpam-841	306	21	y	y	PROPN
ejpam-841	306	22	,	,	PUNCT
ejpam-841	306	23	ci	ci	PROPN
ejpam-841	306	24	,	,	PUNCT
ejpam-841	306	25	i	i	PRON
ejpam-841	306	26	=	=	NOUN
ejpam-841	306	27	0	0	NUM
ejpam-841	306	28	,	,	PUNCT
ejpam-841	306	29	.	.	PUNCT
ejpam-841	306	30	.	.	PUNCT
ejpam-841	307	1	.	.	PUNCT
ejpam-841	308	1	,	,	PUNCT
ejpam-841	308	2	d−2	d−2	PROPN
ejpam-841	308	3	in	in	ADP
ejpam-841	308	4	their	their	PRON
ejpam-841	308	5	notation	notation	NOUN
ejpam-841	308	6	.	.	PUNCT
ejpam-841	309	1	moreover	moreover	ADV
ejpam-841	309	2	,	,	PUNCT
ejpam-841	309	3	in	in	ADP
ejpam-841	309	4	our	our	PRON
ejpam-841	309	5	setting	setting	NOUN
ejpam-841	309	6	,	,	PUNCT
ejpam-841	309	7	we	we	PRON
ejpam-841	309	8	identify	identify	VERB
ejpam-841	309	9	the	the	DET
ejpam-841	309	10	following	follow	VERB
ejpam-841	309	11	items	item	NOUN
ejpam-841	309	12	in	in	ADP
ejpam-841	309	13	their	their	PRON
ejpam-841	309	14	paper	paper	NOUN
ejpam-841	309	15	for	for	ADP
ejpam-841	309	16	singular	singular	ADJ
ejpam-841	309	17	curves	curve	NOUN
ejpam-841	309	18	of	of	ADP
ejpam-841	309	19	odd	odd	ADJ
ejpam-841	309	20	degrees	degree	NOUN
ejpam-841	309	21	:	:	PUNCT
ejpam-841	309	22	ρ	ρ	PROPN
ejpam-841	309	23	=	=	SYM
ejpam-841	309	24	1	1	NUM
ejpam-841	309	25	,	,	PUNCT
ejpam-841	309	26	ℓ	ℓ	NOUN
ejpam-841	309	27	=	=	SYM
ejpam-841	309	28	2	2	NUM
ejpam-841	309	29	,	,	PUNCT
ejpam-841	309	30	σ1	σ1	NOUN
ejpam-841	309	31	=	=	SYM
ejpam-841	309	32	2	2	NUM
ejpam-841	309	33	,	,	PUNCT
ejpam-841	309	34	σ2	σ2	NOUN
ejpam-841	309	35	=	=	SYM
ejpam-841	309	36	1	1	NUM
ejpam-841	309	37	,	,	PUNCT
ejpam-841	309	38	s	s	PART
ejpam-841	309	39	=	=	SYM
ejpam-841	309	40	t2,1	t2,1	PROPN
ejpam-841	309	41	,	,	PUNCT
ejpam-841	309	42	t	t	PROPN
ejpam-841	309	43	=	=	SYM
ejpam-841	309	44	t2,2	t2,2	PROPN
ejpam-841	309	45	,	,	PUNCT
ejpam-841	309	46	t1,1	t1,1	NOUN
ejpam-841	309	47	=	=	PUNCT
ejpam-841	309	48	x	x	SYM
ejpam-841	309	49	,	,	PUNCT
ejpam-841	309	50	t1,2	t1,2	PROPN
ejpam-841	309	51	=	=	SYM
ejpam-841	309	52	y	y	PROPN
ejpam-841	309	53	,	,	PUNCT
ejpam-841	309	54	t1,3	t1,3	PROPN
ejpam-841	309	55	=	=	PUNCT
ejpam-841	309	56	z.	z.	PROPN
ejpam-841	309	57	by	by	ADP
ejpam-841	309	58	theorem	theorem	NOUN
ejpam-841	309	59	3.2	3.2	NUM
ejpam-841	309	60	in	in	ADP
ejpam-841	309	61	[	[	X
ejpam-841	309	62	18	18	NUM
ejpam-841	309	63	]	]	PUNCT
ejpam-841	309	64	,	,	PUNCT
ejpam-841	309	65	a=	a=	PROPN
ejpam-841	309	66	(	(	PUNCT
ejpam-841	309	67	a1	a1	NOUN
ejpam-841	309	68	)	)	PUNCT
ejpam-841	309	69	=	=	SYM
ejpam-841	309	70	0,1	0,1	NUM
ejpam-841	309	71	,	,	PUNCT
ejpam-841	309	72	.	.	PUNCT
ejpam-841	309	73	.	.	PUNCT
ejpam-841	310	1	.	.	PUNCT
ejpam-841	311	1	,	,	PUNCT
ejpam-841	311	2	k−	k−	PROPN
ejpam-841	311	3	1	1	NUM
ejpam-841	311	4	;	;	PUNCT
ejpam-841	311	5	f	f	X
ejpam-841	311	6	(	(	PUNCT
ejpam-841	311	7	a	a	X
ejpam-841	311	8	)	)	PUNCT
ejpam-841	312	1	=	=	SYM
ejpam-841	312	2	f	f	PROPN
ejpam-841	312	3	(	(	PUNCT
ejpam-841	312	4	a1	a1	PROPN
ejpam-841	312	5	)	)	PUNCT
ejpam-841	312	6	=	=	SYM
ejpam-841	313	1	d	d	ADP
ejpam-841	313	2	−	−	PROPN
ejpam-841	313	3	3−	3−	NUM
ejpam-841	313	4	2a1	2a1	NUM
ejpam-841	313	5	;	;	PUNCT
ejpam-841	313	6	r(a	r(a	NUM
ejpam-841	313	7	)	)	PUNCT
ejpam-841	313	8	=	=	SYM
ejpam-841	313	9	r(a1	r(a1	NOUN
ejpam-841	313	10	)	)	PUNCT
ejpam-841	313	11	=	=	SYM
ejpam-841	313	12	1	1	NUM
ejpam-841	313	13	;	;	PUNCT
ejpam-841	313	14	f	f	PROPN
ejpam-841	313	15	(	(	PUNCT
ejpam-841	313	16	;)	;)	PUNCT
ejpam-841	313	17	=	=	PUNCT
ejpam-841	313	18	k−	k−	PROPN
ejpam-841	313	19	1	1	NUM
ejpam-841	313	20	;	;	PUNCT
ejpam-841	313	21	r	r	NOUN
ejpam-841	313	22	(;	(;	X
ejpam-841	313	23	)	)	PUNCT
ejpam-841	313	24	=	=	SYM
ejpam-841	313	25	2	2	NUM
ejpam-841	313	26	;	;	PUNCT
ejpam-841	313	27	t	t	NOUN
ejpam-841	313	28	;	;	PUNCT
ejpam-841	313	29	=	=	SYM
ejpam-841	313	30	1	1	X
ejpam-841	313	31	.	.	PUNCT
ejpam-841	313	32	by	by	ADP
ejpam-841	313	33	definition	definition	NOUN
ejpam-841	313	34	3.5	3.5	NUM
ejpam-841	313	35	and	and	CCONJ
ejpam-841	313	36	the	the	DET
ejpam-841	313	37	description	description	NOUN
ejpam-841	313	38	in	in	ADP
ejpam-841	313	39	[	[	X
ejpam-841	313	40	18	18	NUM
ejpam-841	313	41	]	]	PUNCT
ejpam-841	313	42	,	,	PUNCT
ejpam-841	313	43	the	the	DET
ejpam-841	313	44	generators	generator	NOUN
ejpam-841	313	45	for	for	ADP
ejpam-841	313	46	the	the	DET
ejpam-841	313	47	rees	rees	PROPN
ejpam-841	313	48	algebra	algebra	NOUN
ejpam-841	313	49	are	be	AUX
ejpam-841	313	50	p	p	NOUN
ejpam-841	313	51	,	,	PUNCT
ejpam-841	313	52	q	q	NOUN
ejpam-841	313	53	,	,	PUNCT
ejpam-841	313	54	sylvs	sylvs	NOUN
ejpam-841	313	55	,	,	PUNCT
ejpam-841	313	56	t(p	t(p	PROPN
ejpam-841	313	57	,	,	PUNCT
ejpam-841	313	58	q	q	NOUN
ejpam-841	313	59	)	)	PUNCT
ejpam-841	313	60	,	,	PUNCT
ejpam-841	313	61	f1	f1	NOUN
ejpam-841	313	62	and	and	CCONJ
ejpam-841	313	63	ga1,1	ga1,1	NOUN
ejpam-841	313	64	.	.	PUNCT
ejpam-841	314	1	we	we	PRON
ejpam-841	314	2	shall	shall	AUX
ejpam-841	314	3	now	now	ADV
ejpam-841	314	4	write	write	VERB
ejpam-841	314	5	f1	f1	NOUN
ejpam-841	314	6	and	and	CCONJ
ejpam-841	314	7	ga1,1	ga1,1	NOUN
ejpam-841	314	8	explicitly	explicitly	ADV
ejpam-841	314	9	and	and	CCONJ
ejpam-841	314	10	compare	compare	VERB
ejpam-841	314	11	these	these	DET
ejpam-841	314	12	expressions	expression	NOUN
ejpam-841	314	13	with	with	ADP
ejpam-841	314	14	the	the	DET
ejpam-841	314	15	generators	generator	NOUN
ejpam-841	314	16	listed	list	VERB
ejpam-841	314	17	in	in	ADP
ejpam-841	314	18	the	the	DET
ejpam-841	314	19	statement	statement	NOUN
ejpam-841	314	20	of	of	ADP
ejpam-841	314	21	our	our	PRON
ejpam-841	314	22	theorem	theorem	NOUN
ejpam-841	314	23	.	.	PROPN
ejpam-841	314	24	f1	f1	PROPN
ejpam-841	314	25	=	=	SYM
ejpam-841	314	26	y	y	PROPN
ejpam-841	314	27	∑	∑	PROPN
ejpam-841	314	28	i+	i+	PROPN
ejpam-841	314	29	j	j	PROPN
ejpam-841	315	1	=	=	PROPN
ejpam-841	315	2	k−2	k−2	PROPN
ejpam-841	315	3	x	x	INTJ
ejpam-841	315	4	iz	iz	INTJ
ejpam-841	315	5	j(r2iz	j(r2iz	NOUN
ejpam-841	315	6	+	+	CCONJ
ejpam-841	315	7	r2i+1	r2i+1	PROPN
ejpam-841	315	8	y	y	NOUN
ejpam-841	315	9	)	)	PUNCT
ejpam-841	315	10	+	+	CCONJ
ejpam-841	315	11	x	x	SYM
ejpam-841	315	12	k−1(rd−2	k−1(rd−2	NOUN
ejpam-841	315	13	x	x	PUNCT
ejpam-841	315	14	+	+	CCONJ
ejpam-841	315	15	rd−3	rd−3	PROPN
ejpam-841	315	16	y	y	NOUN
ejpam-841	315	17	)	)	PUNCT
ejpam-841	315	18	=	=	SYM
ejpam-841	316	1	k−2	k−2	PROPN
ejpam-841	316	2	∑	∑	PROPN
ejpam-841	316	3	i=0	i=0	PROPN
ejpam-841	317	1	[	[	X
ejpam-841	317	2	r2i	r2i	ADP
ejpam-841	317	3	x	x	PUNCT
ejpam-841	317	4	izk−1	izk−1	PRON
ejpam-841	317	5	y	y	NOUN
ejpam-841	317	6	+	+	CCONJ
ejpam-841	317	7	r2i+1	r2i+1	PROPN
ejpam-841	317	8	x	x	SYM
ejpam-841	317	9	izk−2−i	izk−2−i	PROPN
ejpam-841	317	10	y2	y2	NOUN
ejpam-841	317	11	]	]	PUNCT
ejpam-841	318	1	+	+	CCONJ
ejpam-841	318	2	rd−2	rd−2	PROPN
ejpam-841	318	3	x	x	PUNCT
ejpam-841	319	1	k	k	PROPN
ejpam-841	319	2	+	+	CCONJ
ejpam-841	319	3	rd−3	rd−3	NOUN
ejpam-841	319	4	x	x	VERB
ejpam-841	319	5	k	k	NOUN
ejpam-841	319	6	y	y	PROPN
ejpam-841	319	7	=	=	PUNCT
ejpam-841	319	8	x	x	SYM
ejpam-841	319	9	k	k	X
ejpam-841	319	10	{	{	PUNCT
ejpam-841	319	11	k−2	k−2	PROPN
ejpam-841	319	12	∑	∑	PROPN
ejpam-841	319	13	i=0	i=0	PROPN
ejpam-841	319	14	[	[	X
ejpam-841	319	15	r2i	r2i	ADP
ejpam-841	319	16	zk−1	zk−1	PROPN
ejpam-841	319	17	y	y	NOUN
ejpam-841	319	18	x	x	PUNCT
ejpam-841	319	19	k−i	k−i	NOUN
ejpam-841	319	20	+	+	CCONJ
ejpam-841	319	21	r2i+1	r2i+1	VERB
ejpam-841	319	22	zk−2−i	zk−2−i	NOUN
ejpam-841	319	23	y2	y2	NOUN
ejpam-841	319	24	x	x	SYM
ejpam-841	319	25	k−i	k−i	NOUN
ejpam-841	319	26	]	]	PUNCT
ejpam-841	320	1	+	+	CCONJ
ejpam-841	320	2	rd−2	rd−2	NOUN
ejpam-841	320	3	+	+	CCONJ
ejpam-841	320	4	rd−3	rd−3	PROPN
ejpam-841	320	5	y	y	NOUN
ejpam-841	320	6	x	x	PUNCT
ejpam-841	320	7	}	}	PUNCT
ejpam-841	320	8	=	=	PUNCT
ejpam-841	320	9	x	x	SYM
ejpam-841	320	10	k	k	PROPN
ejpam-841	320	11	k−1	k−1	PROPN
ejpam-841	320	12	∑	∑	PROPN
ejpam-841	320	13	i=0	i=0	PROPN
ejpam-841	320	14	[	[	X
ejpam-841	320	15	r2i	r2i	ADP
ejpam-841	320	16	(	(	PUNCT
ejpam-841	320	17	z	z	NOUN
ejpam-841	320	18	x	x	NOUN
ejpam-841	320	19	)	)	PUNCT
ejpam-841	320	20	k−1−i	k−1−i	PROPN
ejpam-841	320	21	(	(	PUNCT
ejpam-841	320	22	y	y	NOUN
ejpam-841	320	23	x	x	PROPN
ejpam-841	320	24	)	)	PUNCT
ejpam-841	321	1	+	+	CCONJ
ejpam-841	321	2	r2i+1	r2i+1	PROPN
ejpam-841	321	3	(	(	PUNCT
ejpam-841	321	4	z	z	NOUN
ejpam-841	321	5	x	x	X
ejpam-841	321	6	)	)	PUNCT
ejpam-841	321	7	k−1−i	k−1−i	PROPN
ejpam-841	321	8	]	]	PUNCT
ejpam-841	321	9	,	,	PUNCT
ejpam-841	321	10	since	since	SCONJ
ejpam-841	321	11	y2	y2	PROPN
ejpam-841	321	12	=	=	PROPN
ejpam-841	321	13	xz	xz	PROPN
ejpam-841	321	14	on	on	ADP
ejpam-841	321	15	the	the	DET
ejpam-841	321	16	curve	curve	NOUN
ejpam-841	321	17	=	=	PUNCT
ejpam-841	321	18	x	x	SYM
ejpam-841	321	19	kr	kr	PROPN
ejpam-841	321	20	′′	′′	PROPN
ejpam-841	321	21	(	(	PUNCT
ejpam-841	321	22	y	y	PROPN
ejpam-841	321	23	x	x	PROPN
ejpam-841	321	24	,	,	PUNCT
ejpam-841	321	25	z	z	NOUN
ejpam-841	321	26	x	x	NOUN
ejpam-841	321	27	)	)	PUNCT
ejpam-841	321	28	.	.	PUNCT
ejpam-841	322	1	hence	hence	ADV
ejpam-841	322	2	f1	f1	PROPN
ejpam-841	322	3	is	be	AUX
ejpam-841	322	4	one	one	NUM
ejpam-841	322	5	of	of	ADP
ejpam-841	322	6	our	our	PRON
ejpam-841	322	7	implicit	implicit	ADJ
ejpam-841	322	8	equations	equation	NOUN
ejpam-841	322	9	and	and	CCONJ
ejpam-841	322	10	deg	deg	PROPN
ejpam-841	322	11	(	(	PUNCT
ejpam-841	322	12	f1	f1	NOUN
ejpam-841	322	13	)	)	PUNCT
ejpam-841	322	14	=	=	SYM
ejpam-841	322	15	(	(	PUNCT
ejpam-841	322	16	0	0	NUM
ejpam-841	322	17	,	,	PUNCT
ejpam-841	322	18	k+	k+	NOUN
ejpam-841	322	19	1	1	NUM
ejpam-841	322	20	)	)	PUNCT
ejpam-841	322	21	.	.	PUNCT
ejpam-841	323	1	f2	f2	PROPN
ejpam-841	323	2	=	=	SYM
ejpam-841	324	1	z	z	PROPN
ejpam-841	324	2	∑	∑	PUNCT
ejpam-841	324	3	i+	i+	PROPN
ejpam-841	324	4	j	j	PROPN
ejpam-841	324	5	=	=	PROPN
ejpam-841	324	6	k−2	k−2	PROPN
ejpam-841	324	7	x	x	INTJ
ejpam-841	324	8	iz	iz	INTJ
ejpam-841	324	9	j(r2iz	j(r2iz	NOUN
ejpam-841	324	10	+	+	CCONJ
ejpam-841	324	11	r2i+1	r2i+1	PROPN
ejpam-841	324	12	y	y	NOUN
ejpam-841	324	13	)	)	PUNCT
ejpam-841	325	1	+	+	CCONJ
ejpam-841	325	2	x	x	SYM
ejpam-841	325	3	k−1(rd−2	k−1(rd−2	PROPN
ejpam-841	325	4	y	y	NOUN
ejpam-841	325	5	+	+	CCONJ
ejpam-841	325	6	rd−3z	rd−3z	X
ejpam-841	325	7	)	)	PUNCT
ejpam-841	326	1	=	=	SYM
ejpam-841	326	2	k−2	k−2	PROPN
ejpam-841	326	3	∑	∑	PROPN
ejpam-841	326	4	i=0	i=0	PROPN
ejpam-841	327	1	[	[	X
ejpam-841	327	2	r2i	r2i	ADP
ejpam-841	327	3	x	x	X
ejpam-841	327	4	izk−i	izk−i	NOUN
ejpam-841	327	5	+	+	CCONJ
ejpam-841	327	6	r2i+1	r2i+1	X
ejpam-841	327	7	x	x	X
ejpam-841	327	8	izk−1−i	izk−1−i	ADJ
ejpam-841	327	9	y	y	X
ejpam-841	327	10	]	]	X
ejpam-841	328	1	+	+	CCONJ
ejpam-841	328	2	rd−2	rd−2	PROPN
ejpam-841	328	3	x	x	PUNCT
ejpam-841	328	4	k−1	k−1	PROPN
ejpam-841	328	5	y	y	PROPN
ejpam-841	329	1	+	+	CCONJ
ejpam-841	329	2	rd−3	rd−3	NOUN
ejpam-841	329	3	x	x	SYM
ejpam-841	329	4	k−1z	k−1z	PROPN
ejpam-841	329	5	=	=	PUNCT
ejpam-841	329	6	zk	zk	PROPN
ejpam-841	329	7	{	{	PUNCT
ejpam-841	329	8	k−2	k−2	PROPN
ejpam-841	329	9	∑	∑	PROPN
ejpam-841	329	10	i=0	i=0	PROPN
ejpam-841	329	11	[	[	X
ejpam-841	329	12	r2i	r2i	ADP
ejpam-841	329	13	(	(	PUNCT
ejpam-841	329	14	x	x	NOUN
ejpam-841	329	15	z	z	NOUN
ejpam-841	329	16	)	)	PUNCT
ejpam-841	330	1	i	i	PRON
ejpam-841	330	2	+	+	NUM
ejpam-841	330	3	r2i+1	r2i+1	PROPN
ejpam-841	330	4	(	(	PUNCT
ejpam-841	330	5	x	x	NOUN
ejpam-841	330	6	z	z	NOUN
ejpam-841	330	7	)	)	PUNCT
ejpam-841	331	1	i	i	PRON
ejpam-841	331	2	(	(	PUNCT
ejpam-841	331	3	y	y	PROPN
ejpam-841	331	4	z	z	PROPN
ejpam-841	331	5	)	)	PUNCT
ejpam-841	331	6	]	]	PUNCT
ejpam-841	332	1	+	+	CCONJ
ejpam-841	332	2	rd−2	rd−2	NOUN
ejpam-841	332	3	(	(	PUNCT
ejpam-841	332	4	x	x	NOUN
ejpam-841	332	5	z	z	X
ejpam-841	332	6	)	)	PUNCT
ejpam-841	332	7	k−1	k−1	PROPN
ejpam-841	332	8	(	(	PUNCT
ejpam-841	332	9	y	y	PROPN
ejpam-841	332	10	z	z	PROPN
ejpam-841	332	11	)	)	PUNCT
ejpam-841	333	1	+	+	CCONJ
ejpam-841	333	2	rd−3	rd−3	NOUN
ejpam-841	333	3	(	(	PUNCT
ejpam-841	333	4	x	x	NOUN
ejpam-841	333	5	z	z	X
ejpam-841	333	6	)	)	PUNCT
ejpam-841	333	7	k−1	k−1	PROPN
ejpam-841	333	8	}	}	PUNCT
ejpam-841	333	9	=	=	SYM
ejpam-841	333	10	zk	zk	PROPN
ejpam-841	333	11	k−1	k−1	PROPN
ejpam-841	333	12	∑	∑	PROPN
ejpam-841	333	13	i=0	i=0	PROPN
ejpam-841	333	14	[	[	X
ejpam-841	333	15	r2i	r2i	ADP
ejpam-841	333	16	(	(	PUNCT
ejpam-841	333	17	x	x	NOUN
ejpam-841	333	18	z	z	NOUN
ejpam-841	333	19	)	)	PUNCT
ejpam-841	334	1	i	i	PRON
ejpam-841	334	2	+	+	NUM
ejpam-841	334	3	r2i+1	r2i+1	PROPN
ejpam-841	334	4	(	(	PUNCT
ejpam-841	334	5	x	x	NOUN
ejpam-841	334	6	z	z	NOUN
ejpam-841	334	7	)	)	PUNCT
ejpam-841	335	1	i	i	PRON
ejpam-841	335	2	(	(	PUNCT
ejpam-841	335	3	y	y	PROPN
ejpam-841	335	4	z	z	PROPN
ejpam-841	335	5	)	)	PUNCT
ejpam-841	335	6	]	]	PUNCT
ejpam-841	336	1	=	=	PUNCT
ejpam-841	336	2	zkr	zkr	X
ejpam-841	336	3	′′′	′′′	PROPN
ejpam-841	336	4	(	(	PUNCT
ejpam-841	336	5	y	y	PROPN
ejpam-841	336	6	z	z	PROPN
ejpam-841	336	7	,	,	PUNCT
ejpam-841	336	8	x	x	X
ejpam-841	336	9	z	z	NOUN
ejpam-841	336	10	)	)	PUNCT
ejpam-841	336	11	.	.	PUNCT
ejpam-841	337	1	hence	hence	ADV
ejpam-841	337	2	f2	f2	PROPN
ejpam-841	337	3	is	be	AUX
ejpam-841	337	4	another	another	DET
ejpam-841	337	5	one	one	NUM
ejpam-841	337	6	of	of	ADP
ejpam-841	337	7	our	our	PRON
ejpam-841	337	8	implicit	implicit	ADJ
ejpam-841	337	9	equations	equation	NOUN
ejpam-841	337	10	and	and	CCONJ
ejpam-841	337	11	deg	deg	PROPN
ejpam-841	337	12	(	(	PUNCT
ejpam-841	337	13	f2	f2	PROPN
ejpam-841	337	14	)	)	PUNCT
ejpam-841	337	15	=	=	SYM
ejpam-841	337	16	(	(	PUNCT
ejpam-841	337	17	0	0	NUM
ejpam-841	337	18	,	,	PUNCT
ejpam-841	337	19	k+	k+	NOUN
ejpam-841	337	20	1	1	NUM
ejpam-841	337	21	)	)	PUNCT
ejpam-841	337	22	.	.	PUNCT
ejpam-841	338	1	ga1,1	ga1,1	NOUN
ejpam-841	338	2	=	=	PROPN
ejpam-841	338	3	td−2−2a1	td−2−2a1	PROPN
ejpam-841	338	4	∑	∑	PUNCT
ejpam-841	338	5	i+	i+	PROPN
ejpam-841	338	6	j	j	X
ejpam-841	338	7	=	=	NOUN
ejpam-841	338	8	a1−1	a1−1	NOUN
ejpam-841	338	9	x	x	INTJ
ejpam-841	339	1	iz	iz	INTJ
ejpam-841	339	2	j(r2iz	j(r2iz	NOUN
ejpam-841	340	1	+	+	CCONJ
ejpam-841	340	2	r2i+1	r2i+1	PROPN
ejpam-841	340	3	y	y	NOUN
ejpam-841	340	4	)	)	PUNCT
ejpam-841	341	1	+	+	CCONJ
ejpam-841	341	2	x	x	SYM
ejpam-841	341	3	a1	a1	NOUN
ejpam-841	341	4	t	t	PROPN
ejpam-841	341	5	×	×	PROPN
ejpam-841	341	6	j.	j.	PROPN
ejpam-841	341	7	hoffman	hoffman	PROPN
ejpam-841	341	8	,	,	PUNCT
ejpam-841	341	9	h.	h.	PROPN
ejpam-841	341	10	wang	wang	PROPN
ejpam-841	341	11	,	,	PUNCT
ejpam-841	341	12	x.	x.	PROPN
ejpam-841	341	13	jia	jia	PROPN
ejpam-841	341	14	,	,	PUNCT
ejpam-841	341	15	r.	r.	PROPN
ejpam-841	341	16	goldman	goldman	PROPN
ejpam-841	341	17	/	/	SYM
ejpam-841	341	18	eur	eur	PROPN
ejpam-841	341	19	.	.	PUNCT
ejpam-841	342	1	j.	j.	PROPN
ejpam-841	342	2	pure	pure	PROPN
ejpam-841	342	3	appl	appl	PROPN
ejpam-841	342	4	.	.	PROPN
ejpam-841	342	5	math	math	PROPN
ejpam-841	342	6	,	,	PUNCT
ejpam-841	342	7	3	3	NUM
ejpam-841	342	8	(	(	PUNCT
ejpam-841	342	9	2010	2010	NUM
ejpam-841	342	10	)	)	PUNCT
ejpam-841	342	11	,	,	PUNCT
ejpam-841	342	12	602	602	NUM
ejpam-841	342	13	-	-	SYM
ejpam-841	342	14	632	632	NUM
ejpam-841	342	15	613	613	NUM
ejpam-841	342	16	∑	∑	PROPN
ejpam-841	342	17	i+	i+	PROPN
ejpam-841	342	18	j	j	PROPN
ejpam-841	342	19	=	=	PROPN
ejpam-841	342	20	d−4−2a1	d−4−2a1	NOUN
ejpam-841	342	21	si	si	PROPN
ejpam-841	342	22	t	t	PROPN
ejpam-841	342	23	j	j	PROPN
ejpam-841	342	24	r2a1+i	r2a1+i	PROPN
ejpam-841	342	25	t	t	PROPN
ejpam-841	342	26	+	+	CCONJ
ejpam-841	342	27	x	x	PRON
ejpam-841	342	28	a1sd−3−2a1(rd−2s+	a1sd−3−2a1(rd−2s+	VERB
ejpam-841	342	29	rd−3	rd−3	PROPN
ejpam-841	342	30	t	t	PROPN
ejpam-841	342	31	)	)	PUNCT
ejpam-841	342	32	=	=	PUNCT
ejpam-841	343	1	x	x	X
ejpam-841	343	2	a1{td−2−2a1	a1{td−2−2a1	NOUN
ejpam-841	343	3	a1−1	a1−1	PRON
ejpam-841	343	4	∑	∑	PUNCT
ejpam-841	343	5	i=0	i=0	PROPN
ejpam-841	343	6	[	[	PUNCT
ejpam-841	343	7	r2i	r2i	ADP
ejpam-841	343	8	(	(	PUNCT
ejpam-841	343	9	z	z	NOUN
ejpam-841	343	10	x	x	NOUN
ejpam-841	343	11	)	)	PUNCT
ejpam-841	343	12	a1−i	a1−i	PROPN
ejpam-841	343	13	+	+	SYM
ejpam-841	343	14	r2i+1	r2i+1	PROPN
ejpam-841	343	15	(	(	PUNCT
ejpam-841	343	16	z	z	NOUN
ejpam-841	343	17	x	x	SYM
ejpam-841	343	18	)	)	PUNCT
ejpam-841	343	19	a1−1−i	a1−1−i	PROPN
ejpam-841	343	20	(	(	PUNCT
ejpam-841	343	21	y	y	NOUN
ejpam-841	343	22	x	x	PROPN
ejpam-841	343	23	)	)	PUNCT
ejpam-841	343	24	]	]	PUNCT
ejpam-841	344	1	+	+	CCONJ
ejpam-841	344	2	d−2−2a1	d−2−2a1	PUNCT
ejpam-841	344	3	∑	∑	ADP
ejpam-841	344	4	i=0	i=0	PROPN
ejpam-841	344	5	r2a1+is	r2a1+is	PROPN
ejpam-841	344	6	i	i	PRON
ejpam-841	344	7	td−2−2a1−i	td−2−2a1−i	PROPN
ejpam-841	344	8	}	}	PUNCT
ejpam-841	344	9	ga1,1	ga1,1	NOUN
ejpam-841	344	10	=	=	SYM
ejpam-841	344	11	(	(	PUNCT
ejpam-841	344	12	x	x	SYM
ejpam-841	344	13	a1	a1	NOUN
ejpam-841	344	14	r	r	NOUN
ejpam-841	344	15	′′a1	′′a1	NOUN
ejpam-841	344	16	(	(	PUNCT
ejpam-841	344	17	y	y	NOUN
ejpam-841	344	18	x	x	PROPN
ejpam-841	344	19	,	,	PUNCT
ejpam-841	344	20	z	z	NOUN
ejpam-841	344	21	x	x	PUNCT
ejpam-841	344	22	)	)	PUNCT
ejpam-841	344	23	for	for	ADP
ejpam-841	344	24	a1	a1	NOUN
ejpam-841	344	25	=	=	SYM
ejpam-841	344	26	1	1	NUM
ejpam-841	344	27	,	,	PUNCT
ejpam-841	344	28	·	·	PUNCT
ejpam-841	344	29	·	·	PUNCT
ejpam-841	344	30	·	·	PUNCT
ejpam-841	344	31	,	,	PUNCT
ejpam-841	344	32	k−	k−	PROPN
ejpam-841	344	33	1	1	NUM
ejpam-841	344	34	,	,	PUNCT
ejpam-841	344	35	r	r	NOUN
ejpam-841	344	36	for	for	ADP
ejpam-841	344	37	a1	a1	NOUN
ejpam-841	344	38	=	=	SYM
ejpam-841	344	39	0	0	X
ejpam-841	344	40	.	.	PUNCT
ejpam-841	345	1	hence	hence	ADV
ejpam-841	345	2	ga1,1	ga1,1	NOUN
ejpam-841	345	3	are	be	AUX
ejpam-841	345	4	our	our	PRON
ejpam-841	345	5	moving	move	VERB
ejpam-841	345	6	surfaces	surface	NOUN
ejpam-841	345	7	and	and	CCONJ
ejpam-841	345	8	deg(ga1,1	deg(ga1,1	NOUN
ejpam-841	345	9	)	)	PUNCT
ejpam-841	345	10	=	=	PUNCT
ejpam-841	346	1	(	(	PUNCT
ejpam-841	346	2	d	d	NOUN
ejpam-841	346	3	−	−	PROPN
ejpam-841	346	4	2−	2−	NUM
ejpam-841	346	5	2a1	2a1	NUM
ejpam-841	346	6	,	,	PUNCT
ejpam-841	346	7	a1	a1	NOUN
ejpam-841	346	8	+	+	CCONJ
ejpam-841	346	9	1	1	NUM
ejpam-841	346	10	)	)	PUNCT
ejpam-841	346	11	.	.	PUNCT
ejpam-841	347	1	therefore	therefore	ADV
ejpam-841	347	2	,	,	PUNCT
ejpam-841	347	3	the	the	DET
ejpam-841	347	4	generators	generator	NOUN
ejpam-841	347	5	provided	provide	VERB
ejpam-841	347	6	by	by	ADP
ejpam-841	347	7	our	our	PRON
ejpam-841	347	8	theorem	theorem	NOUN
ejpam-841	347	9	are	be	AUX
ejpam-841	347	10	the	the	DET
ejpam-841	347	11	same	same	ADJ
ejpam-841	347	12	as	as	SCONJ
ejpam-841	347	13	the	the	DET
ejpam-841	347	14	generators	generator	NOUN
ejpam-841	347	15	described	describe	VERB
ejpam-841	347	16	in	in	ADP
ejpam-841	347	17	[	[	X
ejpam-841	347	18	18	18	NUM
ejpam-841	347	19	]	]	PUNCT
ejpam-841	347	20	.	.	PUNCT
ejpam-841	348	1	thus	thus	ADV
ejpam-841	348	2	,	,	PUNCT
ejpam-841	348	3	we	we	PRON
ejpam-841	348	4	have	have	AUX
ejpam-841	348	5	proved	prove	VERB
ejpam-841	348	6	our	our	PRON
ejpam-841	348	7	claim	claim	NOUN
ejpam-841	348	8	.	.	PUNCT
ejpam-841	349	1	3.2	3.2	NUM
ejpam-841	349	2	.	.	PUNCT
ejpam-841	350	1	non	non	ADJ
ejpam-841	350	2	-	-	ADJ
ejpam-841	350	3	singular	singular	ADJ
ejpam-841	350	4	rational	rational	ADJ
ejpam-841	350	5	space	space	NOUN
ejpam-841	350	6	curve	curve	NOUN
ejpam-841	350	7	of	of	ADP
ejpam-841	350	8	type	type	NOUN
ejpam-841	350	9	(	(	PUNCT
ejpam-841	350	10	1	1	NUM
ejpam-841	350	11	,	,	PUNCT
ejpam-841	350	12	1	1	NUM
ejpam-841	350	13	,	,	PUNCT
ejpam-841	350	14	d	d	NOUN
ejpam-841	350	15	−	−	PROPN
ejpam-841	350	16	2	2	NUM
ejpam-841	350	17	)	)	PUNCT
ejpam-841	350	18	now	now	ADV
ejpam-841	350	19	,	,	PUNCT
ejpam-841	350	20	we	we	PRON
ejpam-841	350	21	will	will	AUX
ejpam-841	350	22	find	find	VERB
ejpam-841	350	23	a	a	DET
ejpam-841	350	24	minimal	minimal	ADJ
ejpam-841	350	25	set	set	NOUN
ejpam-841	350	26	of	of	ADP
ejpam-841	350	27	generators	generator	NOUN
ejpam-841	350	28	for	for	ADP
ejpam-841	350	29	the	the	DET
ejpam-841	350	30	rees	rees	PROPN
ejpam-841	350	31	algebra	algebra	NOUN
ejpam-841	350	32	associated	associate	VERB
ejpam-841	350	33	to	to	ADP
ejpam-841	350	34	nonsingular	nonsingular	ADJ
ejpam-841	350	35	rational	rational	ADJ
ejpam-841	350	36	space	space	NOUN
ejpam-841	350	37	curves	curve	NOUN
ejpam-841	350	38	of	of	ADP
ejpam-841	350	39	type	type	NOUN
ejpam-841	350	40	(	(	PUNCT
ejpam-841	350	41	1,1	1,1	NUM
ejpam-841	350	42	,	,	PUNCT
ejpam-841	350	43	d	d	NOUN
ejpam-841	350	44	−	−	PROPN
ejpam-841	350	45	2	2	NUM
ejpam-841	350	46	)	)	PUNCT
ejpam-841	350	47	.	.	PUNCT
ejpam-841	351	1	lemma	lemma	PROPN
ejpam-841	351	2	3	3	X
ejpam-841	351	3	.	.	PUNCT
ejpam-841	352	1	let	let	VERB
ejpam-841	352	2	p	p	PRON
ejpam-841	352	3	,	,	PUNCT
ejpam-841	352	4	q	q	ADJ
ejpam-841	352	5	,	,	PUNCT
ejpam-841	352	6	r	r	NOUN
ejpam-841	352	7	be	be	AUX
ejpam-841	352	8	a	a	DET
ejpam-841	352	9	µ-basis	µ-basis	NOUN
ejpam-841	352	10	for	for	ADP
ejpam-841	352	11	a	a	DET
ejpam-841	352	12	non	non	ADJ
ejpam-841	352	13	-	-	ADJ
ejpam-841	352	14	singular	singular	ADJ
ejpam-841	352	15	rational	rational	ADJ
ejpam-841	352	16	space	space	NOUN
ejpam-841	352	17	curve	curve	NOUN
ejpam-841	352	18	of	of	ADP
ejpam-841	352	19	type	type	NOUN
ejpam-841	352	20	(	(	PUNCT
ejpam-841	352	21	1,1	1,1	NUM
ejpam-841	352	22	,	,	PUNCT
ejpam-841	352	23	d	d	NOUN
ejpam-841	352	24	−	−	PROPN
ejpam-841	352	25	2	2	NUM
ejpam-841	352	26	)	)	PUNCT
ejpam-841	352	27	.	.	PUNCT
ejpam-841	353	1	by	by	ADP
ejpam-841	353	2	a	a	DET
ejpam-841	353	3	linear	linear	ADJ
ejpam-841	353	4	transformation	transformation	NOUN
ejpam-841	353	5	on	on	ADP
ejpam-841	353	6	the	the	DET
ejpam-841	353	7	basis	basis	NOUN
ejpam-841	353	8	elements	element	NOUN
ejpam-841	353	9	p	p	X
ejpam-841	353	10	,	,	PUNCT
ejpam-841	353	11	q	q	ADJ
ejpam-841	353	12	,	,	PUNCT
ejpam-841	353	13	and	and	CCONJ
ejpam-841	353	14	by	by	ADP
ejpam-841	353	15	a	a	DET
ejpam-841	353	16	projective	projective	ADJ
ejpam-841	353	17	change	change	NOUN
ejpam-841	353	18	of	of	ADP
ejpam-841	353	19	coordinates	coordinate	NOUN
ejpam-841	353	20	in	in	ADP
ejpam-841	353	21	x	x	SYM
ejpam-841	353	22	,	,	PUNCT
ejpam-841	353	23	y	y	PROPN
ejpam-841	353	24	,	,	PUNCT
ejpam-841	353	25	z	z	PROPN
ejpam-841	353	26	,	,	PUNCT
ejpam-841	353	27	w	w	PROPN
ejpam-841	353	28	,	,	PUNCT
ejpam-841	353	29	we	we	PRON
ejpam-841	353	30	can	can	AUX
ejpam-841	353	31	adjust	adjust	VERB
ejpam-841	353	32	the	the	DET
ejpam-841	353	33	elements	element	NOUN
ejpam-841	353	34	of	of	ADP
ejpam-841	353	35	the	the	DET
ejpam-841	353	36	µ-basis	µ-basis	NOUN
ejpam-841	353	37	so	so	SCONJ
ejpam-841	353	38	that	that	SCONJ
ejpam-841	353	39	p	p	X
ejpam-841	353	40	=	=	X
ejpam-841	353	41	ys−	ys−	PUNCT
ejpam-841	353	42	x	x	SYM
ejpam-841	353	43	t	t	PROPN
ejpam-841	353	44	,	,	PUNCT
ejpam-841	353	45	q	q	X
ejpam-841	353	46	=	=	SYM
ejpam-841	353	47	ws−	ws−	PUNCT
ejpam-841	353	48	zt	zt	PROPN
ejpam-841	353	49	.	.	PUNCT
ejpam-841	353	50	proof	proof	NOUN
ejpam-841	353	51	.	.	PUNCT
ejpam-841	354	1	first	first	ADV
ejpam-841	354	2	write	write	VERB
ejpam-841	354	3	p	p	PROPN
ejpam-841	354	4	=	=	PUNCT
ejpam-841	354	5	p1s	p1	NOUN
ejpam-841	354	6	+	+	CCONJ
ejpam-841	354	7	p0	p0	PROPN
ejpam-841	354	8	t	t	PROPN
ejpam-841	354	9	and	and	CCONJ
ejpam-841	354	10	q	q	NOUN
ejpam-841	354	11	=	=	NOUN
ejpam-841	354	12	q1s+	q1s+	NOUN
ejpam-841	354	13	q0	q0	PROPN
ejpam-841	354	14	t	t	PROPN
ejpam-841	354	15	where	where	SCONJ
ejpam-841	354	16	p1	p1	PROPN
ejpam-841	354	17	,	,	PUNCT
ejpam-841	354	18	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	354	19	are	be	AUX
ejpam-841	354	20	linear	linear	ADJ
ejpam-841	354	21	forms	form	NOUN
ejpam-841	354	22	in	in	ADP
ejpam-841	354	23	k[x	k[x	PROPN
ejpam-841	354	24	,	,	PUNCT
ejpam-841	354	25	y	y	PROPN
ejpam-841	354	26	,	,	PUNCT
ejpam-841	354	27	z	z	PROPN
ejpam-841	354	28	,	,	PUNCT
ejpam-841	354	29	w	w	PROPN
ejpam-841	354	30	]	]	X
ejpam-841	354	31	.	.	PUNCT
ejpam-841	355	1	since	since	SCONJ
ejpam-841	355	2	the	the	DET
ejpam-841	355	3	curve	curve	NOUN
ejpam-841	355	4	is	be	AUX
ejpam-841	355	5	non	non	ADJ
ejpam-841	355	6	-	-	ADJ
ejpam-841	355	7	singular	singular	ADJ
ejpam-841	355	8	,	,	PUNCT
ejpam-841	355	9	by	by	ADP
ejpam-841	355	10	proposition	proposition	NOUN
ejpam-841	355	11	2	2	NUM
ejpam-841	355	12	,	,	PUNCT
ejpam-841	355	13	v(p1	v(p1	NOUN
ejpam-841	355	14	,	,	PUNCT
ejpam-841	355	15	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	355	16	)	)	PUNCT
ejpam-841	355	17	=	=	SYM
ejpam-841	355	18	;	;	PUNCT
ejpam-841	355	19	.	.	PUNCT
ejpam-841	356	1	hence	hence	ADV
ejpam-841	356	2	the	the	DET
ejpam-841	356	3	polynomials	polynomial	NOUN
ejpam-841	356	4	p1	p1	NOUN
ejpam-841	356	5	,	,	PUNCT
ejpam-841	356	6	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	356	7	are	be	AUX
ejpam-841	356	8	linearly	linearly	ADV
ejpam-841	356	9	independent	independent	ADJ
ejpam-841	356	10	of	of	ADP
ejpam-841	356	11	rank	rank	NOUN
ejpam-841	356	12	4	4	NUM
ejpam-841	356	13	.	.	PUNCT
ejpam-841	357	1	thus	thus	ADV
ejpam-841	357	2	,	,	PUNCT
ejpam-841	357	3	without	without	ADP
ejpam-841	357	4	loss	loss	NOUN
ejpam-841	357	5	of	of	ADP
ejpam-841	357	6	generality	generality	NOUN
ejpam-841	357	7	,	,	PUNCT
ejpam-841	357	8	we	we	PRON
ejpam-841	357	9	can	can	AUX
ejpam-841	357	10	set	set	VERB
ejpam-841	357	11	p	p	NOUN
ejpam-841	357	12	=	=	PUNCT
ejpam-841	357	13	ys−	ys−	PUNCT
ejpam-841	357	14	x	x	SYM
ejpam-841	357	15	t	t	NOUN
ejpam-841	357	16	and	and	CCONJ
ejpam-841	357	17	q	q	NOUN
ejpam-841	357	18	=	=	SYM
ejpam-841	357	19	ws−	ws−	PROPN
ejpam-841	357	20	zt	zt	PROPN
ejpam-841	357	21	.	.	PUNCT
ejpam-841	357	22	now	now	ADV
ejpam-841	357	23	since	since	SCONJ
ejpam-841	357	24	the	the	DET
ejpam-841	357	25	µ-basis	µ-basis	NOUN
ejpam-841	357	26	vanishes	vanish	VERB
ejpam-841	357	27	on	on	ADP
ejpam-841	357	28	the	the	DET
ejpam-841	357	29	curve	curve	NOUN
ejpam-841	357	30	,	,	PUNCT
ejpam-841	357	31	we	we	PRON
ejpam-841	357	32	have	have	VERB
ejpam-841	357	33	the	the	DET
ejpam-841	357	34	following	follow	VERB
ejpam-841	357	35	relations	relation	NOUN
ejpam-841	357	36	for	for	ADP
ejpam-841	357	37	any	any	DET
ejpam-841	357	38	point	point	NOUN
ejpam-841	357	39	on	on	ADP
ejpam-841	357	40	the	the	DET
ejpam-841	357	41	space	space	NOUN
ejpam-841	357	42	curve	curve	NOUN
ejpam-841	357	43	f(s	f(s	PROPN
ejpam-841	357	44	,	,	PUNCT
ejpam-841	357	45	t	t	PROPN
ejpam-841	357	46	)	)	PUNCT
ejpam-841	357	47	t	t	PROPN
ejpam-841	357	48	s	s	PART
ejpam-841	357	49	=	=	SYM
ejpam-841	357	50	y	y	PROPN
ejpam-841	357	51	x	x	PROPN
ejpam-841	357	52	,	,	PUNCT
ejpam-841	357	53	t	t	PROPN
ejpam-841	357	54	s	s	PART
ejpam-841	357	55	=	=	X
ejpam-841	357	56	w	w	PROPN
ejpam-841	357	57	z	z	PROPN
ejpam-841	357	58	.	.	PUNCT
ejpam-841	358	1	(	(	PUNCT
ejpam-841	358	2	7	7	X
ejpam-841	358	3	)	)	PUNCT
ejpam-841	358	4	for	for	ADP
ejpam-841	358	5	i	i	PROPN
ejpam-841	358	6	=	=	SYM
ejpam-841	358	7	0,1	0,1	NUM
ejpam-841	358	8	,	,	PUNCT
ejpam-841	358	9	·	·	PUNCT
ejpam-841	358	10	·	·	PUNCT
ejpam-841	358	11	·	·	PUNCT
ejpam-841	358	12	,	,	PUNCT
ejpam-841	358	13	d	d	X
ejpam-841	358	14	−	−	PROPN
ejpam-841	358	15	2	2	NUM
ejpam-841	358	16	,	,	PUNCT
ejpam-841	358	17	the	the	DET
ejpam-841	358	18	µ-basis	µ-basis	NOUN
ejpam-841	358	19	element	element	NOUN
ejpam-841	358	20	r	r	NOUN
ejpam-841	358	21	can	can	AUX
ejpam-841	358	22	be	be	AUX
ejpam-841	358	23	written	write	VERB
ejpam-841	358	24	as	as	ADP
ejpam-841	358	25	r	r	NOUN
ejpam-841	358	26	=	=	PUNCT
ejpam-841	358	27	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	358	28	+	+	NUM
ejpam-841	358	29	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	358	30	t	t	NOUN
ejpam-841	358	31	+	+	CCONJ
ejpam-841	358	32	·	·	PUNCT
ejpam-841	358	33	·	·	PUNCT
ejpam-841	358	34	·	·	PUNCT
ejpam-841	359	1	+	+	NUM
ejpam-841	359	2	r0	r0	NOUN
ejpam-841	359	3	td−2	td−2	NOUN
ejpam-841	359	4	=	=	SYM
ejpam-841	359	5	d−2	d−2	PROPN
ejpam-841	359	6	∑	∑	PUNCT
ejpam-841	359	7	j=0	j=0	PROPN
ejpam-841	359	8	r	r	NOUN
ejpam-841	359	9	js	js	PROPN
ejpam-841	359	10	j	j	PROPN
ejpam-841	359	11	td−2−	td−2−	NUM
ejpam-841	359	12	j	j	NOUN
ejpam-841	359	13	=	=	SYM
ejpam-841	359	14	sd−2	sd−2	NOUN
ejpam-841	359	15	d−2	d−2	PROPN
ejpam-841	359	16	∑	∑	PUNCT
ejpam-841	359	17	j=0	j=0	PROPN
ejpam-841	359	18	r	r	PROPN
ejpam-841	359	19	j	j	PROPN
ejpam-841	359	20	(	(	PUNCT
ejpam-841	359	21	t	t	PROPN
ejpam-841	359	22	s	s	PART
ejpam-841	359	23	)	)	PUNCT
ejpam-841	359	24	d−2−	d−2−	PROPN
ejpam-841	359	25	j	j	NOUN
ejpam-841	359	26	=	=	SYM
ejpam-841	359	27	sd−2	sd−2	PROPN
ejpam-841	359	28	[	[	PUNCT
ejpam-841	359	29	i−1	i−1	PROPN
ejpam-841	359	30	∑	∑	PUNCT
ejpam-841	359	31	j=0	j=0	PROPN
ejpam-841	359	32	r	r	PROPN
ejpam-841	359	33	j	j	PROPN
ejpam-841	359	34	(	(	PUNCT
ejpam-841	359	35	t	t	PROPN
ejpam-841	359	36	s	s	PART
ejpam-841	359	37	)	)	PUNCT
ejpam-841	359	38	d−2−i	d−2−i	PROPN
ejpam-841	359	39	(	(	PUNCT
ejpam-841	359	40	t	t	PROPN
ejpam-841	359	41	s	s	PART
ejpam-841	359	42	)	)	PUNCT
ejpam-841	359	43	i−	i−	PROPN
ejpam-841	359	44	j	j	PROPN
ejpam-841	359	45	+	+	CCONJ
ejpam-841	359	46	d−2	d−2	PROPN
ejpam-841	359	47	∑	∑	PUNCT
ejpam-841	359	48	j	j	X
ejpam-841	360	1	=	=	NOUN
ejpam-841	360	2	i	i	NOUN
ejpam-841	360	3	r	r	PROPN
ejpam-841	360	4	j	j	PROPN
ejpam-841	360	5	(	(	PUNCT
ejpam-841	360	6	t	t	PROPN
ejpam-841	360	7	s	s	PART
ejpam-841	360	8	)	)	PUNCT
ejpam-841	360	9	d−2−	d−2−	PROPN
ejpam-841	360	10	j	j	PROPN
ejpam-841	360	11	]	]	X
ejpam-841	360	12	,	,	PUNCT
ejpam-841	360	13	where	where	SCONJ
ejpam-841	360	14	i−1	i−1	PROPN
ejpam-841	360	15	∑	∑	PUNCT
ejpam-841	360	16	j=0	j=0	PROPN
ejpam-841	360	17	r	r	PROPN
ejpam-841	360	18	j	j	PROPN
ejpam-841	360	19	(	(	PUNCT
ejpam-841	360	20	t	t	PROPN
ejpam-841	360	21	s	s	PART
ejpam-841	360	22	)	)	PUNCT
ejpam-841	360	23	d−2−i	d−2−i	PROPN
ejpam-841	360	24	(	(	PUNCT
ejpam-841	360	25	t	t	PROPN
ejpam-841	360	26	s	s	PART
ejpam-841	360	27	)	)	PUNCT
ejpam-841	360	28	i−	i−	PROPN
ejpam-841	360	29	j	j	NOUN
ejpam-841	360	30	:	:	PUNCT
ejpam-841	360	31	=	=	SYM
ejpam-841	360	32	0	0	PUNCT
ejpam-841	361	1	if	if	SCONJ
ejpam-841	361	2	i	i	PRON
ejpam-841	361	3	=	=	NOUN
ejpam-841	361	4	0	0	NUM
ejpam-841	361	5	=	=	SYM
ejpam-841	361	6	sd−2	sd−2	PROPN
ejpam-841	361	7	[	[	PUNCT
ejpam-841	361	8	i−1	i−1	PROPN
ejpam-841	361	9	∑	∑	PUNCT
ejpam-841	361	10	j=0	j=0	PROPN
ejpam-841	361	11	r	r	PROPN
ejpam-841	361	12	j	j	PROPN
ejpam-841	361	13	(	(	PUNCT
ejpam-841	361	14	t	t	PROPN
ejpam-841	361	15	s	s	PART
ejpam-841	361	16	)	)	PUNCT
ejpam-841	361	17	d−2−i	d−2−i	PROPN
ejpam-841	361	18	(	(	PUNCT
ejpam-841	361	19	t	t	PROPN
ejpam-841	361	20	s	s	PART
ejpam-841	361	21	)	)	PUNCT
ejpam-841	361	22	i−	i−	PROPN
ejpam-841	361	23	j	j	PROPN
ejpam-841	361	24	+	+	CCONJ
ejpam-841	361	25	d−2−i	d−2−i	PROPN
ejpam-841	361	26	∑	∑	PUNCT
ejpam-841	361	27	j=0	j=0	PROPN
ejpam-841	361	28	ri+	ri+	PROPN
ejpam-841	361	29	j	j	PROPN
ejpam-841	361	30	(	(	PUNCT
ejpam-841	361	31	t	t	PROPN
ejpam-841	361	32	s	s	PART
ejpam-841	361	33	)	)	PUNCT
ejpam-841	361	34	d−2−i−	d−2−i−	PROPN
ejpam-841	361	35	j	j	PROPN
ejpam-841	361	36	]	]	X
ejpam-841	361	37	.	.	PUNCT
ejpam-841	362	1	for	for	ADP
ejpam-841	362	2	i	i	PROPN
ejpam-841	362	3	=	=	SYM
ejpam-841	362	4	0,1	0,1	NUM
ejpam-841	362	5	,	,	PUNCT
ejpam-841	362	6	·	·	PUNCT
ejpam-841	362	7	·	·	PUNCT
ejpam-841	362	8	·	·	PUNCT
ejpam-841	362	9	,	,	PUNCT
ejpam-841	362	10	d	d	X
ejpam-841	362	11	−	−	PROPN
ejpam-841	362	12	2	2	NUM
ejpam-841	362	13	,	,	PUNCT
ejpam-841	362	14	let	let	VERB
ejpam-841	362	15	r	r	NOUN
ejpam-841	362	16	′i	′i	NOUN
ejpam-841	362	17	(	(	PUNCT
ejpam-841	362	18	a	a	DET
ejpam-841	362	19	,	,	PUNCT
ejpam-841	362	20	b	b	NOUN
ejpam-841	362	21	)	)	PUNCT
ejpam-841	362	22	=	=	SYM
ejpam-841	362	23	i−1	i−1	PROPN
ejpam-841	362	24	∑	∑	PUNCT
ejpam-841	362	25	j=0	j=0	PROPN
ejpam-841	362	26	r	r	NOUN
ejpam-841	362	27	j(b	j(b	PROPN
ejpam-841	362	28	)	)	PUNCT
ejpam-841	362	29	d−2−i(a)i−	d−2−i(a)i−	NUM
ejpam-841	362	30	j+	j+	NUM
ejpam-841	362	31	d−2−i	d−2−i	PROPN
ejpam-841	362	32	∑	∑	PUNCT
ejpam-841	362	33	j=0	j=0	PROPN
ejpam-841	362	34	ri+	ri+	PROPN
ejpam-841	362	35	j(b	j(b	PROPN
ejpam-841	362	36	)	)	PUNCT
ejpam-841	363	1	d−2−i−	d−2−i−	PROPN
ejpam-841	363	2	j	j	PROPN
ejpam-841	363	3	,	,	PUNCT
ejpam-841	363	4	where	where	SCONJ
ejpam-841	363	5	i−1	i−1	PROPN
ejpam-841	363	6	∑	∑	PUNCT
ejpam-841	363	7	j=0	j=0	PROPN
ejpam-841	363	8	r	r	NOUN
ejpam-841	363	9	j(b	j(b	PROPN
ejpam-841	363	10	)	)	PUNCT
ejpam-841	363	11	d−2−i(a)i−	d−2−i(a)i−	PRON
ejpam-841	364	1	j	j	NOUN
ejpam-841	364	2	:	:	PUNCT
ejpam-841	365	1	=	=	SYM
ejpam-841	365	2	0	0	PUNCT
ejpam-841	366	1	if	if	SCONJ
ejpam-841	366	2	i	i	PRON
ejpam-841	366	3	=	=	NOUN
ejpam-841	366	4	0	0	X
ejpam-841	366	5	.	.	PUNCT
ejpam-841	367	1	j.	j.	PROPN
ejpam-841	367	2	hoffman	hoffman	PROPN
ejpam-841	367	3	,	,	PUNCT
ejpam-841	367	4	h.	h.	PROPN
ejpam-841	367	5	wang	wang	PROPN
ejpam-841	367	6	,	,	PUNCT
ejpam-841	367	7	x.	x.	PROPN
ejpam-841	367	8	jia	jia	PROPN
ejpam-841	367	9	,	,	PUNCT
ejpam-841	367	10	r.	r.	PROPN
ejpam-841	367	11	goldman	goldman	PROPN
ejpam-841	367	12	/	/	SYM
ejpam-841	367	13	eur	eur	PROPN
ejpam-841	367	14	.	.	PUNCT
ejpam-841	368	1	j.	j.	PROPN
ejpam-841	368	2	pure	pure	PROPN
ejpam-841	368	3	appl	appl	PROPN
ejpam-841	368	4	.	.	PROPN
ejpam-841	368	5	math	math	PROPN
ejpam-841	368	6	,	,	PUNCT
ejpam-841	368	7	3	3	NUM
ejpam-841	368	8	(	(	PUNCT
ejpam-841	368	9	2010	2010	NUM
ejpam-841	368	10	)	)	PUNCT
ejpam-841	368	11	,	,	PUNCT
ejpam-841	368	12	602	602	NUM
ejpam-841	368	13	-	-	SYM
ejpam-841	368	14	632	632	NUM
ejpam-841	368	15	614	614	NUM
ejpam-841	368	16	then	then	ADV
ejpam-841	368	17	r	r	NOUN
ejpam-841	368	18	′i	′i	NOUN
ejpam-841	368	19	(	(	PUNCT
ejpam-841	368	20	y	y	NOUN
ejpam-841	368	21	x	x	X
ejpam-841	368	22	,	,	PUNCT
ejpam-841	368	23	w	w	PROPN
ejpam-841	368	24	z	z	NOUN
ejpam-841	368	25	)	)	PUNCT
ejpam-841	369	1	=	=	SYM
ejpam-841	369	2	i−1	i−1	PROPN
ejpam-841	369	3	∑	∑	PUNCT
ejpam-841	369	4	j=0	j=0	PROPN
ejpam-841	369	5	r	r	PROPN
ejpam-841	369	6	j	j	PROPN
ejpam-841	369	7	(	(	PUNCT
ejpam-841	369	8	w	w	PROPN
ejpam-841	369	9	z	z	NOUN
ejpam-841	369	10	)	)	PUNCT
ejpam-841	369	11	d−2−i	d−2−i	PROPN
ejpam-841	369	12	(	(	PUNCT
ejpam-841	369	13	y	y	NOUN
ejpam-841	369	14	x	x	PROPN
ejpam-841	369	15	)	)	PUNCT
ejpam-841	369	16	i−	i−	PROPN
ejpam-841	369	17	j	j	PROPN
ejpam-841	369	18	+	+	CCONJ
ejpam-841	369	19	d−2−i	d−2−i	PROPN
ejpam-841	369	20	∑	∑	PUNCT
ejpam-841	369	21	j=0	j=0	PROPN
ejpam-841	369	22	ri+	ri+	PROPN
ejpam-841	369	23	j	j	PROPN
ejpam-841	369	24	(	(	PUNCT
ejpam-841	369	25	w	w	PROPN
ejpam-841	369	26	z	z	NOUN
ejpam-841	369	27	)	)	PUNCT
ejpam-841	370	1	d−2−i−	d−2−i−	PROPN
ejpam-841	370	2	j	j	PROPN
ejpam-841	370	3	,	,	PUNCT
ejpam-841	370	4	(	(	PUNCT
ejpam-841	370	5	8)	8)	NUM
ejpam-841	370	6	where	where	SCONJ
ejpam-841	370	7	∑i−1	∑i−1	PROPN
ejpam-841	370	8	j=0	j=0	PROPN
ejpam-841	370	9	r	r	PROPN
ejpam-841	370	10	j	j	PROPN
ejpam-841	370	11	(	(	PUNCT
ejpam-841	370	12	w	w	PROPN
ejpam-841	370	13	z	z	NOUN
ejpam-841	370	14	)	)	PUNCT
ejpam-841	370	15	d−2−i	d−2−i	PROPN
ejpam-841	370	16	(	(	PUNCT
ejpam-841	370	17	y	y	NOUN
ejpam-841	370	18	x	x	PROPN
ejpam-841	370	19	)	)	PUNCT
ejpam-841	370	20	i−	i−	PROPN
ejpam-841	370	21	j	j	NOUN
ejpam-841	370	22	:	:	PUNCT
ejpam-841	371	1	=	=	SYM
ejpam-841	371	2	0	0	PUNCT
ejpam-841	372	1	if	if	SCONJ
ejpam-841	372	2	i	i	PRON
ejpam-841	372	3	=	=	NOUN
ejpam-841	372	4	0	0	X
ejpam-841	372	5	.	.	PUNCT
ejpam-841	373	1	moreover	moreover	ADV
ejpam-841	373	2	,	,	PUNCT
ejpam-841	373	3	for	for	ADP
ejpam-841	373	4	all	all	DET
ejpam-841	373	5	i	i	PRON
ejpam-841	373	6	=	=	NOUN
ejpam-841	373	7	1	1	NUM
ejpam-841	373	8	,	,	PUNCT
ejpam-841	373	9	.	.	PUNCT
ejpam-841	373	10	.	.	PUNCT
ejpam-841	373	11	.	.	PUNCT
ejpam-841	374	1	,	,	PUNCT
ejpam-841	375	1	d	d	X
ejpam-841	375	2	−	−	PROPN
ejpam-841	375	3	3	3	NUM
ejpam-841	375	4	,	,	PUNCT
ejpam-841	375	5	and	and	CCONJ
ejpam-841	375	6	for	for	ADP
ejpam-841	375	7	all	all	DET
ejpam-841	375	8	0≤	0≤	ADJ
ejpam-841	375	9	j	j	NOUN
ejpam-841	375	10	≤	≤	X
ejpam-841	376	1	i	i	PRON
ejpam-841	376	2	,	,	PUNCT
ejpam-841	376	3	we	we	PRON
ejpam-841	376	4	can	can	AUX
ejpam-841	376	5	also	also	ADV
ejpam-841	376	6	write	write	VERB
ejpam-841	376	7	r	r	NOUN
ejpam-841	376	8	=	=	PUNCT
ejpam-841	376	9	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	376	10	+	+	NUM
ejpam-841	376	11	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	376	12	t	t	NOUN
ejpam-841	376	13	+	+	CCONJ
ejpam-841	376	14	·	·	PUNCT
ejpam-841	376	15	·	·	PUNCT
ejpam-841	376	16	·	·	PUNCT
ejpam-841	376	17	+	+	NUM
ejpam-841	376	18	r0	r0	NOUN
ejpam-841	376	19	td−2	td−2	NOUN
ejpam-841	376	20	=	=	PUNCT
ejpam-841	376	21	d−2	d−2	PROPN
ejpam-841	376	22	∑	∑	PUNCT
ejpam-841	376	23	ℓ=0	ℓ=0	PRON
ejpam-841	376	24	rℓs	rℓ	NOUN
ejpam-841	376	25	ℓ	ℓ	NOUN
ejpam-841	376	26	td−2−ℓ	td−2−ℓ	NOUN
ejpam-841	376	27	=	=	SYM
ejpam-841	377	1	si	si	PROPN
ejpam-841	377	2	[	[	PUNCT
ejpam-841	377	3	i−1	i−1	PROPN
ejpam-841	377	4	∑	∑	PUNCT
ejpam-841	377	5	ℓ=0	ℓ=0	PROPN
ejpam-841	377	6	rℓ	rℓ	PROPN
ejpam-841	377	7	t	t	PROPN
ejpam-841	377	8	d−2−i	d−2−i	PROPN
ejpam-841	377	9	(	(	PUNCT
ejpam-841	377	10	t	t	PROPN
ejpam-841	377	11	s	s	PART
ejpam-841	377	12	)	)	PUNCT
ejpam-841	377	13	i−ℓ+	i−ℓ+	PROPN
ejpam-841	377	14	d−2	d−2	PROPN
ejpam-841	377	15	∑	∑	PART
ejpam-841	377	16	ℓ=i	ℓ=i	PROPN
ejpam-841	377	17	rℓ	rℓ	ADP
ejpam-841	377	18	t	t	NOUN
ejpam-841	377	19	d−2−ℓsℓ−i	d−2−ℓsℓ−i	PROPN
ejpam-841	377	20	]	]	X
ejpam-841	377	21	=	=	SYM
ejpam-841	377	22	si{td−2−i	si{td−2−i	NOUN
ejpam-841	377	23	[	[	PUNCT
ejpam-841	377	24	j−1	j−1	PROPN
ejpam-841	377	25	∑	∑	PART
ejpam-841	377	26	ℓ=0	ℓ=0	SYM
ejpam-841	377	27	rℓ	rℓ	PROPN
ejpam-841	377	28	(	(	PUNCT
ejpam-841	377	29	t	t	PROPN
ejpam-841	377	30	s	s	PART
ejpam-841	377	31	)	)	PUNCT
ejpam-841	377	32	i−	i−	PROPN
ejpam-841	377	33	j	j	PROPN
ejpam-841	377	34	(	(	PUNCT
ejpam-841	377	35	t	t	PROPN
ejpam-841	377	36	s	s	PART
ejpam-841	377	37	)	)	PUNCT
ejpam-841	377	38	j−ℓ+	j−ℓ+	PROPN
ejpam-841	377	39	i−1	i−1	PROPN
ejpam-841	377	40	∑	∑	PROPN
ejpam-841	377	41	ℓ=	ℓ=	PROPN
ejpam-841	377	42	j	j	PROPN
ejpam-841	377	43	rℓ	rℓ	PROPN
ejpam-841	377	44	(	(	PUNCT
ejpam-841	377	45	t	t	PROPN
ejpam-841	377	46	s	s	PART
ejpam-841	377	47	)	)	PUNCT
ejpam-841	377	48	i−ℓ	i−ℓ	PROPN
ejpam-841	377	49	]	]	PUNCT
ejpam-841	378	1	+	+	CCONJ
ejpam-841	378	2	d−2	d−2	PROPN
ejpam-841	378	3	∑	∑	PART
ejpam-841	378	4	ℓ=i	ℓ=i	PROPN
ejpam-841	378	5	rℓ	rℓ	ADP
ejpam-841	378	6	t	t	NOUN
ejpam-841	378	7	d−2−ℓsℓ−i	d−2−ℓsℓ−i	PROPN
ejpam-841	378	8	}	}	PUNCT
ejpam-841	378	9	,	,	PUNCT
ejpam-841	378	10	where	where	SCONJ
ejpam-841	378	11	i−1	i−1	PROPN
ejpam-841	378	12	∑	∑	PROPN
ejpam-841	378	13	ℓ=	ℓ=	PROPN
ejpam-841	378	14	j	j	PROPN
ejpam-841	378	15	rℓ	rℓ	PROPN
ejpam-841	378	16	(	(	PUNCT
ejpam-841	378	17	t	t	PROPN
ejpam-841	378	18	s	s	PART
ejpam-841	378	19	)	)	PUNCT
ejpam-841	378	20	i−ℓ	i−ℓ	VERB
ejpam-841	378	21	:	:	PUNCT
ejpam-841	378	22	=	=	SYM
ejpam-841	378	23	0	0	NUM
ejpam-841	378	24	,	,	PUNCT
ejpam-841	378	25	if	if	SCONJ
ejpam-841	378	26	j	j	PROPN
ejpam-841	378	27	=	=	PUNCT
ejpam-841	379	1	i	i	PROPN
ejpam-841	379	2	=	=	PUNCT
ejpam-841	379	3	si{td−2−i	si{td−2−i	NOUN
ejpam-841	379	4	[	[	PUNCT
ejpam-841	379	5	j−1	j−1	PROPN
ejpam-841	379	6	∑	∑	PART
ejpam-841	379	7	ℓ=0	ℓ=0	SYM
ejpam-841	379	8	rℓ	rℓ	PROPN
ejpam-841	379	9	(	(	PUNCT
ejpam-841	379	10	t	t	PROPN
ejpam-841	379	11	s	s	PART
ejpam-841	379	12	)	)	PUNCT
ejpam-841	379	13	i−	i−	PROPN
ejpam-841	379	14	j	j	PROPN
ejpam-841	379	15	(	(	PUNCT
ejpam-841	379	16	t	t	PROPN
ejpam-841	379	17	s	s	PART
ejpam-841	379	18	)	)	PUNCT
ejpam-841	379	19	j−ℓ+	j−ℓ+	PROPN
ejpam-841	379	20	i−1−	i−1−	PROPN
ejpam-841	379	21	j	j	PROPN
ejpam-841	379	22	∑	∑	PUNCT
ejpam-841	379	23	ℓ=0	ℓ=0	PROPN
ejpam-841	379	24	r	r	PROPN
ejpam-841	379	25	j+ℓ	j+ℓ	PROPN
ejpam-841	379	26	(	(	PUNCT
ejpam-841	379	27	t	t	PROPN
ejpam-841	379	28	s	s	PART
ejpam-841	379	29	)	)	PUNCT
ejpam-841	379	30	i−	i−	PROPN
ejpam-841	379	31	j−ℓ	j−ℓ	PROPN
ejpam-841	379	32	]	]	X
ejpam-841	380	1	+	+	X
ejpam-841	380	2	d−2−i	d−2−i	ADJ
ejpam-841	380	3	∑	∑	PRON
ejpam-841	380	4	ℓ=0	ℓ=0	PROPN
ejpam-841	380	5	ri+ℓ	ri+ℓ	PROPN
ejpam-841	380	6	t	t	PROPN
ejpam-841	380	7	d−2−i−ℓsℓ	d−2−i−ℓsℓ	PROPN
ejpam-841	380	8	}	}	PUNCT
ejpam-841	380	9	.	.	PUNCT
ejpam-841	381	1	for	for	ADP
ejpam-841	381	2	all	all	DET
ejpam-841	381	3	i	i	PRON
ejpam-841	381	4	=	=	NOUN
ejpam-841	381	5	1	1	NUM
ejpam-841	381	6	,	,	PUNCT
ejpam-841	381	7	.	.	PUNCT
ejpam-841	381	8	.	.	PUNCT
ejpam-841	381	9	.	.	PUNCT
ejpam-841	382	1	,	,	PUNCT
ejpam-841	383	1	d	d	X
ejpam-841	383	2	−	−	PROPN
ejpam-841	383	3	3	3	NUM
ejpam-841	383	4	,	,	PUNCT
ejpam-841	383	5	and	and	CCONJ
ejpam-841	383	6	for	for	ADP
ejpam-841	383	7	all	all	DET
ejpam-841	383	8	0≤	0≤	ADJ
ejpam-841	383	9	j	j	NOUN
ejpam-841	383	10	≤	≤	PROPN
ejpam-841	384	1	i	i	PRON
ejpam-841	384	2	,	,	PUNCT
ejpam-841	384	3	let	let	VERB
ejpam-841	384	4	r	r	NOUN
ejpam-841	384	5	′j	′j	NOUN
ejpam-841	384	6	,	,	PUNCT
ejpam-841	384	7	i−	i−	PROPN
ejpam-841	384	8	j(a	j(a	PROPN
ejpam-841	384	9	,	,	PUNCT
ejpam-841	384	10	b	b	X
ejpam-841	384	11	)	)	PUNCT
ejpam-841	384	12	=	=	PUNCT
ejpam-841	385	1	td−2−i	td−2−i	X
ejpam-841	385	2	[	[	PUNCT
ejpam-841	385	3	j−1	j−1	X
ejpam-841	385	4	∑	∑	PUNCT
ejpam-841	385	5	ℓ=0	ℓ=0	PRON
ejpam-841	385	6	rℓ(b	rℓ(b	ADJ
ejpam-841	385	7	)	)	PUNCT
ejpam-841	385	8	i−	i−	ADP
ejpam-841	385	9	j(a	j(a	PROPN
ejpam-841	385	10	)	)	PUNCT
ejpam-841	385	11	j−ℓ+	j−ℓ+	PROPN
ejpam-841	385	12	i−1−	i−1−	PROPN
ejpam-841	385	13	j	j	PROPN
ejpam-841	385	14	∑	∑	PUNCT
ejpam-841	385	15	ℓ=0	ℓ=0	PROPN
ejpam-841	385	16	r	r	NOUN
ejpam-841	385	17	j+ℓ(b	j+ℓ(b	ADJ
ejpam-841	385	18	)	)	PUNCT
ejpam-841	385	19	i−	i−	PROPN
ejpam-841	385	20	j−ℓ	j−ℓ	PROPN
ejpam-841	385	21	]	]	X
ejpam-841	385	22	+	+	X
ejpam-841	385	23	d−2−i	d−2−i	ADJ
ejpam-841	385	24	∑	∑	PRON
ejpam-841	385	25	ℓ=0	ℓ=0	PROPN
ejpam-841	385	26	ri+ℓ	ri+ℓ	PROPN
ejpam-841	385	27	t	t	PROPN
ejpam-841	385	28	d−2−i−ℓsℓ	d−2−i−ℓsℓ	NOUN
ejpam-841	385	29	,	,	PUNCT
ejpam-841	385	30	where	where	SCONJ
ejpam-841	385	31	∑i−1−	∑i−1−	PROPN
ejpam-841	385	32	j	j	PROPN
ejpam-841	385	33	ℓ=0	ℓ=0	PROPN
ejpam-841	385	34	r	r	NOUN
ejpam-841	385	35	j+ℓ(b	j+ℓ(b	ADJ
ejpam-841	385	36	)	)	PUNCT
ejpam-841	385	37	i−	i−	ADJ
ejpam-841	385	38	j−ℓ	j−ℓ	NOUN
ejpam-841	385	39	:	:	PUNCT
ejpam-841	385	40	=	=	SYM
ejpam-841	385	41	0	0	NUM
ejpam-841	385	42	,	,	PUNCT
ejpam-841	385	43	if	if	SCONJ
ejpam-841	385	44	j	j	PROPN
ejpam-841	385	45	=	=	PROPN
ejpam-841	385	46	i.	i.	PROPN
ejpam-841	385	47	then	then	ADV
ejpam-841	385	48	r	r	NOUN
ejpam-841	385	49	′j	′j	NOUN
ejpam-841	385	50	,	,	PUNCT
ejpam-841	385	51	i−	i−	PROPN
ejpam-841	385	52	j	j	PROPN
ejpam-841	385	53	(	(	PUNCT
ejpam-841	385	54	y	y	PROPN
ejpam-841	385	55	x	x	PROPN
ejpam-841	385	56	,	,	PUNCT
ejpam-841	385	57	w	w	PROPN
ejpam-841	385	58	z	z	NOUN
ejpam-841	385	59	)	)	PUNCT
ejpam-841	386	1	=	=	PUNCT
ejpam-841	387	1	td−2−i	td−2−i	X
ejpam-841	387	2	[	[	PUNCT
ejpam-841	387	3	j−1	j−1	PROPN
ejpam-841	387	4	∑	∑	PART
ejpam-841	387	5	ℓ=0	ℓ=0	SYM
ejpam-841	387	6	rℓ	rℓ	PROPN
ejpam-841	387	7	(	(	PUNCT
ejpam-841	387	8	w	w	PROPN
ejpam-841	387	9	z	z	NOUN
ejpam-841	387	10	)	)	PUNCT
ejpam-841	387	11	i−	i−	PROPN
ejpam-841	387	12	j	j	PROPN
ejpam-841	387	13	(	(	PUNCT
ejpam-841	387	14	y	y	PROPN
ejpam-841	387	15	x	x	PROPN
ejpam-841	387	16	)	)	PUNCT
ejpam-841	387	17	j−ℓ+	j−ℓ+	PROPN
ejpam-841	387	18	i−1−	i−1−	PROPN
ejpam-841	387	19	j	j	PROPN
ejpam-841	387	20	∑	∑	PUNCT
ejpam-841	387	21	ℓ=0	ℓ=0	PROPN
ejpam-841	387	22	r	r	PROPN
ejpam-841	387	23	j+ℓ	j+ℓ	PROPN
ejpam-841	387	24	(	(	PUNCT
ejpam-841	387	25	w	w	PROPN
ejpam-841	387	26	z	z	NOUN
ejpam-841	387	27	)	)	PUNCT
ejpam-841	387	28	i−	i−	PROPN
ejpam-841	387	29	j−ℓ	j−ℓ	PROPN
ejpam-841	387	30	]	]	X
ejpam-841	387	31	+	+	X
ejpam-841	387	32	d−2−i	d−2−i	ADJ
ejpam-841	387	33	∑	∑	PRON
ejpam-841	387	34	ℓ=0	ℓ=0	PROPN
ejpam-841	387	35	ri+ℓ	ri+ℓ	PROPN
ejpam-841	387	36	t	t	PROPN
ejpam-841	387	37	d−2−i−ℓsℓ	d−2−i−ℓsℓ	PROPN
ejpam-841	387	38	,	,	PUNCT
ejpam-841	387	39	(	(	PUNCT
ejpam-841	387	40	9	9	NUM
ejpam-841	387	41	)	)	PUNCT
ejpam-841	387	42	where	where	SCONJ
ejpam-841	387	43	∑i−1−	∑i−1−	PROPN
ejpam-841	387	44	j	j	PROPN
ejpam-841	387	45	ℓ=0	ℓ=0	PROPN
ejpam-841	387	46	r	r	PROPN
ejpam-841	387	47	j+ℓ	j+ℓ	PROPN
ejpam-841	387	48	(	(	PUNCT
ejpam-841	387	49	w	w	PROPN
ejpam-841	387	50	z	z	NOUN
ejpam-841	387	51	)	)	PUNCT
ejpam-841	387	52	i−	i−	PROPN
ejpam-841	387	53	j−ℓ	j−ℓ	NOUN
ejpam-841	387	54	:	:	PUNCT
ejpam-841	387	55	=	=	SYM
ejpam-841	387	56	0	0	NUM
ejpam-841	387	57	,	,	PUNCT
ejpam-841	387	58	if	if	SCONJ
ejpam-841	387	59	j	j	PROPN
ejpam-841	387	60	=	=	PROPN
ejpam-841	387	61	i.	i.	PROPN
ejpam-841	387	62	theorem	theorem	VERB
ejpam-841	387	63	3	3	NUM
ejpam-841	387	64	.	.	PUNCT
ejpam-841	387	65	a	a	DET
ejpam-841	387	66	minimal	minimal	ADJ
ejpam-841	387	67	set	set	NOUN
ejpam-841	387	68	of	of	ADP
ejpam-841	387	69	generators	generator	NOUN
ejpam-841	387	70	for	for	ADP
ejpam-841	387	71	the	the	DET
ejpam-841	387	72	rees	rees	PROPN
ejpam-841	387	73	algebra	algebra	PROPN
ejpam-841	387	74	associated	associate	VERB
ejpam-841	387	75	to	to	ADP
ejpam-841	387	76	a	a	DET
ejpam-841	387	77	non	non	ADJ
ejpam-841	387	78	-	-	ADJ
ejpam-841	387	79	singular	singular	ADJ
ejpam-841	387	80	rational	rational	ADJ
ejpam-841	387	81	space	space	NOUN
ejpam-841	387	82	curve	curve	NOUN
ejpam-841	387	83	type	type	NOUN
ejpam-841	387	84	(	(	PUNCT
ejpam-841	387	85	1,1	1,1	NUM
ejpam-841	387	86	,	,	PUNCT
ejpam-841	387	87	d	d	NOUN
ejpam-841	387	88	−	−	PROPN
ejpam-841	387	89	2	2	NUM
ejpam-841	387	90	)	)	PUNCT
ejpam-841	387	91	are	be	AUX
ejpam-841	387	92	given	give	VERB
ejpam-841	387	93	by	by	ADP
ejpam-841	387	94	the	the	DET
ejpam-841	387	95	following	follow	VERB
ejpam-841	387	96	3	3	NUM
ejpam-841	387	97	+	+	SYM
ejpam-841	387	98	d	d	NOUN
ejpam-841	387	99	+	+	CCONJ
ejpam-841	387	100	d(d	d(d	PROPN
ejpam-841	387	101	−	−	PROPN
ejpam-841	387	102	3	3	NUM
ejpam-841	387	103	)	)	SYM
ejpam-841	387	104	2	2	NUM
ejpam-841	387	105	polynomials	polynomial	NOUN
ejpam-841	387	106	:	:	PUNCT
ejpam-841	388	1	1	1	NUM
ejpam-841	388	2	.	.	X
ejpam-841	388	3	three	three	NUM
ejpam-841	388	4	µ-basis	µ-basis	NOUN
ejpam-841	388	5	elements	element	NOUN
ejpam-841	388	6	:	:	PUNCT
ejpam-841	388	7	p	p	X
ejpam-841	388	8	,	,	PUNCT
ejpam-841	388	9	q	q	INTJ
ejpam-841	388	10	,	,	PUNCT
ejpam-841	388	11	r	r	NOUN
ejpam-841	388	12	where	where	SCONJ
ejpam-841	388	13	deg(p	deg(p	NOUN
ejpam-841	388	14	)	)	PUNCT
ejpam-841	389	1	=	=	SYM
ejpam-841	389	2	deg(q	deg(q	NOUN
ejpam-841	389	3	)	)	PUNCT
ejpam-841	389	4	=	=	SYM
ejpam-841	389	5	(	(	PUNCT
ejpam-841	389	6	1,1	1,1	NUM
ejpam-841	389	7	)	)	PUNCT
ejpam-841	389	8	,	,	PUNCT
ejpam-841	389	9	and	and	CCONJ
ejpam-841	389	10	deg(r	deg(r	PROPN
ejpam-841	389	11	)	)	PUNCT
ejpam-841	389	12	=	=	PUNCT
ejpam-841	390	1	(	(	PUNCT
ejpam-841	390	2	d	d	NOUN
ejpam-841	390	3	−	−	PROPN
ejpam-841	390	4	2,1	2,1	NUM
ejpam-841	390	5	)	)	PUNCT
ejpam-841	390	6	;	;	PUNCT
ejpam-841	391	1	2	2	X
ejpam-841	391	2	.	.	X
ejpam-841	391	3	d	d	PRON
ejpam-841	391	4	implicit	implicit	ADJ
ejpam-841	391	5	equations	equation	NOUN
ejpam-841	391	6	:	:	PUNCT
ejpam-841	391	7	sylvs	sylv	NOUN
ejpam-841	391	8	,	,	PUNCT
ejpam-841	391	9	t(p	t(p	PROPN
ejpam-841	391	10	,	,	PUNCT
ejpam-841	391	11	q	q	NOUN
ejpam-841	391	12	)	)	PUNCT
ejpam-841	391	13	of	of	ADP
ejpam-841	391	14	degree	degree	NOUN
ejpam-841	391	15	2	2	NUM
ejpam-841	391	16	,	,	PUNCT
ejpam-841	391	17	and	and	CCONJ
ejpam-841	391	18	d−1	d−1	PROPN
ejpam-841	391	19	implicit	implicit	ADJ
ejpam-841	391	20	equations	equation	NOUN
ejpam-841	391	21	x	x	SYM
ejpam-841	391	22	izd−2−i	izd−2−i	NOUN
ejpam-841	391	23	r	r	NOUN
ejpam-841	391	24	′i	′i	NOUN
ejpam-841	391	25	(	(	PUNCT
ejpam-841	391	26	y	y	NOUN
ejpam-841	391	27	x	x	X
ejpam-841	391	28	,	,	PUNCT
ejpam-841	391	29	w	w	PROPN
ejpam-841	391	30	z	z	NOUN
ejpam-841	391	31	)	)	PUNCT
ejpam-841	391	32	of	of	ADP
ejpam-841	391	33	degree	degree	NOUN
ejpam-841	391	34	d	d	NOUN
ejpam-841	391	35	−	−	PROPN
ejpam-841	391	36	1	1	NUM
ejpam-841	391	37	for	for	ADP
ejpam-841	391	38	i	i	PRON
ejpam-841	391	39	=	=	NOUN
ejpam-841	391	40	0,1	0,1	NUM
ejpam-841	391	41	,	,	PUNCT
ejpam-841	391	42	.	.	PUNCT
ejpam-841	391	43	.	.	PUNCT
ejpam-841	391	44	.	.	PUNCT
ejpam-841	392	1	,	,	PUNCT
ejpam-841	393	1	d	d	X
ejpam-841	393	2	−	−	PROPN
ejpam-841	393	3	2	2	NUM
ejpam-841	393	4	;	;	PUNCT
ejpam-841	393	5	j.	j.	PROPN
ejpam-841	393	6	hoffman	hoffman	PROPN
ejpam-841	393	7	,	,	PUNCT
ejpam-841	393	8	h.	h.	PROPN
ejpam-841	393	9	wang	wang	PROPN
ejpam-841	393	10	,	,	PUNCT
ejpam-841	393	11	x.	x.	PROPN
ejpam-841	393	12	jia	jia	PROPN
ejpam-841	393	13	,	,	PUNCT
ejpam-841	393	14	r.	r.	PROPN
ejpam-841	393	15	goldman	goldman	PROPN
ejpam-841	393	16	/	/	SYM
ejpam-841	393	17	eur	eur	PROPN
ejpam-841	393	18	.	.	PUNCT
ejpam-841	394	1	j.	j.	PROPN
ejpam-841	394	2	pure	pure	PROPN
ejpam-841	394	3	appl	appl	PROPN
ejpam-841	394	4	.	.	PROPN
ejpam-841	394	5	math	math	PROPN
ejpam-841	394	6	,	,	PUNCT
ejpam-841	394	7	3	3	NUM
ejpam-841	394	8	(	(	PUNCT
ejpam-841	394	9	2010	2010	NUM
ejpam-841	394	10	)	)	PUNCT
ejpam-841	394	11	,	,	PUNCT
ejpam-841	394	12	602	602	NUM
ejpam-841	394	13	-	-	SYM
ejpam-841	394	14	632	632	NUM
ejpam-841	394	15	615	615	NUM
ejpam-841	394	16	3	3	NUM
ejpam-841	394	17	.	.	PUNCT
ejpam-841	395	1	d(d−3	d(d−3	NOUN
ejpam-841	395	2	)	)	PUNCT
ejpam-841	395	3	2	2	NUM
ejpam-841	395	4	moving	move	VERB
ejpam-841	395	5	planes	plane	NOUN
ejpam-841	395	6	:	:	PUNCT
ejpam-841	395	7	x	x	PUNCT
ejpam-841	395	8	jz	jz	PROPN
ejpam-841	395	9	i−	i−	PROPN
ejpam-841	395	10	j	j	PROPN
ejpam-841	395	11	r	r	NOUN
ejpam-841	395	12	′j	′j	NOUN
ejpam-841	395	13	,	,	PUNCT
ejpam-841	395	14	i−	i−	PROPN
ejpam-841	395	15	j	j	PROPN
ejpam-841	395	16	(	(	PUNCT
ejpam-841	395	17	y	y	PROPN
ejpam-841	395	18	x	x	PROPN
ejpam-841	395	19	,	,	PUNCT
ejpam-841	395	20	w	w	PROPN
ejpam-841	395	21	z	z	NOUN
ejpam-841	395	22	)	)	PUNCT
ejpam-841	395	23	of	of	ADP
ejpam-841	395	24	degree	degree	NOUN
ejpam-841	395	25	(	(	PUNCT
ejpam-841	395	26	d−2−	d−2−	PROPN
ejpam-841	395	27	i	i	PROPN
ejpam-841	395	28	,	,	PUNCT
ejpam-841	395	29	i+1	i+1	VERB
ejpam-841	395	30	)	)	PUNCT
ejpam-841	395	31	for	for	ADP
ejpam-841	395	32	all	all	DET
ejpam-841	395	33	i	i	PRON
ejpam-841	395	34	=	=	NOUN
ejpam-841	395	35	1	1	NUM
ejpam-841	395	36	,	,	PUNCT
ejpam-841	395	37	,	,	PUNCT
ejpam-841	395	38	.	.	PUNCT
ejpam-841	395	39	.	.	PUNCT
ejpam-841	396	1	.	.	PUNCT
ejpam-841	397	1	,	,	PUNCT
ejpam-841	397	2	d−3	d−3	PROPN
ejpam-841	397	3	and	and	CCONJ
ejpam-841	397	4	0≤	0≤	NUM
ejpam-841	397	5	j	j	PROPN
ejpam-841	397	6	≤	≤	PROPN
ejpam-841	398	1	i	i	PRON
ejpam-841	398	2	;	;	PUNCT
ejpam-841	398	3	where	where	SCONJ
ejpam-841	398	4	r	r	NOUN
ejpam-841	398	5	′i	′i	NOUN
ejpam-841	398	6	(	(	PUNCT
ejpam-841	398	7	y	y	NOUN
ejpam-841	398	8	x	x	X
ejpam-841	398	9	,	,	PUNCT
ejpam-841	398	10	w	w	PROPN
ejpam-841	398	11	z	z	NOUN
ejpam-841	398	12	)	)	PUNCT
ejpam-841	398	13	and	and	CCONJ
ejpam-841	398	14	r	r	NOUN
ejpam-841	398	15	′j	′j	NOUN
ejpam-841	398	16	,	,	PUNCT
ejpam-841	398	17	i−	i−	PROPN
ejpam-841	398	18	j	j	PROPN
ejpam-841	398	19	(	(	PUNCT
ejpam-841	398	20	y	y	PROPN
ejpam-841	398	21	x	x	PROPN
ejpam-841	398	22	,	,	PUNCT
ejpam-841	398	23	w	w	PROPN
ejpam-841	398	24	z	z	NOUN
ejpam-841	398	25	)	)	PUNCT
ejpam-841	398	26	are	be	AUX
ejpam-841	398	27	defined	define	VERB
ejpam-841	398	28	in	in	ADP
ejpam-841	398	29	equations	equation	NOUN
ejpam-841	398	30	(	(	PUNCT
ejpam-841	398	31	8)	8)	NUM
ejpam-841	398	32	and	and	CCONJ
ejpam-841	398	33	(	(	PUNCT
ejpam-841	398	34	9	9	NUM
ejpam-841	398	35	)	)	PUNCT
ejpam-841	398	36	.	.	PUNCT
ejpam-841	399	1	proof	proof	NOUN
ejpam-841	399	2	.	.	PUNCT
ejpam-841	400	1	we	we	PRON
ejpam-841	400	2	will	will	AUX
ejpam-841	400	3	apply	apply	VERB
ejpam-841	400	4	the	the	DET
ejpam-841	400	5	results	result	NOUN
ejpam-841	400	6	of	of	ADP
ejpam-841	400	7	kustin	kustin	NOUN
ejpam-841	400	8	,	,	PUNCT
ejpam-841	400	9	polini	polini	NOUN
ejpam-841	400	10	and	and	CCONJ
ejpam-841	400	11	ulrich	ulrich	PROPN
ejpam-841	400	12	[	[	X
ejpam-841	400	13	18	18	NUM
ejpam-841	400	14	]	]	PUNCT
ejpam-841	400	15	,	,	PUNCT
ejpam-841	400	16	by	by	ADP
ejpam-841	400	17	listing	list	VERB
ejpam-841	400	18	the	the	DET
ejpam-841	400	19	set	set	NOUN
ejpam-841	400	20	of	of	ADP
ejpam-841	400	21	minimal	minimal	ADJ
ejpam-841	400	22	generators	generator	NOUN
ejpam-841	400	23	in	in	ADP
ejpam-841	400	24	their	their	PRON
ejpam-841	400	25	paper	paper	NOUN
ejpam-841	400	26	,	,	PUNCT
ejpam-841	400	27	and	and	CCONJ
ejpam-841	400	28	comparing	compare	VERB
ejpam-841	400	29	these	these	DET
ejpam-841	400	30	generators	generator	NOUN
ejpam-841	400	31	with	with	ADP
ejpam-841	400	32	the	the	DET
ejpam-841	400	33	generators	generator	NOUN
ejpam-841	400	34	listed	list	VERB
ejpam-841	400	35	in	in	ADP
ejpam-841	400	36	our	our	PRON
ejpam-841	400	37	theorem	theorem	NOUN
ejpam-841	400	38	.	.	PUNCT
ejpam-841	401	1	first	first	ADV
ejpam-841	401	2	,	,	PUNCT
ejpam-841	401	3	we	we	PRON
ejpam-841	401	4	note	note	VERB
ejpam-841	401	5	that	that	SCONJ
ejpam-841	401	6	the	the	DET
ejpam-841	401	7	notation	notation	NOUN
ejpam-841	401	8	x	x	PUNCT
ejpam-841	401	9	,	,	PUNCT
ejpam-841	401	10	y	y	PROPN
ejpam-841	401	11	,	,	PUNCT
ejpam-841	401	12	z	z	PROPN
ejpam-841	401	13	,	,	PUNCT
ejpam-841	401	14	w	w	PROPN
ejpam-841	401	15	,	,	PUNCT
ejpam-841	401	16	t	t	PROPN
ejpam-841	401	17	,	,	PUNCT
ejpam-841	401	18	s	s	PROPN
ejpam-841	401	19	,	,	PUNCT
ejpam-841	401	20	ri	ri	PROPN
ejpam-841	401	21	,	,	PUNCT
ejpam-841	401	22	i	i	PRON
ejpam-841	401	23	=	=	NOUN
ejpam-841	401	24	0	0	NUM
ejpam-841	401	25	,	,	PUNCT
ejpam-841	401	26	.	.	PUNCT
ejpam-841	401	27	.	.	PUNCT
ejpam-841	401	28	.	.	PUNCT
ejpam-841	402	1	,	,	PUNCT
ejpam-841	403	1	d	d	X
ejpam-841	403	2	−	−	PROPN
ejpam-841	403	3	2	2	NUM
ejpam-841	403	4	in	in	ADP
ejpam-841	403	5	this	this	DET
ejpam-841	403	6	paper	paper	NOUN
ejpam-841	403	7	is	be	AUX
ejpam-841	403	8	the	the	DET
ejpam-841	403	9	same	same	ADJ
ejpam-841	403	10	as	as	ADP
ejpam-841	403	11	t1	t1	NOUN
ejpam-841	403	12	,	,	PUNCT
ejpam-841	403	13	t2	t2	NOUN
ejpam-841	403	14	,	,	PUNCT
ejpam-841	403	15	t3	t3	PROPN
ejpam-841	403	16	,	,	PUNCT
ejpam-841	403	17	t4	t4	PROPN
ejpam-841	403	18	,	,	PUNCT
ejpam-841	403	19	x	x	INTJ
ejpam-841	403	20	,	,	PUNCT
ejpam-841	403	21	y	y	PROPN
ejpam-841	403	22	,	,	PUNCT
ejpam-841	403	23	ci	ci	PROPN
ejpam-841	403	24	,	,	PUNCT
ejpam-841	403	25	i	i	PRON
ejpam-841	403	26	=	=	NOUN
ejpam-841	403	27	0	0	NUM
ejpam-841	403	28	,	,	PUNCT
ejpam-841	403	29	.	.	PUNCT
ejpam-841	403	30	.	.	PUNCT
ejpam-841	404	1	.	.	PUNCT
ejpam-841	405	1	,	,	PUNCT
ejpam-841	405	2	d−2	d−2	PROPN
ejpam-841	405	3	in	in	ADP
ejpam-841	405	4	their	their	PRON
ejpam-841	405	5	notation	notation	NOUN
ejpam-841	405	6	.	.	PUNCT
ejpam-841	406	1	moreover	moreover	ADV
ejpam-841	406	2	,	,	PUNCT
ejpam-841	406	3	in	in	ADP
ejpam-841	406	4	our	our	PRON
ejpam-841	406	5	setting	setting	NOUN
ejpam-841	406	6	,	,	PUNCT
ejpam-841	406	7	we	we	PRON
ejpam-841	406	8	identify	identify	VERB
ejpam-841	406	9	the	the	DET
ejpam-841	406	10	following	follow	VERB
ejpam-841	406	11	items	item	NOUN
ejpam-841	406	12	in	in	ADP
ejpam-841	406	13	their	their	PRON
ejpam-841	406	14	paper	paper	NOUN
ejpam-841	406	15	for	for	ADP
ejpam-841	406	16	non	non	ADJ
ejpam-841	406	17	-	-	ADJ
ejpam-841	406	18	singular	singular	ADJ
ejpam-841	406	19	curves	curve	NOUN
ejpam-841	406	20	:	:	PUNCT
ejpam-841	406	21	ρ	ρ	PROPN
ejpam-841	406	22	=	=	SYM
ejpam-841	406	23	2	2	NUM
ejpam-841	406	24	,	,	PUNCT
ejpam-841	406	25	ℓ=	ℓ=	NOUN
ejpam-841	406	26	3	3	NUM
ejpam-841	406	27	,	,	PUNCT
ejpam-841	407	1	σ1	σ1	PROPN
ejpam-841	407	2	=	=	PROPN
ejpam-841	407	3	σ2	σ2	PROPN
ejpam-841	407	4	=	=	PROPN
ejpam-841	407	5	σ3	σ3	PROPN
ejpam-841	407	6	=	=	NOUN
ejpam-841	407	7	1	1	NUM
ejpam-841	407	8	,	,	PUNCT
ejpam-841	407	9	[	[	X
ejpam-841	407	10	t3,1	t3,1	PROPN
ejpam-841	407	11	,	,	PUNCT
ejpam-841	407	12	t3,2	t3,2	NOUN
ejpam-841	407	13	;	;	PUNCT
ejpam-841	407	14	t1,1	t1,1	NUM
ejpam-841	407	15	,	,	PUNCT
ejpam-841	407	16	t1,2	t1,2	ADJ
ejpam-841	407	17	,	,	PUNCT
ejpam-841	407	18	t1,3	t1,3	NOUN
ejpam-841	407	19	;	;	PUNCT
ejpam-841	407	20	t2,1	t2,1	PROPN
ejpam-841	407	21	,	,	PUNCT
ejpam-841	407	22	t2,2	t2,2	PROPN
ejpam-841	407	23	]	]	X
ejpam-841	408	1	=	=	PUNCT
ejpam-841	409	1	[	[	X
ejpam-841	409	2	s	s	X
ejpam-841	409	3	,	,	PUNCT
ejpam-841	409	4	t	t	PROPN
ejpam-841	409	5	;	;	PUNCT
ejpam-841	409	6	x	x	X
ejpam-841	409	7	,	,	PUNCT
ejpam-841	409	8	y	y	PROPN
ejpam-841	409	9	,	,	PUNCT
ejpam-841	409	10	z	z	PROPN
ejpam-841	409	11	;	;	PUNCT
ejpam-841	409	12	z	z	X
ejpam-841	409	13	,	,	PUNCT
ejpam-841	409	14	w	w	PROPN
ejpam-841	409	15	]	]	PUNCT
ejpam-841	409	16	.	.	PUNCT
ejpam-841	410	1	by	by	ADP
ejpam-841	410	2	theorem	theorem	NOUN
ejpam-841	410	3	3.2	3.2	NUM
ejpam-841	410	4	in	in	ADP
ejpam-841	410	5	[	[	X
ejpam-841	410	6	18	18	NUM
ejpam-841	410	7	]	]	PUNCT
ejpam-841	410	8	,	,	PUNCT
ejpam-841	410	9	we	we	PRON
ejpam-841	410	10	have	have	AUX
ejpam-841	410	11	a=	a=	VERB
ejpam-841	410	12	(	(	PUNCT
ejpam-841	410	13	a1	a1	NOUN
ejpam-841	410	14	,	,	PUNCT
ejpam-841	410	15	a2	a2	PROPN
ejpam-841	410	16	)	)	PUNCT
ejpam-841	410	17	,	,	PUNCT
ejpam-841	410	18	0≤	0≤	NUM
ejpam-841	410	19	a1	a1	NOUN
ejpam-841	410	20	+	+	CCONJ
ejpam-841	410	21	a2	a2	PROPN
ejpam-841	410	22	≤	≤	PROPN
ejpam-841	411	1	d	d	ADP
ejpam-841	411	2	−	−	PROPN
ejpam-841	411	3	3	3	NUM
ejpam-841	411	4	;	;	PUNCT
ejpam-841	411	5	f	f	PROPN
ejpam-841	411	6	(	(	PUNCT
ejpam-841	411	7	a1	a1	PROPN
ejpam-841	411	8	)	)	PUNCT
ejpam-841	411	9	=	=	SYM
ejpam-841	412	1	d	d	ADP
ejpam-841	412	2	−	−	PROPN
ejpam-841	412	3	3−	3−	NUM
ejpam-841	412	4	a1	a1	NOUN
ejpam-841	412	5	;	;	PUNCT
ejpam-841	412	6	f	f	PROPN
ejpam-841	412	7	(	(	PUNCT
ejpam-841	412	8	a1	a1	PROPN
ejpam-841	412	9	,	,	PUNCT
ejpam-841	412	10	a2	a2	NOUN
ejpam-841	412	11	)	)	PUNCT
ejpam-841	412	12	=	=	PUNCT
ejpam-841	413	1	d	d	ADP
ejpam-841	413	2	−	−	NUM
ejpam-841	413	3	3−	3−	NUM
ejpam-841	413	4	a1−	a1−	PROPN
ejpam-841	413	5	a2	a2	PROPN
ejpam-841	413	6	;	;	PUNCT
ejpam-841	413	7	r(a1	r(a1	NOUN
ejpam-841	413	8	)	)	PUNCT
ejpam-841	413	9	=	=	SYM
ejpam-841	413	10	1	1	NUM
ejpam-841	413	11	;	;	PUNCT
ejpam-841	413	12	r(a1	r(a1	NOUN
ejpam-841	413	13	,	,	PUNCT
ejpam-841	413	14	a2	a2	NOUN
ejpam-841	413	15	)	)	PUNCT
ejpam-841	413	16	=	=	SYM
ejpam-841	414	1	1	1	NUM
ejpam-841	414	2	;	;	PUNCT
ejpam-841	414	3	f	f	PROPN
ejpam-841	414	4	(	(	PUNCT
ejpam-841	414	5	;)	;)	PUNCT
ejpam-841	414	6	=	=	PUNCT
ejpam-841	415	1	d	d	NOUN
ejpam-841	415	2	−	−	PROPN
ejpam-841	415	3	3	3	NUM
ejpam-841	415	4	;	;	PUNCT
ejpam-841	415	5	r	r	NOUN
ejpam-841	415	6	(;	(;	X
ejpam-841	415	7	)	)	PUNCT
ejpam-841	415	8	=	=	SYM
ejpam-841	416	1	1	1	NUM
ejpam-841	416	2	;	;	PUNCT
ejpam-841	416	3	t	t	NOUN
ejpam-841	416	4	;	;	PUNCT
ejpam-841	416	5	=	=	SYM
ejpam-841	416	6	1	1	X
ejpam-841	416	7	.	.	PUNCT
ejpam-841	416	8	by	by	ADP
ejpam-841	416	9	definition	definition	NOUN
ejpam-841	416	10	3.5	3.5	NUM
ejpam-841	416	11	and	and	CCONJ
ejpam-841	416	12	the	the	DET
ejpam-841	416	13	description	description	NOUN
ejpam-841	416	14	in	in	ADP
ejpam-841	416	15	[	[	X
ejpam-841	416	16	18	18	NUM
ejpam-841	416	17	]	]	PUNCT
ejpam-841	416	18	,	,	PUNCT
ejpam-841	416	19	the	the	DET
ejpam-841	416	20	generators	generator	NOUN
ejpam-841	416	21	for	for	ADP
ejpam-841	416	22	the	the	DET
ejpam-841	416	23	rees	rees	PROPN
ejpam-841	416	24	algebra	algebra	NOUN
ejpam-841	416	25	are	be	AUX
ejpam-841	416	26	p	p	NOUN
ejpam-841	416	27	,	,	PUNCT
ejpam-841	416	28	q	q	NOUN
ejpam-841	416	29	,	,	PUNCT
ejpam-841	416	30	sylvs	sylvs	NOUN
ejpam-841	416	31	,	,	PUNCT
ejpam-841	416	32	t(p	t(p	PROPN
ejpam-841	416	33	,	,	PUNCT
ejpam-841	416	34	q	q	NOUN
ejpam-841	416	35	)	)	PUNCT
ejpam-841	416	36	,	,	PUNCT
ejpam-841	416	37	f1	f1	NOUN
ejpam-841	416	38	,	,	PUNCT
ejpam-841	416	39	ga1,1	ga1,1	NOUN
ejpam-841	416	40	,	,	PUNCT
ejpam-841	416	41	and	and	CCONJ
ejpam-841	416	42	h(a1	h(a1	NOUN
ejpam-841	416	43	,	,	PUNCT
ejpam-841	416	44	a2	a2	PROPN
ejpam-841	416	45	)	)	PUNCT
ejpam-841	416	46	.	.	PUNCT
ejpam-841	417	1	we	we	PRON
ejpam-841	417	2	shall	shall	AUX
ejpam-841	417	3	now	now	ADV
ejpam-841	417	4	write	write	VERB
ejpam-841	417	5	f1	f1	NOUN
ejpam-841	417	6	,	,	PUNCT
ejpam-841	417	7	ga1,1	ga1,1	NOUN
ejpam-841	417	8	,	,	PUNCT
ejpam-841	417	9	and	and	CCONJ
ejpam-841	417	10	h(a1	h(a1	NOUN
ejpam-841	417	11	,	,	PUNCT
ejpam-841	417	12	a2	a2	PROPN
ejpam-841	417	13	)	)	PUNCT
ejpam-841	417	14	explicitly	explicitly	ADV
ejpam-841	417	15	and	and	CCONJ
ejpam-841	417	16	compare	compare	VERB
ejpam-841	417	17	these	these	DET
ejpam-841	417	18	expressions	expression	NOUN
ejpam-841	417	19	with	with	ADP
ejpam-841	417	20	the	the	DET
ejpam-841	417	21	generators	generator	NOUN
ejpam-841	417	22	listed	list	VERB
ejpam-841	417	23	in	in	ADP
ejpam-841	417	24	the	the	DET
ejpam-841	417	25	statement	statement	NOUN
ejpam-841	417	26	of	of	ADP
ejpam-841	417	27	our	our	PRON
ejpam-841	417	28	theorem	theorem	NOUN
ejpam-841	417	29	.	.	PUNCT
ejpam-841	418	1	we	we	PRON
ejpam-841	418	2	have	have	VERB
ejpam-841	418	3	f1	f1	NOUN
ejpam-841	418	4	=	=	SYM
ejpam-841	418	5	y	y	PROPN
ejpam-841	418	6	∑	∑	PROPN
ejpam-841	418	7	i+	i+	PROPN
ejpam-841	418	8	j	j	PROPN
ejpam-841	418	9	=	=	PROPN
ejpam-841	418	10	d−4	d−4	PROPN
ejpam-841	418	11	x	x	PROPN
ejpam-841	418	12	i	i	PRON
ejpam-841	418	13	y	y	PROPN
ejpam-841	418	14	j(ri	j(ri	PROPN
ejpam-841	418	15	y	y	PROPN
ejpam-841	418	16	)	)	PUNCT
ejpam-841	419	1	+	+	CCONJ
ejpam-841	419	2	x	x	SYM
ejpam-841	419	3	d−3(rd−2	d−3(rd−2	NOUN
ejpam-841	419	4	x	x	PUNCT
ejpam-841	420	1	+	+	CCONJ
ejpam-841	420	2	rd−3	rd−3	PROPN
ejpam-841	420	3	y	y	NOUN
ejpam-841	420	4	)	)	PUNCT
ejpam-841	420	5	=	=	SYM
ejpam-841	420	6	d−4	d−4	PROPN
ejpam-841	420	7	∑	∑	PUNCT
ejpam-841	420	8	i=0	i=0	PROPN
ejpam-841	420	9	ri	ri	X
ejpam-841	420	10	x	x	PUNCT
ejpam-841	420	11	i	i	PRON
ejpam-841	420	12	yd−2−i	yd−2−i	VERB
ejpam-841	420	13	+	+	CCONJ
ejpam-841	420	14	rd−2	rd−2	NOUN
ejpam-841	420	15	x	x	PUNCT
ejpam-841	421	1	d−2	d−2	PROPN
ejpam-841	421	2	+	+	X
ejpam-841	421	3	rd−3	rd−3	NOUN
ejpam-841	421	4	x	x	SYM
ejpam-841	421	5	d−3	d−3	PROPN
ejpam-841	421	6	y	y	NOUN
ejpam-841	421	7	=	=	PUNCT
ejpam-841	421	8	x	x	PUNCT
ejpam-841	421	9	d−2	d−2	NOUN
ejpam-841	421	10	d−2	d−2	PROPN
ejpam-841	421	11	∑	∑	PUNCT
ejpam-841	421	12	i=0	i=0	PROPN
ejpam-841	421	13	ri	ri	PROPN
ejpam-841	421	14	(	(	PUNCT
ejpam-841	421	15	y	y	PROPN
ejpam-841	421	16	x	x	PROPN
ejpam-841	421	17	)	)	PUNCT
ejpam-841	421	18	d−2−i	d−2−i	NOUN
ejpam-841	422	1	=	=	NOUN
ejpam-841	422	2	x	x	SYM
ejpam-841	422	3	d−2r	d−2r	ADP
ejpam-841	422	4	′d−2	′d−2	PROPN
ejpam-841	422	5	(	(	PUNCT
ejpam-841	422	6	y	y	PROPN
ejpam-841	422	7	x	x	X
ejpam-841	422	8	,	,	PUNCT
ejpam-841	422	9	w	w	PROPN
ejpam-841	422	10	z	z	NOUN
ejpam-841	422	11	)	)	PUNCT
ejpam-841	422	12	.	.	PUNCT
ejpam-841	423	1	hence	hence	ADV
ejpam-841	423	2	f1	f1	PROPN
ejpam-841	423	3	is	be	AUX
ejpam-841	423	4	one	one	NUM
ejpam-841	423	5	of	of	ADP
ejpam-841	423	6	our	our	PRON
ejpam-841	423	7	implicit	implicit	ADJ
ejpam-841	423	8	equations	equation	NOUN
ejpam-841	423	9	and	and	CCONJ
ejpam-841	423	10	deg	deg	PROPN
ejpam-841	423	11	(	(	PUNCT
ejpam-841	423	12	f1	f1	NOUN
ejpam-841	423	13	)	)	PUNCT
ejpam-841	423	14	=	=	SYM
ejpam-841	424	1	(	(	PUNCT
ejpam-841	424	2	0	0	NUM
ejpam-841	424	3	,	,	PUNCT
ejpam-841	424	4	d	d	NOUN
ejpam-841	424	5	−	−	PROPN
ejpam-841	424	6	1	1	NUM
ejpam-841	424	7	)	)	PUNCT
ejpam-841	424	8	.	.	PUNCT
ejpam-841	425	1	ga1,1	ga1,1	NOUN
ejpam-841	425	2	=	=	PUNCT
ejpam-841	425	3	wd−2−a1	wd−2−a1	VERB
ejpam-841	425	4	∑	∑	PUNCT
ejpam-841	425	5	i+	i+	PROPN
ejpam-841	425	6	j	j	X
ejpam-841	425	7	=	=	VERB
ejpam-841	425	8	a1−1	a1−1	ADJ
ejpam-841	425	9	x	x	VERB
ejpam-841	426	1	i	i	PRON
ejpam-841	426	2	y	y	PROPN
ejpam-841	426	3	j(ri	j(ri	PROPN
ejpam-841	426	4	y	y	PROPN
ejpam-841	426	5	)	)	PUNCT
ejpam-841	427	1	+	+	CCONJ
ejpam-841	427	2	x	x	PUNCT
ejpam-841	427	3	a1	a1	NOUN
ejpam-841	427	4	w	w	PROPN
ejpam-841	427	5	×	×	PROPN
ejpam-841	427	6	∑	∑	PUNCT
ejpam-841	427	7	i+	i+	PROPN
ejpam-841	427	8	j	j	NOUN
ejpam-841	427	9	=	=	NOUN
ejpam-841	427	10	d−4−a1	d−4−a1	X
ejpam-841	427	11	z	z	PROPN
ejpam-841	427	12	iw	iw	PROPN
ejpam-841	428	1	j	j	PROPN
ejpam-841	429	1	ra1+iw	ra1+iw	NOUN
ejpam-841	430	1	+	+	NOUN
ejpam-841	430	2	x	x	SYM
ejpam-841	430	3	a1zd−3−a1	a1zd−3−a1	PROPN
ejpam-841	430	4	(	(	PUNCT
ejpam-841	430	5	rd−2z	rd−2z	NOUN
ejpam-841	430	6	+	+	CCONJ
ejpam-841	430	7	rd−3w	rd−3w	NOUN
ejpam-841	430	8	)	)	PUNCT
ejpam-841	430	9	=	=	NOUN
ejpam-841	430	10	a1−1	a1−1	PROPN
ejpam-841	430	11	∑	∑	PUNCT
ejpam-841	430	12	i=0	i=0	PROPN
ejpam-841	430	13	ri	ri	X
ejpam-841	431	1	x	x	PUNCT
ejpam-841	431	2	i	i	PRON
ejpam-841	431	3	ya1−iwd−2−a1	ya1−iwd−2−a1	NOUN
ejpam-841	431	4	+	+	CCONJ
ejpam-841	431	5	d−4−a1	d−4−a1	NOUN
ejpam-841	431	6	∑	∑	PUNCT
ejpam-841	431	7	i=0	i=0	PROPN
ejpam-841	431	8	ra1+i	ra1+i	VERB
ejpam-841	431	9	x	x	PUNCT
ejpam-841	431	10	a1z	a1z	PROPN
ejpam-841	431	11	iwd−2−a1−i	iwd−2−a1−i	NOUN
ejpam-841	431	12	+	+	CCONJ
ejpam-841	431	13	rd−2	rd−2	PROPN
ejpam-841	431	14	x	x	PUNCT
ejpam-841	431	15	a1zd−2−a1	a1zd−2−a1	CCONJ
ejpam-841	432	1	+	+	CCONJ
ejpam-841	432	2	rd−3	rd−3	NOUN
ejpam-841	432	3	x	x	X
ejpam-841	432	4	a1zd−3−a1	a1zd−3−a1	PROPN
ejpam-841	433	1	w	w	NOUN
ejpam-841	433	2	=	=	NOUN
ejpam-841	433	3	x	x	SYM
ejpam-841	433	4	a1zd−2−a1	a1zd−2−a1	ADP
ejpam-841	433	5	[	[	PUNCT
ejpam-841	433	6	a1−1	a1−1	PROPN
ejpam-841	433	7	∑	∑	PUNCT
ejpam-841	433	8	i=0	i=0	PROPN
ejpam-841	433	9	ri	ri	PROPN
ejpam-841	433	10	(	(	PUNCT
ejpam-841	433	11	w	w	PROPN
ejpam-841	433	12	z	z	NOUN
ejpam-841	433	13	)	)	PUNCT
ejpam-841	433	14	d−2−a1	d−2−a1	NOUN
ejpam-841	433	15	(	(	PUNCT
ejpam-841	433	16	y	y	NOUN
ejpam-841	433	17	x	x	PROPN
ejpam-841	433	18	)	)	PUNCT
ejpam-841	434	1	a1−i	a1−i	PROPN
ejpam-841	434	2	+	+	CCONJ
ejpam-841	434	3	d−2−a1	d−2−a1	NOUN
ejpam-841	434	4	∑	∑	PUNCT
ejpam-841	434	5	i=0	i=0	ADJ
ejpam-841	434	6	ra1+i	ra1+i	X
ejpam-841	434	7	(	(	PUNCT
ejpam-841	434	8	w	w	PROPN
ejpam-841	434	9	z	z	NOUN
ejpam-841	434	10	)	)	PUNCT
ejpam-841	434	11	d−2−a1−i	d−2−a1−i	ADP
ejpam-841	434	12	]	]	X
ejpam-841	435	1	=	=	PUNCT
ejpam-841	435	2	x	x	SYM
ejpam-841	435	3	a1zd−2−a1	a1zd−2−a1	ADP
ejpam-841	435	4	r	r	NOUN
ejpam-841	435	5	′a1	′a1	NOUN
ejpam-841	435	6	(	(	PUNCT
ejpam-841	435	7	y	y	NOUN
ejpam-841	435	8	x	x	X
ejpam-841	435	9	,	,	PUNCT
ejpam-841	435	10	w	w	PROPN
ejpam-841	435	11	z	z	NOUN
ejpam-841	435	12	)	)	PUNCT
ejpam-841	435	13	.	.	PUNCT
ejpam-841	436	1	j.	j.	PROPN
ejpam-841	436	2	hoffman	hoffman	PROPN
ejpam-841	436	3	,	,	PUNCT
ejpam-841	436	4	h.	h.	PROPN
ejpam-841	436	5	wang	wang	PROPN
ejpam-841	436	6	,	,	PUNCT
ejpam-841	436	7	x.	x.	PROPN
ejpam-841	436	8	jia	jia	PROPN
ejpam-841	436	9	,	,	PUNCT
ejpam-841	436	10	r.	r.	PROPN
ejpam-841	436	11	goldman	goldman	PROPN
ejpam-841	436	12	/	/	SYM
ejpam-841	436	13	eur	eur	PROPN
ejpam-841	436	14	.	.	PUNCT
ejpam-841	437	1	j.	j.	PROPN
ejpam-841	437	2	pure	pure	PROPN
ejpam-841	437	3	appl	appl	PROPN
ejpam-841	437	4	.	.	PROPN
ejpam-841	437	5	math	math	PROPN
ejpam-841	437	6	,	,	PUNCT
ejpam-841	437	7	3	3	NUM
ejpam-841	437	8	(	(	PUNCT
ejpam-841	437	9	2010	2010	NUM
ejpam-841	437	10	)	)	PUNCT
ejpam-841	437	11	,	,	PUNCT
ejpam-841	437	12	602	602	NUM
ejpam-841	437	13	-	-	SYM
ejpam-841	437	14	632	632	NUM
ejpam-841	437	15	616	616	NUM
ejpam-841	437	16	hence	hence	ADV
ejpam-841	437	17	each	each	DET
ejpam-841	437	18	ga1,1	ga1,1	NOUN
ejpam-841	437	19	is	be	AUX
ejpam-841	437	20	one	one	NUM
ejpam-841	437	21	of	of	ADP
ejpam-841	437	22	our	our	PRON
ejpam-841	437	23	implicit	implicit	ADJ
ejpam-841	437	24	equations	equation	NOUN
ejpam-841	437	25	and	and	CCONJ
ejpam-841	437	26	deg(ga1,1	deg(ga1,1	NOUN
ejpam-841	437	27	)	)	PUNCT
ejpam-841	437	28	=	=	SYM
ejpam-841	438	1	(	(	PUNCT
ejpam-841	438	2	0	0	NUM
ejpam-841	438	3	,	,	PUNCT
ejpam-841	438	4	d−1	d−1	PROPN
ejpam-841	438	5	)	)	PUNCT
ejpam-841	438	6	,	,	PUNCT
ejpam-841	438	7	a1	a1	NOUN
ejpam-841	438	8	=	=	SYM
ejpam-841	438	9	0,1	0,1	NUM
ejpam-841	438	10	,	,	PUNCT
ejpam-841	438	11	.	.	PUNCT
ejpam-841	438	12	.	.	PUNCT
ejpam-841	439	1	.	.	PUNCT
ejpam-841	440	1	,	,	PUNCT
ejpam-841	440	2	d−	d−	PROPN
ejpam-841	440	3	3	3	NUM
ejpam-841	440	4	.	.	PUNCT
ejpam-841	440	5	ha1,a2	ha1,a2	NOUN
ejpam-841	441	1	=	=	PUNCT
ejpam-841	441	2	td−2−a1−a2[wa2	td−2−a1−a2[wa2	PROPN
ejpam-841	441	3	∑	∑	PUNCT
ejpam-841	441	4	i+	i+	ADV
ejpam-841	441	5	j	j	X
ejpam-841	441	6	=	=	VERB
ejpam-841	441	7	a1−1	a1−1	ADJ
ejpam-841	441	8	x	x	VERB
ejpam-841	442	1	i	i	PRON
ejpam-841	442	2	y	y	PROPN
ejpam-841	442	3	j(ri	j(ri	PROPN
ejpam-841	442	4	y	y	PROPN
ejpam-841	442	5	)	)	PUNCT
ejpam-841	443	1	+	+	CCONJ
ejpam-841	443	2	x	x	X
ejpam-841	443	3	a1	a1	VERB
ejpam-841	443	4	∑	∑	PROPN
ejpam-841	443	5	i+	i+	PROPN
ejpam-841	443	6	j	j	NOUN
ejpam-841	444	1	=	=	PROPN
ejpam-841	445	1	a2−1	a2−1	NOUN
ejpam-841	445	2	z	z	PROPN
ejpam-841	445	3	iw	iw	PROPN
ejpam-841	445	4	j	j	PROPN
ejpam-841	446	1	ra1+iw	ra1+iw	PROPN
ejpam-841	446	2	]	]	X
ejpam-841	446	3	+	+	NOUN
ejpam-841	446	4	x	x	SYM
ejpam-841	446	5	a1za2	a1za2	X
ejpam-841	446	6	d−2−a1−a2	d−2−a1−a2	PROPN
ejpam-841	446	7	∑	∑	PUNCT
ejpam-841	446	8	i=0	i=0	PROPN
ejpam-841	446	9	ra1+a2+i	ra1+a2+i	PROPN
ejpam-841	446	10	t	t	NOUN
ejpam-841	446	11	d−2−a1−a2−isi	d−2−a1−a2−isi	X
ejpam-841	446	12	=	=	SYM
ejpam-841	446	13	td−2−a1−a2	td−2−a1−a2	X
ejpam-841	446	14	[	[	PUNCT
ejpam-841	446	15	a1−1	a1−1	NOUN
ejpam-841	446	16	∑	∑	PUNCT
ejpam-841	446	17	i=0	i=0	PROPN
ejpam-841	446	18	ri	ri	X
ejpam-841	447	1	x	x	PUNCT
ejpam-841	447	2	i	i	PRON
ejpam-841	447	3	ya1−iwa2	ya1−iwa2	PROPN
ejpam-841	448	1	+	+	CCONJ
ejpam-841	448	2	a2−1	a2−1	ADP
ejpam-841	448	3	∑	∑	ADV
ejpam-841	448	4	i=0	i=0	ADJ
ejpam-841	448	5	ra2+i	ra2+i	NOUN
ejpam-841	448	6	x	x	NOUN
ejpam-841	448	7	a1z	a1z	PROPN
ejpam-841	448	8	iwa2−i	iwa2−i	PROPN
ejpam-841	448	9	]	]	X
ejpam-841	449	1	+	+	NOUN
ejpam-841	449	2	x	x	SYM
ejpam-841	449	3	a1za2	a1za2	X
ejpam-841	449	4	d−2−a1−a2	d−2−a1−a2	PROPN
ejpam-841	449	5	∑	∑	PUNCT
ejpam-841	449	6	i=0	i=0	PROPN
ejpam-841	449	7	ra1+a2+i	ra1+a2+i	PROPN
ejpam-841	449	8	t	t	NOUN
ejpam-841	449	9	d−2−a1−a2−isi	d−2−a1−a2−isi	VERB
ejpam-841	449	10	=	=	SYM
ejpam-841	449	11	x	x	SYM
ejpam-841	449	12	a1za2{td−2−a1−a2	a1za2{td−2−a1−a2	X
ejpam-841	449	13	[	[	PUNCT
ejpam-841	449	14	a1−1	a1−1	PROPN
ejpam-841	449	15	∑	∑	PUNCT
ejpam-841	449	16	i=0	i=0	PROPN
ejpam-841	449	17	ri	ri	PROPN
ejpam-841	449	18	(	(	PUNCT
ejpam-841	449	19	w	w	PROPN
ejpam-841	449	20	z	z	PROPN
ejpam-841	449	21	)	)	PUNCT
ejpam-841	449	22	a2	a2	PROPN
ejpam-841	449	23	(	(	PUNCT
ejpam-841	449	24	y	y	NOUN
ejpam-841	449	25	x	x	PROPN
ejpam-841	449	26	)	)	PUNCT
ejpam-841	449	27	a1−i	a1−i	PROPN
ejpam-841	450	1	+	+	CCONJ
ejpam-841	450	2	a2−1	a2−1	ADP
ejpam-841	450	3	∑	∑	ADV
ejpam-841	450	4	i=0	i=0	ADJ
ejpam-841	450	5	ra2+i	ra2+i	NOUN
ejpam-841	450	6	(	(	PUNCT
ejpam-841	450	7	w	w	PROPN
ejpam-841	450	8	z	z	NOUN
ejpam-841	450	9	)	)	PUNCT
ejpam-841	450	10	a2−i	a2−i	PROPN
ejpam-841	450	11	]	]	PUNCT
ejpam-841	450	12	+	+	CCONJ
ejpam-841	450	13	d−2−a1−a2	d−2−a1−a2	PROPN
ejpam-841	450	14	∑	∑	PUNCT
ejpam-841	450	15	i=0	i=0	ADJ
ejpam-841	450	16	ra1+a2+i	ra1+a2+i	PROPN
ejpam-841	450	17	t	t	NOUN
ejpam-841	450	18	d−2−a1−a2−isi	d−2−a1−a2−isi	PROPN
ejpam-841	450	19	}	}	PUNCT
ejpam-841	450	20	=	=	SYM
ejpam-841	450	21	(	(	PUNCT
ejpam-841	450	22	x	x	SYM
ejpam-841	450	23	a1za2	a1za2	X
ejpam-841	450	24	r	r	NOUN
ejpam-841	450	25	′a1,a2	′a1,a2	NOUN
ejpam-841	450	26	(	(	PUNCT
ejpam-841	450	27	y	y	NOUN
ejpam-841	450	28	x	x	X
ejpam-841	450	29	,	,	PUNCT
ejpam-841	450	30	w	w	PROPN
ejpam-841	450	31	z	z	NOUN
ejpam-841	450	32	)	)	PUNCT
ejpam-841	450	33	,	,	PUNCT
ejpam-841	450	34	a1	a1	NOUN
ejpam-841	450	35	+	+	X
ejpam-841	450	36	a2	a2	PROPN
ejpam-841	450	37	6=	6=	ADP
ejpam-841	450	38	0	0	NUM
ejpam-841	450	39	,	,	PUNCT
ejpam-841	450	40	0	0	NUM
ejpam-841	450	41	<	<	X
ejpam-841	450	42	a1	a1	NOUN
ejpam-841	450	43	+	+	CCONJ
ejpam-841	450	44	a2	a2	PROPN
ejpam-841	450	45	≤	≤	PROPN
ejpam-841	451	1	d	d	ADP
ejpam-841	451	2	−	−	PROPN
ejpam-841	451	3	3	3	NUM
ejpam-841	451	4	,	,	PUNCT
ejpam-841	451	5	r	r	NOUN
ejpam-841	451	6	a1	a1	NOUN
ejpam-841	451	7	+	+	CCONJ
ejpam-841	451	8	a2	a2	PROPN
ejpam-841	451	9	=	=	SYM
ejpam-841	451	10	0	0	X
ejpam-841	451	11	.	.	PUNCT
ejpam-841	452	1	hence	hence	ADV
ejpam-841	452	2	ha1,a2	ha1,a2	NOUN
ejpam-841	452	3	are	be	AUX
ejpam-841	452	4	our	our	PRON
ejpam-841	452	5	moving	move	VERB
ejpam-841	452	6	surfaces	surface	NOUN
ejpam-841	452	7	and	and	CCONJ
ejpam-841	452	8	deg(ha1,a2	deg(ha1,a2	NOUN
ejpam-841	452	9	)	)	PUNCT
ejpam-841	452	10	=	=	PUNCT
ejpam-841	453	1	(	(	PUNCT
ejpam-841	453	2	d	d	NOUN
ejpam-841	453	3	−	−	PROPN
ejpam-841	453	4	2−	2−	NUM
ejpam-841	453	5	a1	a1	NOUN
ejpam-841	453	6	−	−	PROPN
ejpam-841	453	7	a2	a2	PROPN
ejpam-841	453	8	,	,	PUNCT
ejpam-841	453	9	a1	a1	NOUN
ejpam-841	453	10	+	+	CCONJ
ejpam-841	453	11	a2	a2	PROPN
ejpam-841	453	12	+	+	CCONJ
ejpam-841	453	13	1	1	NUM
ejpam-841	453	14	)	)	PUNCT
ejpam-841	453	15	,	,	PUNCT
ejpam-841	453	16	0	0	NUM
ejpam-841	453	17	≤	≤	NOUN
ejpam-841	453	18	a1	a1	NOUN
ejpam-841	453	19	+	+	CCONJ
ejpam-841	453	20	a2	a2	PROPN
ejpam-841	453	21	≤	≤	PROPN
ejpam-841	454	1	d	d	ADP
ejpam-841	454	2	−	−	PROPN
ejpam-841	454	3	3	3	NUM
ejpam-841	454	4	.	.	PUNCT
ejpam-841	454	5	therefore	therefore	ADV
ejpam-841	454	6	,	,	PUNCT
ejpam-841	454	7	the	the	DET
ejpam-841	454	8	generators	generator	NOUN
ejpam-841	454	9	provided	provide	VERB
ejpam-841	454	10	by	by	ADP
ejpam-841	454	11	our	our	PRON
ejpam-841	454	12	theorem	theorem	NOUN
ejpam-841	454	13	are	be	AUX
ejpam-841	454	14	the	the	DET
ejpam-841	454	15	same	same	ADJ
ejpam-841	454	16	as	as	SCONJ
ejpam-841	454	17	the	the	DET
ejpam-841	454	18	generators	generator	NOUN
ejpam-841	454	19	described	describe	VERB
ejpam-841	454	20	in	in	ADP
ejpam-841	454	21	[	[	X
ejpam-841	454	22	18	18	NUM
ejpam-841	454	23	]	]	PUNCT
ejpam-841	454	24	.	.	PUNCT
ejpam-841	455	1	thus	thus	ADV
ejpam-841	455	2	,	,	PUNCT
ejpam-841	455	3	we	we	PRON
ejpam-841	455	4	have	have	AUX
ejpam-841	455	5	proved	prove	VERB
ejpam-841	455	6	our	our	PRON
ejpam-841	455	7	claim	claim	NOUN
ejpam-841	455	8	.	.	PUNCT
ejpam-841	456	1	3.3	3.3	NUM
ejpam-841	456	2	.	.	PUNCT
ejpam-841	456	3	algorithm	algorithm	PROPN
ejpam-841	456	4	based	base	VERB
ejpam-841	456	5	on	on	ADP
ejpam-841	456	6	theorems	theorem	NOUN
ejpam-841	456	7	1	1	NUM
ejpam-841	456	8	,	,	PUNCT
ejpam-841	456	9	2	2	NUM
ejpam-841	456	10	,	,	PUNCT
ejpam-841	456	11	and	and	CCONJ
ejpam-841	456	12	3	3	NUM
ejpam-841	456	13	,	,	PUNCT
ejpam-841	456	14	we	we	PRON
ejpam-841	456	15	now	now	ADV
ejpam-841	456	16	provide	provide	VERB
ejpam-841	456	17	a	a	DET
ejpam-841	456	18	simple	simple	ADJ
ejpam-841	456	19	algorithm	algorithm	NOUN
ejpam-841	456	20	to	to	PART
ejpam-841	456	21	find	find	VERB
ejpam-841	456	22	a	a	DET
ejpam-841	456	23	minimal	minimal	ADJ
ejpam-841	456	24	set	set	NOUN
ejpam-841	456	25	of	of	ADP
ejpam-841	456	26	generators	generator	NOUN
ejpam-841	456	27	for	for	ADP
ejpam-841	456	28	the	the	DET
ejpam-841	456	29	rees	rees	PROPN
ejpam-841	456	30	algebra	algebra	NOUN
ejpam-841	456	31	associated	associate	VERB
ejpam-841	456	32	to	to	ADP
ejpam-841	456	33	rational	rational	ADJ
ejpam-841	456	34	space	space	NOUN
ejpam-841	456	35	curves	curve	NOUN
ejpam-841	456	36	of	of	ADP
ejpam-841	456	37	type	type	NOUN
ejpam-841	456	38	(	(	PUNCT
ejpam-841	456	39	1,1	1,1	NUM
ejpam-841	456	40	,	,	PUNCT
ejpam-841	456	41	d	d	NOUN
ejpam-841	456	42	−	−	PROPN
ejpam-841	456	43	2	2	NUM
ejpam-841	456	44	)	)	PUNCT
ejpam-841	456	45	in	in	ADP
ejpam-841	456	46	projective	projective	ADJ
ejpam-841	456	47	3	3	NUM
ejpam-841	456	48	-	-	PUNCT
ejpam-841	456	49	space	space	NOUN
ejpam-841	456	50	based	base	VERB
ejpam-841	456	51	solely	solely	ADV
ejpam-841	456	52	on	on	ADP
ejpam-841	456	53	a	a	DET
ejpam-841	456	54	µ-basis	µ-basis	NOUN
ejpam-841	456	55	of	of	ADP
ejpam-841	456	56	the	the	DET
ejpam-841	456	57	curve	curve	NOUN
ejpam-841	456	58	.	.	PUNCT
ejpam-841	457	1	algorithm	algorithm	NOUN
ejpam-841	457	2	4	4	NUM
ejpam-841	457	3	.	.	PUNCT
ejpam-841	458	1	given	give	VERB
ejpam-841	458	2	a	a	DET
ejpam-841	458	3	rational	rational	ADJ
ejpam-841	458	4	space	space	NOUN
ejpam-841	458	5	curve	curve	NOUN
ejpam-841	458	6	c	c	PROPN
ejpam-841	458	7	as	as	ADP
ejpam-841	458	8	the	the	DET
ejpam-841	458	9	image	image	NOUN
ejpam-841	458	10	of	of	ADP
ejpam-841	458	11	a	a	DET
ejpam-841	458	12	generic	generic	ADJ
ejpam-841	458	13	1	1	NUM
ejpam-841	458	14	-	-	SYM
ejpam-841	458	15	1	1	NUM
ejpam-841	458	16	rational	rational	ADJ
ejpam-841	458	17	parametrization	parametrization	NOUN
ejpam-841	458	18	f(s	f(s	AUX
ejpam-841	458	19	,	,	PUNCT
ejpam-841	458	20	t	t	PROPN
ejpam-841	458	21	)	)	PUNCT
ejpam-841	458	22	as	as	ADP
ejpam-841	458	23	in	in	ADP
ejpam-841	458	24	equation	equation	NOUN
ejpam-841	458	25	(	(	PUNCT
ejpam-841	458	26	1	1	NUM
ejpam-841	458	27	)	)	PUNCT
ejpam-841	458	28	,	,	PUNCT
ejpam-841	458	29	we	we	PRON
ejpam-841	458	30	compute	compute	VERB
ejpam-841	458	31	a	a	DET
ejpam-841	458	32	minimal	minimal	ADJ
ejpam-841	458	33	set	set	NOUN
ejpam-841	458	34	of	of	ADP
ejpam-841	458	35	generators	generator	NOUN
ejpam-841	458	36	for	for	ADP
ejpam-841	458	37	the	the	DET
ejpam-841	458	38	associated	associated	ADJ
ejpam-841	458	39	rees	rees	PROPN
ejpam-841	458	40	algebra	algebra	NOUN
ejpam-841	458	41	.	.	PUNCT
ejpam-841	459	1	1	1	X
ejpam-841	459	2	.	.	X
ejpam-841	459	3	compute	compute	VERB
ejpam-841	459	4	a	a	DET
ejpam-841	459	5	µ-basis	µ-basis	NOUN
ejpam-841	459	6	p	p	NOUN
ejpam-841	459	7	,	,	PUNCT
ejpam-841	459	8	q	q	NOUN
ejpam-841	459	9	,	,	PUNCT
ejpam-841	459	10	r	r	NOUN
ejpam-841	459	11	,	,	PUNCT
ejpam-841	459	12	and	and	CCONJ
ejpam-841	459	13	write	write	VERB
ejpam-841	459	14	in	in	ADP
ejpam-841	459	15	the	the	DET
ejpam-841	459	16	following	follow	VERB
ejpam-841	459	17	form	form	NOUN
ejpam-841	459	18	:	:	PUNCT
ejpam-841	459	19	p	p	X
ejpam-841	459	20	=	=	PUNCT
ejpam-841	459	21	p1s+	p1s+	PROPN
ejpam-841	459	22	p0	p0	NOUN
ejpam-841	459	23	t	t	PROPN
ejpam-841	459	24	,	,	PUNCT
ejpam-841	459	25	q	q	NOUN
ejpam-841	460	1	=	=	PUNCT
ejpam-841	460	2	q1s+	q1s+	NOUN
ejpam-841	460	3	q0	q0	PROPN
ejpam-841	460	4	t	t	PROPN
ejpam-841	460	5	,	,	PUNCT
ejpam-841	460	6	r	r	NOUN
ejpam-841	460	7	=	=	SYM
ejpam-841	460	8	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	460	9	+	+	NUM
ejpam-841	460	10	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	460	11	t	t	NOUN
ejpam-841	460	12	+	+	CCONJ
ejpam-841	460	13	·	·	PUNCT
ejpam-841	460	14	·	·	PUNCT
ejpam-841	460	15	·	·	PUNCT
ejpam-841	460	16	+	+	NUM
ejpam-841	460	17	r0	r0	NOUN
ejpam-841	460	18	td−2	td−2	NOUN
ejpam-841	460	19	.	.	PUNCT
ejpam-841	461	1	2	2	X
ejpam-841	461	2	.	.	X
ejpam-841	461	3	compute	compute	PROPN
ejpam-841	461	4	v(p1	v(p1	NOUN
ejpam-841	461	5	,	,	PUNCT
ejpam-841	461	6	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	461	7	)	)	PUNCT
ejpam-841	461	8	.	.	PUNCT
ejpam-841	462	1	j.	j.	PROPN
ejpam-841	462	2	hoffman	hoffman	PROPN
ejpam-841	462	3	,	,	PUNCT
ejpam-841	462	4	h.	h.	PROPN
ejpam-841	462	5	wang	wang	PROPN
ejpam-841	462	6	,	,	PUNCT
ejpam-841	462	7	x.	x.	PROPN
ejpam-841	462	8	jia	jia	PROPN
ejpam-841	462	9	,	,	PUNCT
ejpam-841	462	10	r.	r.	PROPN
ejpam-841	462	11	goldman	goldman	PROPN
ejpam-841	462	12	/	/	SYM
ejpam-841	462	13	eur	eur	PROPN
ejpam-841	462	14	.	.	PUNCT
ejpam-841	463	1	j.	j.	PROPN
ejpam-841	463	2	pure	pure	PROPN
ejpam-841	463	3	appl	appl	PROPN
ejpam-841	463	4	.	.	PROPN
ejpam-841	463	5	math	math	PROPN
ejpam-841	463	6	,	,	PUNCT
ejpam-841	463	7	3	3	NUM
ejpam-841	463	8	(	(	PUNCT
ejpam-841	463	9	2010	2010	NUM
ejpam-841	463	10	)	)	PUNCT
ejpam-841	463	11	,	,	PUNCT
ejpam-841	463	12	602	602	NUM
ejpam-841	463	13	-	-	SYM
ejpam-841	463	14	632	632	NUM
ejpam-841	463	15	617	617	NUM
ejpam-841	463	16	3	3	NUM
ejpam-841	463	17	.	.	PUNCT
ejpam-841	464	1	if	if	SCONJ
ejpam-841	464	2	v(p1	v(p1	NOUN
ejpam-841	464	3	,	,	PUNCT
ejpam-841	464	4	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	464	5	)	)	PUNCT
ejpam-841	464	6	6=	6=	NUM
ejpam-841	464	7	;	;	PUNCT
ejpam-841	464	8	,	,	PUNCT
ejpam-841	464	9	and	and	CCONJ
ejpam-841	464	10	if	if	SCONJ
ejpam-841	464	11	p1	p1	PROPN
ejpam-841	464	12	=	=	PUNCT
ejpam-841	464	13	bq1−	bq1−	VERB
ejpam-841	464	14	aq0	aq0	NOUN
ejpam-841	464	15	for	for	ADP
ejpam-841	464	16	some	some	PRON
ejpam-841	464	17	a	a	PRON
ejpam-841	464	18	,	,	PUNCT
ejpam-841	464	19	b	b	NOUN
ejpam-841	464	20	∈k	∈k	NOUN
ejpam-841	464	21	and	and	CCONJ
ejpam-841	464	22	a	a	DET
ejpam-841	464	23	6=	6=	NUM
ejpam-841	464	24	0	0	NUM
ejpam-841	464	25	,	,	PUNCT
ejpam-841	464	26	then	then	ADV
ejpam-841	464	27	let	let	VERB
ejpam-841	464	28	x	x	PUNCT
ejpam-841	464	29	=	=	PUNCT
ejpam-841	464	30	−	−	PROPN
ejpam-841	464	31	p0	p0	NOUN
ejpam-841	464	32	−	−	PROPN
ejpam-841	464	33	bq0	bq0	NOUN
ejpam-841	464	34	a	a	PRON
ejpam-841	464	35	,	,	PUNCT
ejpam-841	464	36	y	y	PROPN
ejpam-841	464	37	=	=	SYM
ejpam-841	464	38	−q0	−q0	PROPN
ejpam-841	464	39	,	,	PUNCT
ejpam-841	464	40	z	z	NOUN
ejpam-841	464	41	=	=	SYM
ejpam-841	464	42	q1	q1	PROPN
ejpam-841	464	43	,	,	PUNCT
ejpam-841	464	44	w	w	PROPN
ejpam-841	464	45	=	=	SYM
ejpam-841	464	46	w	w	NOUN
ejpam-841	464	47	otherwise	otherwise	ADV
ejpam-841	464	48	,	,	PUNCT
ejpam-841	464	49	find	find	VERB
ejpam-841	464	50	non	non	ADJ
ejpam-841	464	51	-	-	ADJ
ejpam-841	464	52	zero	zero	ADJ
ejpam-841	464	53	constant	constant	ADJ
ejpam-841	464	54	a	a	DET
ejpam-841	464	55	,	,	PUNCT
ejpam-841	464	56	b	b	NOUN
ejpam-841	464	57	,	,	PUNCT
ejpam-841	464	58	c	c	NOUN
ejpam-841	464	59	∈k	∈k	PROPN
ejpam-841	464	60	,	,	PUNCT
ejpam-841	465	1	such	such	ADJ
ejpam-841	465	2	that	that	SCONJ
ejpam-841	465	3	p0	p0	NOUN
ejpam-841	465	4	=	=	SYM
ejpam-841	465	5	−aq0	−aq0	PROPN
ejpam-841	465	6	+	+	NUM
ejpam-841	465	7	bq1	bq1	PROPN
ejpam-841	465	8	+	+	X
ejpam-841	465	9	cp1	cp1	ADJ
ejpam-841	465	10	,	,	PUNCT
ejpam-841	465	11	and	and	CCONJ
ejpam-841	465	12	let	let	VERB
ejpam-841	465	13	x	x	PUNCT
ejpam-841	465	14	=	=	SYM
ejpam-841	465	15	(	(	PUNCT
ejpam-841	465	16	ac	ac	PROPN
ejpam-841	465	17	−	−	PROPN
ejpam-841	465	18	b)q0−	b)q0−	PROPN
ejpam-841	465	19	bcq1	bcq1	PROPN
ejpam-841	466	1	−	−	PROPN
ejpam-841	466	2	c2p1	c2p1	PROPN
ejpam-841	466	3	,	,	PUNCT
ejpam-841	466	4	y	y	PROPN
ejpam-841	466	5	=	=	SYM
ejpam-841	466	6	bq1	bq1	PROPN
ejpam-841	466	7	+	+	X
ejpam-841	466	8	cp1	cp1	ADJ
ejpam-841	466	9	,	,	PUNCT
ejpam-841	466	10	z	z	NOUN
ejpam-841	466	11	=	=	SYM
ejpam-841	466	12	−aq1−	−aq1−	NOUN
ejpam-841	466	13	p1	p1	NOUN
ejpam-841	466	14	,	,	PUNCT
ejpam-841	466	15	w	w	PROPN
ejpam-841	466	16	=	=	SYM
ejpam-841	466	17	w	w	PROPN
ejpam-841	466	18	,	,	PUNCT
ejpam-841	466	19	so	so	SCONJ
ejpam-841	466	20	that	that	SCONJ
ejpam-841	466	21	p	p	NOUN
ejpam-841	466	22	=	=	PUNCT
ejpam-841	466	23	y	y	PROPN
ejpam-841	466	24	s−	s−	PROPN
ejpam-841	466	25	x	x	PUNCT
ejpam-841	466	26	t	t	PROPN
ejpam-841	466	27	,	,	PUNCT
ejpam-841	466	28	q	q	NOUN
ejpam-841	466	29	=	=	SYM
ejpam-841	466	30	zs−	zs−	NUM
ejpam-841	466	31	y	y	PROPN
ejpam-841	466	32	t	t	PROPN
ejpam-841	466	33	,	,	PUNCT
ejpam-841	466	34	r	r	NOUN
ejpam-841	466	35	=	=	SYM
ejpam-841	466	36	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	466	37	+	+	NUM
ejpam-841	466	38	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	466	39	t	t	NOUN
ejpam-841	466	40	+	+	CCONJ
ejpam-841	466	41	·	·	PUNCT
ejpam-841	466	42	·	·	PUNCT
ejpam-841	466	43	·	·	PUNCT
ejpam-841	467	1	+	+	NUM
ejpam-841	467	2	r0td−2	r0td−2	NOUN
ejpam-841	467	3	,	,	PUNCT
ejpam-841	467	4	where	where	SCONJ
ejpam-841	467	5	ri	ri	PROPN
ejpam-841	467	6	are	be	AUX
ejpam-841	467	7	linear	linear	ADJ
ejpam-841	467	8	in	in	ADP
ejpam-841	467	9	x	x	SYM
ejpam-841	467	10	,	,	PUNCT
ejpam-841	467	11	y	y	PROPN
ejpam-841	467	12	,	,	PUNCT
ejpam-841	467	13	z	z	PROPN
ejpam-841	467	14	,	,	PUNCT
ejpam-841	467	15	w.	w.	PROPN
ejpam-841	467	16	(	(	PUNCT
ejpam-841	467	17	1	1	NUM
ejpam-841	467	18	)	)	PUNCT
ejpam-841	467	19	if	if	SCONJ
ejpam-841	467	20	d	d	PROPN
ejpam-841	467	21	=	=	SYM
ejpam-841	467	22	2k	2k	NUM
ejpam-841	467	23	,	,	PUNCT
ejpam-841	467	24	then	then	ADV
ejpam-841	467	25	a	a	DET
ejpam-841	467	26	minimal	minimal	ADJ
ejpam-841	467	27	set	set	NOUN
ejpam-841	467	28	of	of	ADP
ejpam-841	467	29	generators	generator	NOUN
ejpam-841	467	30	for	for	ADP
ejpam-841	467	31	the	the	DET
ejpam-841	467	32	associated	associated	ADJ
ejpam-841	467	33	rees	ree	NOUN
ejpam-841	467	34	algebra	algebra	NOUN
ejpam-841	467	35	is	be	AUX
ejpam-841	467	36	p	p	NOUN
ejpam-841	467	37	,	,	PUNCT
ejpam-841	467	38	q	q	ADJ
ejpam-841	467	39	,	,	PUNCT
ejpam-841	467	40	r	r	NOUN
ejpam-841	467	41	,	,	PUNCT
ejpam-841	467	42	sylvs	sylvs	NOUN
ejpam-841	467	43	,	,	PUNCT
ejpam-841	467	44	t(p	t(p	PROPN
ejpam-841	467	45	,	,	PUNCT
ejpam-841	467	46	q	q	NOUN
ejpam-841	467	47	)	)	PUNCT
ejpam-841	467	48	,	,	PUNCT
ejpam-841	467	49	x	x	PRON
ejpam-841	467	50	ir′i	ir′i	VERB
ejpam-841	467	51	�	�	PROPN
ejpam-841	467	52	y	y	PROPN
ejpam-841	467	53	x	x	PROPN
ejpam-841	467	54	,	,	PUNCT
ejpam-841	467	55	z	z	NOUN
ejpam-841	467	56	x	x	SYM
ejpam-841	467	57	�	�	PROPN
ejpam-841	467	58	,	,	PUNCT
ejpam-841	467	59	i	i	PRON
ejpam-841	467	60	=	=	NOUN
ejpam-841	467	61	0	0	NUM
ejpam-841	467	62	,	,	PUNCT
ejpam-841	467	63	.	.	PUNCT
ejpam-841	467	64	.	.	PUNCT
ejpam-841	468	1	.	.	PUNCT
ejpam-841	469	1	,	,	PUNCT
ejpam-841	469	2	k−	k−	PROPN
ejpam-841	469	3	1	1	NUM
ejpam-841	469	4	,	,	PUNCT
ejpam-841	469	5	where	where	SCONJ
ejpam-841	469	6	x	x	PRON
ejpam-841	469	7	ir′i	ir′i	VERB
ejpam-841	469	8	�	�	PROPN
ejpam-841	469	9	y	y	PROPN
ejpam-841	469	10	x	x	PROPN
ejpam-841	469	11	,	,	PUNCT
ejpam-841	469	12	z	z	NOUN
ejpam-841	469	13	x	x	SYM
ejpam-841	469	14	�	�	PROPN
ejpam-841	469	15	=	=	NOUN
ejpam-841	469	16	td−2−2i	td−2−2i	VERB
ejpam-841	469	17	i−1	i−1	PROPN
ejpam-841	469	18	∑	∑	PUNCT
ejpam-841	469	19	j=0	j=0	PROPN
ejpam-841	469	20	�	�	PROPN
ejpam-841	469	21	r2	r2	PROPN
ejpam-841	469	22	j	j	PROPN
ejpam-841	469	23	�	�	PROPN
ejpam-841	469	24	z	z	PROPN
ejpam-841	469	25	x	x	SYM
ejpam-841	469	26	�	�	PROPN
ejpam-841	469	27	i−	i−	PROPN
ejpam-841	469	28	j	j	PROPN
ejpam-841	469	29	+	+	CCONJ
ejpam-841	469	30	r2	r2	PROPN
ejpam-841	469	31	j+1	j+1	PUNCT
ejpam-841	469	32	�	�	PROPN
ejpam-841	469	33	z	z	NOUN
ejpam-841	469	34	x	x	SYM
ejpam-841	469	35	�	�	PROPN
ejpam-841	469	36	i−	i−	PROPN
ejpam-841	469	37	j−1	j−1	PROPN
ejpam-841	469	38	�	�	PROPN
ejpam-841	469	39	y	y	PROPN
ejpam-841	469	40	x	x	SYM
ejpam-841	469	41	�	�	PROPN
ejpam-841	469	42	�	�	PROPN
ejpam-841	469	43	+	+	CCONJ
ejpam-841	469	44	d−2−2i	d−2−2i	PROPN
ejpam-841	469	45	∑	∑	PUNCT
ejpam-841	469	46	j=0	j=0	PROPN
ejpam-841	469	47	r2i+	r2i+	PROPN
ejpam-841	469	48	js	js	ADP
ejpam-841	469	49	j	j	PROPN
ejpam-841	469	50	td−2−2i−	td−2−2i−	PROPN
ejpam-841	469	51	j	j	PROPN
ejpam-841	469	52	.	.	PUNCT
ejpam-841	470	1	(	(	PUNCT
ejpam-841	470	2	2	2	X
ejpam-841	470	3	)	)	PUNCT
ejpam-841	470	4	if	if	SCONJ
ejpam-841	470	5	d	d	NOUN
ejpam-841	470	6	=	=	SYM
ejpam-841	470	7	2k+	2k+	NUM
ejpam-841	470	8	1	1	NUM
ejpam-841	470	9	,	,	PUNCT
ejpam-841	470	10	then	then	ADV
ejpam-841	470	11	a	a	DET
ejpam-841	470	12	minimal	minimal	ADJ
ejpam-841	470	13	set	set	NOUN
ejpam-841	470	14	of	of	ADP
ejpam-841	470	15	generators	generator	NOUN
ejpam-841	470	16	for	for	ADP
ejpam-841	470	17	the	the	DET
ejpam-841	470	18	associated	associated	ADJ
ejpam-841	470	19	rees	ree	NOUN
ejpam-841	470	20	algebra	algebra	NOUN
ejpam-841	470	21	is	be	AUX
ejpam-841	470	22	p	p	NOUN
ejpam-841	470	23	,	,	PUNCT
ejpam-841	470	24	q	q	ADJ
ejpam-841	470	25	,	,	PUNCT
ejpam-841	470	26	r	r	NOUN
ejpam-841	470	27	,	,	PUNCT
ejpam-841	470	28	sylvs	sylvs	NOUN
ejpam-841	470	29	,	,	PUNCT
ejpam-841	470	30	t(p	t(p	PROPN
ejpam-841	470	31	,	,	PUNCT
ejpam-841	470	32	q	q	NOUN
ejpam-841	470	33	)	)	PUNCT
ejpam-841	470	34	,	,	PUNCT
ejpam-841	470	35	r′′	r′′	VERB
ejpam-841	470	36	,	,	PUNCT
ejpam-841	470	37	r′′′	r′′′	PROPN
ejpam-841	470	38	,	,	PUNCT
ejpam-841	470	39	x	x	PRON
ejpam-841	470	40	ir′i	ir′i	VERB
ejpam-841	470	41	�	�	PROPN
ejpam-841	470	42	y	y	PROPN
ejpam-841	470	43	x	x	PROPN
ejpam-841	470	44	,	,	PUNCT
ejpam-841	470	45	z	z	NOUN
ejpam-841	470	46	x	x	SYM
ejpam-841	470	47	�	�	PROPN
ejpam-841	470	48	,	,	PUNCT
ejpam-841	470	49	i	i	PRON
ejpam-841	470	50	=	=	NOUN
ejpam-841	470	51	1	1	NUM
ejpam-841	470	52	,	,	PUNCT
ejpam-841	470	53	.	.	PUNCT
ejpam-841	470	54	.	.	PUNCT
ejpam-841	471	1	.	.	PUNCT
ejpam-841	472	1	,	,	PUNCT
ejpam-841	472	2	k−	k−	PROPN
ejpam-841	472	3	1	1	NUM
ejpam-841	472	4	,	,	PUNCT
ejpam-841	472	5	where	where	SCONJ
ejpam-841	472	6	r′′	r′′	VERB
ejpam-841	472	7	�	�	PROPN
ejpam-841	472	8	y	y	PROPN
ejpam-841	472	9	x	x	PROPN
ejpam-841	472	10	,	,	PUNCT
ejpam-841	472	11	z	z	NOUN
ejpam-841	472	12	x	x	SYM
ejpam-841	472	13	�	�	PROPN
ejpam-841	472	14	=	=	SYM
ejpam-841	472	15	k−1	k−1	PROPN
ejpam-841	472	16	∑	∑	PUNCT
ejpam-841	472	17	j=0	j=0	PROPN
ejpam-841	472	18	�	�	PROPN
ejpam-841	472	19	r2	r2	PROPN
ejpam-841	472	20	j	j	PROPN
ejpam-841	472	21	�	�	PROPN
ejpam-841	472	22	z	z	PROPN
ejpam-841	472	23	x	x	SYM
ejpam-841	472	24	�	�	PROPN
ejpam-841	472	25	k−1−	k−1−	PROPN
ejpam-841	472	26	j	j	PROPN
ejpam-841	472	27	�	�	PROPN
ejpam-841	472	28	y	y	PROPN
ejpam-841	472	29	x	x	PROPN
ejpam-841	472	30	�	�	PROPN
ejpam-841	472	31	+	+	NOUN
ejpam-841	472	32	r2	r2	PROPN
ejpam-841	472	33	j+1	j+1	PUNCT
ejpam-841	472	34	�	�	PROPN
ejpam-841	472	35	z	z	PROPN
ejpam-841	472	36	x	x	SYM
ejpam-841	472	37	�	�	PROPN
ejpam-841	472	38	k−1−	k−1−	PROPN
ejpam-841	472	39	j	j	PROPN
ejpam-841	472	40	�	�	PROPN
ejpam-841	472	41	r′′′	r′′′	PROPN
ejpam-841	472	42	�	�	PROPN
ejpam-841	472	43	y	y	PROPN
ejpam-841	472	44	z	z	PROPN
ejpam-841	472	45	,	,	PUNCT
ejpam-841	472	46	x	x	PUNCT
ejpam-841	472	47	z	z	X
ejpam-841	472	48	�	�	PROPN
ejpam-841	472	49	=	=	SYM
ejpam-841	472	50	k−1	k−1	PROPN
ejpam-841	472	51	∑	∑	PUNCT
ejpam-841	472	52	j=0	j=0	PROPN
ejpam-841	472	53	�	�	PROPN
ejpam-841	472	54	r2	r2	PROPN
ejpam-841	472	55	j	j	PROPN
ejpam-841	472	56	�	�	PROPN
ejpam-841	472	57	x	x	PROPN
ejpam-841	472	58	z	z	PROPN
ejpam-841	472	59	�	�	PROPN
ejpam-841	472	60	j	j	PROPN
ejpam-841	472	61	+	+	CCONJ
ejpam-841	472	62	r2	r2	PROPN
ejpam-841	472	63	j+1	j+1	PUNCT
ejpam-841	472	64	�	�	PROPN
ejpam-841	473	1	x	x	PUNCT
ejpam-841	473	2	z	z	PROPN
ejpam-841	473	3	�	�	PROPN
ejpam-841	473	4	j	j	PROPN
ejpam-841	473	5	�	�	PROPN
ejpam-841	473	6	y	y	PROPN
ejpam-841	473	7	z	z	PROPN
ejpam-841	473	8	�	�	PROPN
ejpam-841	473	9	�	�	PROPN
ejpam-841	473	10	r′′i	r′′i	PROPN
ejpam-841	473	11	(	(	PUNCT
ejpam-841	473	12	y	y	NOUN
ejpam-841	473	13	x	x	INTJ
ejpam-841	473	14	,	,	PUNCT
ejpam-841	473	15	z	z	NOUN
ejpam-841	473	16	x	x	SYM
ejpam-841	473	17	)	)	PUNCT
ejpam-841	473	18	=	=	NOUN
ejpam-841	473	19	td−2−2i	td−2−2i	PUNCT
ejpam-841	473	20	i−1	i−1	PROPN
ejpam-841	473	21	∑	∑	PUNCT
ejpam-841	473	22	j=0	j=0	PROPN
ejpam-841	473	23	�	�	PROPN
ejpam-841	473	24	r2	r2	PROPN
ejpam-841	473	25	j	j	PROPN
ejpam-841	473	26	�	�	PROPN
ejpam-841	473	27	z	z	PROPN
ejpam-841	473	28	x	x	SYM
ejpam-841	473	29	�	�	PROPN
ejpam-841	473	30	i−	i−	PROPN
ejpam-841	473	31	j	j	PROPN
ejpam-841	473	32	+	+	CCONJ
ejpam-841	473	33	r2	r2	PROPN
ejpam-841	473	34	j+1	j+1	PUNCT
ejpam-841	473	35	�	�	PROPN
ejpam-841	473	36	z	z	NOUN
ejpam-841	473	37	x	x	SYM
ejpam-841	473	38	�	�	PROPN
ejpam-841	473	39	i−	i−	PROPN
ejpam-841	473	40	j−1	j−1	PROPN
ejpam-841	473	41	�	�	PROPN
ejpam-841	473	42	y	y	PROPN
ejpam-841	473	43	x	x	SYM
ejpam-841	473	44	�	�	PROPN
ejpam-841	473	45	�	�	PROPN
ejpam-841	473	46	+	+	CCONJ
ejpam-841	473	47	d−2−2i	d−2−2i	PROPN
ejpam-841	473	48	∑	∑	PUNCT
ejpam-841	473	49	j=0	j=0	PROPN
ejpam-841	473	50	r2i+	r2i+	PROPN
ejpam-841	473	51	js	js	ADP
ejpam-841	473	52	j	j	PROPN
ejpam-841	473	53	td−2−2i−	td−2−2i−	PROPN
ejpam-841	473	54	j	j	PROPN
ejpam-841	473	55	.	.	PUNCT
ejpam-841	474	1	j.	j.	PROPN
ejpam-841	474	2	hoffman	hoffman	PROPN
ejpam-841	474	3	,	,	PUNCT
ejpam-841	474	4	h.	h.	PROPN
ejpam-841	474	5	wang	wang	PROPN
ejpam-841	474	6	,	,	PUNCT
ejpam-841	474	7	x.	x.	PROPN
ejpam-841	474	8	jia	jia	PROPN
ejpam-841	474	9	,	,	PUNCT
ejpam-841	474	10	r.	r.	PROPN
ejpam-841	474	11	goldman	goldman	PROPN
ejpam-841	474	12	/	/	SYM
ejpam-841	474	13	eur	eur	PROPN
ejpam-841	474	14	.	.	PUNCT
ejpam-841	475	1	j.	j.	PROPN
ejpam-841	475	2	pure	pure	PROPN
ejpam-841	475	3	appl	appl	PROPN
ejpam-841	475	4	.	.	PROPN
ejpam-841	475	5	math	math	PROPN
ejpam-841	475	6	,	,	PUNCT
ejpam-841	475	7	3	3	NUM
ejpam-841	475	8	(	(	PUNCT
ejpam-841	475	9	2010	2010	NUM
ejpam-841	475	10	)	)	PUNCT
ejpam-841	475	11	,	,	PUNCT
ejpam-841	475	12	602	602	NUM
ejpam-841	475	13	-	-	SYM
ejpam-841	475	14	632	632	NUM
ejpam-841	475	15	618	618	NUM
ejpam-841	475	16	4	4	NUM
ejpam-841	475	17	.	.	PUNCT
ejpam-841	476	1	if	if	SCONJ
ejpam-841	476	2	v(p1	v(p1	NOUN
ejpam-841	476	3	,	,	PUNCT
ejpam-841	476	4	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	476	5	)	)	PUNCT
ejpam-841	476	6	=	=	SYM
ejpam-841	476	7	;	;	PUNCT
ejpam-841	476	8	,	,	PUNCT
ejpam-841	476	9	then	then	ADV
ejpam-841	476	10	let	let	VERB
ejpam-841	476	11	x	x	SYM
ejpam-841	476	12	=	=	SYM
ejpam-841	476	13	−p0	−p0	PROPN
ejpam-841	476	14	,	,	PUNCT
ejpam-841	476	15	y	y	PROPN
ejpam-841	476	16	=	=	SYM
ejpam-841	476	17	p1	p1	PROPN
ejpam-841	476	18	,	,	PUNCT
ejpam-841	476	19	z	z	NOUN
ejpam-841	476	20	=	=	SYM
ejpam-841	476	21	−q0	−q0	PROPN
ejpam-841	476	22	,	,	PUNCT
ejpam-841	476	23	w	w	PROPN
ejpam-841	476	24	=	=	SYM
ejpam-841	476	25	q1	q1	PROPN
ejpam-841	476	26	,	,	PUNCT
ejpam-841	476	27	so	so	SCONJ
ejpam-841	476	28	that	that	SCONJ
ejpam-841	476	29	p	p	NOUN
ejpam-841	476	30	=	=	PUNCT
ejpam-841	476	31	y	y	PROPN
ejpam-841	476	32	s−	s−	PROPN
ejpam-841	476	33	x	x	PUNCT
ejpam-841	476	34	t	t	PROPN
ejpam-841	476	35	,	,	PUNCT
ejpam-841	476	36	q	q	NOUN
ejpam-841	476	37	=	=	SYM
ejpam-841	476	38	ws−	ws−	PROPN
ejpam-841	476	39	z	z	PROPN
ejpam-841	476	40	t	t	PROPN
ejpam-841	476	41	,	,	PUNCT
ejpam-841	476	42	r	r	NOUN
ejpam-841	476	43	=	=	SYM
ejpam-841	476	44	rd−2sd−2	rd−2sd−2	NOUN
ejpam-841	476	45	+	+	NUM
ejpam-841	476	46	rd−3sd−3	rd−3sd−3	NOUN
ejpam-841	476	47	t	t	NOUN
ejpam-841	476	48	+	+	CCONJ
ejpam-841	476	49	·	·	PUNCT
ejpam-841	476	50	·	·	PUNCT
ejpam-841	476	51	·	·	PUNCT
ejpam-841	476	52	+	+	NUM
ejpam-841	476	53	r0	r0	NOUN
ejpam-841	476	54	td−2	td−2	NOUN
ejpam-841	476	55	,	,	PUNCT
ejpam-841	476	56	where	where	SCONJ
ejpam-841	476	57	ri	ri	PROPN
ejpam-841	476	58	are	be	AUX
ejpam-841	476	59	linear	linear	ADJ
ejpam-841	476	60	in	in	ADP
ejpam-841	476	61	x	x	SYM
ejpam-841	476	62	,	,	PUNCT
ejpam-841	476	63	y	y	PROPN
ejpam-841	476	64	,	,	PUNCT
ejpam-841	476	65	z	z	PROPN
ejpam-841	476	66	,	,	PUNCT
ejpam-841	476	67	w.	w.	PROPN
ejpam-841	476	68	a	a	DET
ejpam-841	476	69	minimal	minimal	ADJ
ejpam-841	476	70	set	set	NOUN
ejpam-841	476	71	of	of	ADP
ejpam-841	476	72	generators	generator	NOUN
ejpam-841	476	73	for	for	ADP
ejpam-841	476	74	the	the	DET
ejpam-841	476	75	associated	associated	ADJ
ejpam-841	476	76	rees	ree	NOUN
ejpam-841	476	77	algebra	algebra	NOUN
ejpam-841	476	78	is	be	AUX
ejpam-841	476	79	p	p	NOUN
ejpam-841	476	80	,	,	PUNCT
ejpam-841	476	81	q	q	ADJ
ejpam-841	476	82	,	,	PUNCT
ejpam-841	476	83	r	r	NOUN
ejpam-841	476	84	,	,	PUNCT
ejpam-841	476	85	sylvs	sylvs	NOUN
ejpam-841	476	86	,	,	PUNCT
ejpam-841	476	87	t(p	t(p	PROPN
ejpam-841	476	88	,	,	PUNCT
ejpam-841	476	89	q	q	NOUN
ejpam-841	476	90	)	)	PUNCT
ejpam-841	476	91	,	,	PUNCT
ejpam-841	476	92	x	x	SYM
ejpam-841	476	93	ℓz	ℓz	NOUN
ejpam-841	476	94	d−2−ℓr′ℓ	d−2−ℓr′ℓ	VERB
ejpam-841	476	95	�	�	PROPN
ejpam-841	476	96	y	y	PROPN
ejpam-841	476	97	x	x	PROPN
ejpam-841	476	98	,	,	PUNCT
ejpam-841	476	99	w	w	PROPN
ejpam-841	476	100	z	z	PROPN
ejpam-841	476	101	�	�	PROPN
ejpam-841	476	102	,	,	PUNCT
ejpam-841	476	103	x	x	PROPN
ejpam-841	476	104	j	j	PROPN
ejpam-841	476	105	z	z	PROPN
ejpam-841	476	106	i−	i−	PROPN
ejpam-841	476	107	jr′j	jr′j	PROPN
ejpam-841	476	108	,	,	PUNCT
ejpam-841	476	109	i−	i−	PROPN
ejpam-841	476	110	j	j	PROPN
ejpam-841	476	111	�	�	PROPN
ejpam-841	477	1	y	y	PROPN
ejpam-841	477	2	x	x	PROPN
ejpam-841	477	3	,	,	PUNCT
ejpam-841	477	4	w	w	PROPN
ejpam-841	477	5	z	z	PROPN
ejpam-841	477	6	�	�	PROPN
ejpam-841	477	7	,	,	PUNCT
ejpam-841	477	8	0≤	0≤	NUM
ejpam-841	477	9	ℓ	ℓ	NOUN
ejpam-841	477	10	≤	≤	NUM
ejpam-841	478	1	d	d	ADP
ejpam-841	478	2	−	−	PROPN
ejpam-841	478	3	2	2	NUM
ejpam-841	478	4	,	,	PUNCT
ejpam-841	478	5	0≤	0≤	NUM
ejpam-841	479	1	j	j	NOUN
ejpam-841	479	2	≤	≤	X
ejpam-841	480	1	i	i	NOUN
ejpam-841	480	2	=	=	NOUN
ejpam-841	480	3	1	1	NUM
ejpam-841	480	4	,	,	PUNCT
ejpam-841	480	5	.	.	PUNCT
ejpam-841	480	6	.	.	PUNCT
ejpam-841	480	7	.	.	PUNCT
ejpam-841	481	1	,	,	PUNCT
ejpam-841	482	1	d	d	X
ejpam-841	482	2	−	−	PROPN
ejpam-841	482	3	3	3	NUM
ejpam-841	482	4	,	,	PUNCT
ejpam-841	483	1	where	where	SCONJ
ejpam-841	483	2	r′ℓ	r′ℓ	NOUN
ejpam-841	483	3	�	�	NOUN
ejpam-841	483	4	y	y	PROPN
ejpam-841	483	5	x	x	PROPN
ejpam-841	483	6	,	,	PUNCT
ejpam-841	483	7	w	w	PROPN
ejpam-841	483	8	z	z	PROPN
ejpam-841	483	9	�	�	PROPN
ejpam-841	483	10	=	=	SYM
ejpam-841	483	11	ℓ−1	ℓ−1	PROPN
ejpam-841	483	12	∑	∑	PUNCT
ejpam-841	483	13	j=0	j=0	PROPN
ejpam-841	483	14	r	r	PROPN
ejpam-841	483	15	j	j	PROPN
ejpam-841	483	16	�	�	PROPN
ejpam-841	483	17	w	w	PROPN
ejpam-841	483	18	z	z	PROPN
ejpam-841	483	19	�	�	PROPN
ejpam-841	483	20	d−2−ℓ	d−2−ℓ	NOUN
ejpam-841	483	21	�	�	PROPN
ejpam-841	483	22	y	y	PROPN
ejpam-841	483	23	x	x	SYM
ejpam-841	483	24	�	�	PROPN
ejpam-841	483	25	ℓ−	ℓ−	PROPN
ejpam-841	483	26	j	j	PROPN
ejpam-841	483	27	+	+	CCONJ
ejpam-841	483	28	d−2−ℓ	d−2−ℓ	NOUN
ejpam-841	483	29	∑	∑	ADP
ejpam-841	483	30	j=0	j=0	PROPN
ejpam-841	483	31	rℓ+	rℓ+	PROPN
ejpam-841	483	32	j	j	PROPN
ejpam-841	483	33	�	�	PROPN
ejpam-841	483	34	w	w	PROPN
ejpam-841	483	35	z	z	PROPN
ejpam-841	483	36	�	�	PROPN
ejpam-841	483	37	d−2−ℓ−	d−2−ℓ−	PROPN
ejpam-841	483	38	j	j	NOUN
ejpam-841	483	39	,	,	PUNCT
ejpam-841	483	40	r′j	r′j	PROPN
ejpam-841	483	41	,	,	PUNCT
ejpam-841	483	42	i−	i−	PROPN
ejpam-841	483	43	j	j	PROPN
ejpam-841	483	44	�	�	PROPN
ejpam-841	483	45	y	y	PROPN
ejpam-841	483	46	x	x	PROPN
ejpam-841	483	47	,	,	PUNCT
ejpam-841	483	48	w	w	PROPN
ejpam-841	483	49	z	z	PROPN
ejpam-841	483	50	�	�	PROPN
ejpam-841	483	51	=	=	SYM
ejpam-841	483	52	td−2−i	td−2−i	NOUN
ejpam-841	483	53			NOUN
ejpam-841	483	54			VERB
ejpam-841	484	1	j−1	j−1	PROPN
ejpam-841	484	2	∑	∑	PUNCT
ejpam-841	484	3	ℓ=0	ℓ=0	PROPN
ejpam-841	484	4	rℓ	rℓ	NOUN
ejpam-841	484	5	�	�	PROPN
ejpam-841	484	6	w	w	PROPN
ejpam-841	484	7	z	z	PROPN
ejpam-841	484	8	�	�	PROPN
ejpam-841	484	9	i−	i−	PROPN
ejpam-841	484	10	j	j	PROPN
ejpam-841	484	11	�	�	PROPN
ejpam-841	484	12	y	y	PROPN
ejpam-841	484	13	z	z	PROPN
ejpam-841	484	14	�	�	PROPN
ejpam-841	484	15	j−ℓ	j−ℓ	PROPN
ejpam-841	484	16	+	+	CCONJ
ejpam-841	484	17	i−1−	i−1−	PROPN
ejpam-841	484	18	j	j	PROPN
ejpam-841	484	19	∑	∑	PUNCT
ejpam-841	484	20	ℓ=0	ℓ=0	PROPN
ejpam-841	484	21	r	r	PROPN
ejpam-841	484	22	j+ℓ	j+ℓ	NUM
ejpam-841	484	23	�	�	PROPN
ejpam-841	484	24	w	w	PROPN
ejpam-841	484	25	z	z	PROPN
ejpam-841	484	26	�	�	PROPN
ejpam-841	484	27	i−	i−	PROPN
ejpam-841	484	28	j−ℓ	j−ℓ	NOUN
ejpam-841	484	29			PROPN
ejpam-841	484	30			VERB
ejpam-841	484	31	+	+	NUM
ejpam-841	484	32	d−2−i	d−2−i	ADJ
ejpam-841	484	33	∑	∑	PRON
ejpam-841	484	34	ℓ=0	ℓ=0	PROPN
ejpam-841	484	35	ri+ℓ	ri+ℓ	PROPN
ejpam-841	484	36	t	t	PROPN
ejpam-841	484	37	d−2−i−ℓsℓ.	d−2−i−ℓsℓ.	VERB
ejpam-841	484	38	below	below	ADV
ejpam-841	484	39	we	we	PRON
ejpam-841	484	40	present	present	VERB
ejpam-841	484	41	two	two	NUM
ejpam-841	484	42	examples	example	NOUN
ejpam-841	484	43	to	to	PART
ejpam-841	484	44	show	show	VERB
ejpam-841	484	45	how	how	SCONJ
ejpam-841	484	46	to	to	PART
ejpam-841	484	47	use	use	VERB
ejpam-841	484	48	algorithm	algorithm	NOUN
ejpam-841	484	49	4	4	NUM
ejpam-841	484	50	to	to	PART
ejpam-841	484	51	find	find	VERB
ejpam-841	484	52	a	a	DET
ejpam-841	484	53	minimal	minimal	ADJ
ejpam-841	484	54	set	set	NOUN
ejpam-841	484	55	of	of	ADP
ejpam-841	484	56	generators	generator	NOUN
ejpam-841	484	57	for	for	ADP
ejpam-841	484	58	the	the	DET
ejpam-841	484	59	associated	associated	ADJ
ejpam-841	484	60	rees	rees	PROPN
ejpam-841	484	61	algebra	algebra	NOUN
ejpam-841	484	62	both	both	PRON
ejpam-841	484	63	for	for	ADP
ejpam-841	484	64	singular	singular	NOUN
ejpam-841	484	65	and	and	CCONJ
ejpam-841	484	66	for	for	ADP
ejpam-841	484	67	non	non	ADJ
ejpam-841	484	68	-	-	ADJ
ejpam-841	484	69	singular	singular	ADJ
ejpam-841	484	70	rational	rational	ADJ
ejpam-841	484	71	space	space	NOUN
ejpam-841	484	72	curves	curve	NOUN
ejpam-841	484	73	.	.	PUNCT
ejpam-841	485	1	example	example	NOUN
ejpam-841	485	2	1	1	NUM
ejpam-841	485	3	.	.	X
ejpam-841	486	1	consider	consider	VERB
ejpam-841	486	2	the	the	DET
ejpam-841	486	3	rational	rational	ADJ
ejpam-841	486	4	quintic	quintic	ADJ
ejpam-841	486	5	space	space	NOUN
ejpam-841	486	6	curve	curve	NOUN
ejpam-841	486	7	given	give	VERB
ejpam-841	486	8	by	by	ADP
ejpam-841	486	9	f(s	f(	NOUN
ejpam-841	486	10	,	,	PUNCT
ejpam-841	486	11	t	t	PROPN
ejpam-841	486	12	)	)	PUNCT
ejpam-841	486	13	=	=	PUNCT
ejpam-841	486	14	(	(	PUNCT
ejpam-841	486	15	s4	s4	PROPN
ejpam-841	486	16	t	t	PROPN
ejpam-841	486	17	+	+	CCONJ
ejpam-841	486	18	s3	s3	PROPN
ejpam-841	486	19	t2	t2	NOUN
ejpam-841	486	20	−	−	PROPN
ejpam-841	486	21	2s2t3	2s2t3	NOUN
ejpam-841	486	22	,	,	PUNCT
ejpam-841	486	23	s5	s5	NOUN
ejpam-841	486	24	+	+	X
ejpam-841	486	25	5s4	5s4	NUM
ejpam-841	486	26	t	t	NOUN
ejpam-841	486	27	+	+	X
ejpam-841	486	28	6s3t2	6s3t2	NUM
ejpam-841	486	29	−	−	PROPN
ejpam-841	486	30	4s2	4s2	NUM
ejpam-841	486	31	t3	t3	PROPN
ejpam-841	486	32	−	−	PROPN
ejpam-841	487	1	8st4	8st4	NUM
ejpam-841	487	2	,	,	PUNCT
ejpam-841	487	3	s4	s4	PROPN
ejpam-841	487	4	t	t	PROPN
ejpam-841	487	5	−	−	PROPN
ejpam-841	487	6	3s2	3s2	NUM
ejpam-841	487	7	t3	t3	NOUN
ejpam-841	488	1	+	+	CCONJ
ejpam-841	489	1	2st4	2st4	NUM
ejpam-841	489	2	,	,	PUNCT
ejpam-841	489	3	t5	t5	PROPN
ejpam-841	489	4	)	)	PUNCT
ejpam-841	489	5	.	.	PUNCT
ejpam-841	490	1	compute	compute	VERB
ejpam-841	490	2	a	a	DET
ejpam-841	490	3	µ-basis	µ-basis	NOUN
ejpam-841	490	4	for	for	ADP
ejpam-841	490	5	f(s	f(	NOUN
ejpam-841	490	6	,	,	PUNCT
ejpam-841	490	7	t	t	PROPN
ejpam-841	490	8	):	):	PUNCT
ejpam-841	491	1	p	p	X
ejpam-841	491	2	=	=	PROPN
ejpam-841	491	3	xs−	xs−	PROPN
ejpam-841	491	4	(	(	PUNCT
ejpam-841	491	5	4z	4z	NOUN
ejpam-841	491	6	+	+	CCONJ
ejpam-841	491	7	y	y	PROPN
ejpam-841	491	8	−	−	PROPN
ejpam-841	491	9	8x)t	8x)t	NUM
ejpam-841	491	10	,	,	PUNCT
ejpam-841	491	11	q	q	NOUN
ejpam-841	491	12	=	=	SYM
ejpam-841	491	13	(	(	PUNCT
ejpam-841	491	14	x	x	NOUN
ejpam-841	491	15	−	−	PROPN
ejpam-841	491	16	z)s−	z)s−	PROPN
ejpam-841	491	17	x	x	SYM
ejpam-841	491	18	t	t	PROPN
ejpam-841	491	19	,	,	PUNCT
ejpam-841	491	20	r	r	NOUN
ejpam-841	491	21	=	=	SYM
ejpam-841	491	22	(	(	PUNCT
ejpam-841	491	23	s3	s3	PROPN
ejpam-841	491	24	+	+	CCONJ
ejpam-841	491	25	s2	s2	PROPN
ejpam-841	491	26	t	t	NOUN
ejpam-841	491	27	−	−	NUM
ejpam-841	491	28	2st2)w	2st2)w	NUM
ejpam-841	491	29	+	+	CCONJ
ejpam-841	491	30	t3(z	t3(z	ADP
ejpam-841	491	31	−	−	NOUN
ejpam-841	491	32	x	x	NOUN
ejpam-841	491	33	)	)	PUNCT
ejpam-841	491	34	.	.	PUNCT
ejpam-841	492	1	the	the	DET
ejpam-841	492	2	point	point	NOUN
ejpam-841	492	3	(	(	PUNCT
ejpam-841	492	4	0,0,0,1	0,0,0,1	NOUN
ejpam-841	492	5	)	)	PUNCT
ejpam-841	492	6	is	be	AUX
ejpam-841	492	7	a	a	DET
ejpam-841	492	8	singular	singular	ADJ
ejpam-841	492	9	point	point	NOUN
ejpam-841	492	10	of	of	ADP
ejpam-841	492	11	order	order	NOUN
ejpam-841	492	12	3	3	X
ejpam-841	492	13	.	.	PUNCT
ejpam-841	493	1	by	by	ADP
ejpam-841	493	2	a	a	DET
ejpam-841	493	3	projective	projective	ADJ
ejpam-841	493	4	changes	change	NOUN
ejpam-841	493	5	of	of	ADP
ejpam-841	493	6	coordinates	coordinate	NOUN
ejpam-841	493	7	,	,	PUNCT
ejpam-841	493	8	let	let	VERB
ejpam-841	493	9	x	x	X
ejpam-841	493	10	=	=	PUNCT
ejpam-841	493	11	−8x	−8x	PUNCT
ejpam-841	494	1	+	+	NUM
ejpam-841	494	2	y	y	PROPN
ejpam-841	494	3	+	+	CCONJ
ejpam-841	494	4	4z	4z	NOUN
ejpam-841	494	5	,	,	PUNCT
ejpam-841	494	6	y	y	PROPN
ejpam-841	494	7	=	=	PUNCT
ejpam-841	494	8	x	x	PROPN
ejpam-841	494	9	,	,	PUNCT
ejpam-841	494	10	z	z	NOUN
ejpam-841	494	11	=	=	PUNCT
ejpam-841	494	12	x	x	X
ejpam-841	494	13	−	−	PROPN
ejpam-841	494	14	z	z	NOUN
ejpam-841	494	15	,	,	PUNCT
ejpam-841	494	16	w	w	PROPN
ejpam-841	494	17	=	=	SYM
ejpam-841	494	18	w.	w.	PROPN
ejpam-841	494	19	then	then	ADV
ejpam-841	494	20	the	the	DET
ejpam-841	494	21	new	new	ADJ
ejpam-841	494	22	µ-basis	µ-basis	NOUN
ejpam-841	494	23	can	can	AUX
ejpam-841	494	24	be	be	AUX
ejpam-841	494	25	written	write	VERB
ejpam-841	494	26	as	as	ADP
ejpam-841	494	27	p	p	NOUN
ejpam-841	494	28	=	=	PUNCT
ejpam-841	494	29	y	y	PROPN
ejpam-841	494	30	s−	s−	PROPN
ejpam-841	494	31	x	x	PUNCT
ejpam-841	494	32	t	t	PROPN
ejpam-841	494	33	,	,	PUNCT
ejpam-841	494	34	q	q	NOUN
ejpam-841	494	35	=	=	SYM
ejpam-841	494	36	zs−	zs−	NUM
ejpam-841	494	37	y	y	PROPN
ejpam-841	494	38	t	t	PROPN
ejpam-841	494	39	,	,	PUNCT
ejpam-841	494	40	r	r	NOUN
ejpam-841	494	41	=	=	NOUN
ejpam-841	494	42	ws3	ws3	VERB
ejpam-841	494	43	+	+	NOUN
ejpam-841	494	44	ws2	ws2	NOUN
ejpam-841	494	45	t	t	PROPN
ejpam-841	494	46	−	−	PROPN
ejpam-841	494	47	2wst2	2wst2	NUM
ejpam-841	494	48	−	−	PROPN
ejpam-841	494	49	z	z	PROPN
ejpam-841	494	50	t3	t3	PROPN
ejpam-841	494	51	.	.	PUNCT
ejpam-841	495	1	the	the	DET
ejpam-841	495	2	minimal	minimal	ADJ
ejpam-841	495	3	generators	generator	NOUN
ejpam-841	495	4	for	for	ADP
ejpam-841	495	5	the	the	DET
ejpam-841	495	6	rees	rees	PROPN
ejpam-841	495	7	algebra	algebra	NOUN
ejpam-841	495	8	associated	associate	VERB
ejpam-841	495	9	to	to	ADP
ejpam-841	495	10	this	this	DET
ejpam-841	495	11	curve	curve	NOUN
ejpam-841	495	12	are	be	AUX
ejpam-841	495	13	p	p	NOUN
ejpam-841	495	14	,	,	PUNCT
ejpam-841	495	15	q	q	ADJ
ejpam-841	495	16	,	,	PUNCT
ejpam-841	495	17	r	r	NOUN
ejpam-841	495	18	and	and	CCONJ
ejpam-841	495	19	the	the	DET
ejpam-841	495	20	following	follow	VERB
ejpam-841	495	21	polynomials	polynomial	NOUN
ejpam-841	495	22	:	:	PUNCT
ejpam-841	495	23	sylvs	sylv	NOUN
ejpam-841	495	24	,	,	PUNCT
ejpam-841	495	25	t(p	t(p	PROPN
ejpam-841	495	26	,	,	PUNCT
ejpam-841	495	27	q	q	NOUN
ejpam-841	495	28	)	)	PUNCT
ejpam-841	495	29	=	=	PUNCT
ejpam-841	496	1	x	x	SYM
ejpam-841	496	2	z	z	NOUN
ejpam-841	496	3	−	−	NOUN
ejpam-841	496	4	y	y	NOUN
ejpam-841	496	5	2	2	NUM
ejpam-841	496	6	=	=	SYM
ejpam-841	496	7	−9x2	−9x2	PROPN
ejpam-841	496	8	+	+	NOUN
ejpam-841	496	9	x	x	SYM
ejpam-841	496	10	y	y	PROPN
ejpam-841	496	11	+	+	CCONJ
ejpam-841	496	12	12xz−	12xz−	NUM
ejpam-841	496	13	yz	yz	PROPN
ejpam-841	496	14	−	−	PROPN
ejpam-841	496	15	4z2	4z2	NUM
ejpam-841	496	16	;	;	PUNCT
ejpam-841	496	17	j.	j.	PROPN
ejpam-841	496	18	hoffman	hoffman	PROPN
ejpam-841	496	19	,	,	PUNCT
ejpam-841	496	20	h.	h.	PROPN
ejpam-841	496	21	wang	wang	PROPN
ejpam-841	496	22	,	,	PUNCT
ejpam-841	496	23	x.	x.	PROPN
ejpam-841	496	24	jia	jia	PROPN
ejpam-841	496	25	,	,	PUNCT
ejpam-841	496	26	r.	r.	PROPN
ejpam-841	496	27	goldman	goldman	PROPN
ejpam-841	496	28	/	/	SYM
ejpam-841	496	29	eur	eur	PROPN
ejpam-841	496	30	.	.	PUNCT
ejpam-841	497	1	j.	j.	PROPN
ejpam-841	497	2	pure	pure	PROPN
ejpam-841	497	3	appl	appl	PROPN
ejpam-841	497	4	.	.	PROPN
ejpam-841	497	5	math	math	PROPN
ejpam-841	497	6	,	,	PUNCT
ejpam-841	497	7	3	3	NUM
ejpam-841	497	8	(	(	PUNCT
ejpam-841	497	9	2010	2010	NUM
ejpam-841	497	10	)	)	PUNCT
ejpam-841	497	11	,	,	PUNCT
ejpam-841	497	12	602	602	NUM
ejpam-841	497	13	-	-	SYM
ejpam-841	497	14	632	632	NUM
ejpam-841	497	15	619	619	NUM
ejpam-841	497	16	x	x	SYM
ejpam-841	497	17	2r	2r	NUM
ejpam-841	497	18	′′	′′	PROPN
ejpam-841	497	19	(	(	PUNCT
ejpam-841	497	20	y	y	PROPN
ejpam-841	497	21	x	x	PROPN
ejpam-841	497	22	,	,	PUNCT
ejpam-841	497	23	z	z	NOUN
ejpam-841	497	24	x	x	SYM
ejpam-841	497	25	)	)	PUNCT
ejpam-841	497	26	=	=	PUNCT
ejpam-841	498	1	x	x	SYM
ejpam-841	498	2	2{[1	2{[1	NUM
ejpam-841	498	3	+	+	CCONJ
ejpam-841	498	4	(	(	PUNCT
ejpam-841	498	5	y	y	NOUN
ejpam-841	498	6	x	x	PROPN
ejpam-841	498	7	)	)	PUNCT
ejpam-841	499	1	−	−	PROPN
ejpam-841	499	2	2	2	NUM
ejpam-841	499	3	(	(	PUNCT
ejpam-841	499	4	z	z	NOUN
ejpam-841	499	5	x	x	NOUN
ejpam-841	499	6	)	)	PUNCT
ejpam-841	499	7	]	]	X
ejpam-841	500	1	w	w	X
ejpam-841	500	2	−	−	PROPN
ejpam-841	501	1	(	(	PUNCT
ejpam-841	501	2	y	y	NOUN
ejpam-841	501	3	x	x	PROPN
ejpam-841	501	4	)	)	PUNCT
ejpam-841	501	5	(	(	PUNCT
ejpam-841	501	6	z	z	NOUN
ejpam-841	501	7	x	x	SYM
ejpam-841	501	8	)	)	PUNCT
ejpam-841	501	9	z	z	X
ejpam-841	501	10	}	}	PUNCT
ejpam-841	501	11	=	=	SYM
ejpam-841	501	12	−x3	−x3	PROPN
ejpam-841	501	13	+	+	PROPN
ejpam-841	501	14	2x2z	2x2z	NUM
ejpam-841	501	15	−	−	NOUN
ejpam-841	501	16	xz2	xz2	PUNCT
ejpam-841	502	1	+	+	CCONJ
ejpam-841	502	2	72x2w	72x2w	NUM
ejpam-841	502	3	−	−	NOUN
ejpam-841	502	4	17x	17x	NOUN
ejpam-841	502	5	yw+	yw+	VERB
ejpam-841	503	1	y2w	y2w	NOUN
ejpam-841	503	2	−	−	PROPN
ejpam-841	504	1	84xzw+	84xzw+	NUM
ejpam-841	504	2	10yzw	10yzw	NOUN
ejpam-841	504	3	+	+	CCONJ
ejpam-841	504	4	24z2w	24z2w	NUM
ejpam-841	504	5	;	;	PUNCT
ejpam-841	504	6	z2r	z2r	NUM
ejpam-841	504	7	′′′	′′′	NOUN
ejpam-841	504	8	(	(	PUNCT
ejpam-841	504	9	y	y	PROPN
ejpam-841	504	10	z	z	PROPN
ejpam-841	504	11	,	,	PUNCT
ejpam-841	504	12	x	x	X
ejpam-841	504	13	z	z	NOUN
ejpam-841	504	14	)	)	PUNCT
ejpam-841	504	15	=	=	SYM
ejpam-841	504	16	z2	z2	NOUN
ejpam-841	504	17	{	{	PUNCT
ejpam-841	504	18	[	[	X
ejpam-841	504	19	(	(	PUNCT
ejpam-841	504	20	y	y	PROPN
ejpam-841	504	21	z	z	PROPN
ejpam-841	504	22	)	)	PUNCT
ejpam-841	504	23	(	(	PUNCT
ejpam-841	504	24	x	x	SYM
ejpam-841	504	25	z	z	NOUN
ejpam-841	504	26	)	)	PUNCT
ejpam-841	505	1	+	+	CCONJ
ejpam-841	505	2	(	(	PUNCT
ejpam-841	505	3	x	x	SYM
ejpam-841	505	4	z	z	NOUN
ejpam-841	505	5	)	)	PUNCT
ejpam-841	505	6	−	−	PROPN
ejpam-841	506	1	2	2	NUM
ejpam-841	506	2	(	(	PUNCT
ejpam-841	506	3	y	y	PROPN
ejpam-841	506	4	z	z	PROPN
ejpam-841	506	5	)	)	PUNCT
ejpam-841	507	1	]	]	X
ejpam-841	507	2	w	w	X
ejpam-841	507	3	−	−	PROPN
ejpam-841	507	4	z	z	NOUN
ejpam-841	507	5	}	}	PUNCT
ejpam-841	507	6	=	=	SYM
ejpam-841	507	7	−x3	−x3	PROPN
ejpam-841	507	8	+	+	CCONJ
ejpam-841	507	9	3x2z	3x2z	NUM
ejpam-841	507	10	−	−	PROPN
ejpam-841	507	11	3xz2	3xz2	NUM
ejpam-841	508	1	+	+	CCONJ
ejpam-841	508	2	z3	z3	PROPN
ejpam-841	508	3	−	−	PROPN
ejpam-841	509	1	18x2w	18x2w	NUM
ejpam-841	510	1	+	+	CCONJ
ejpam-841	510	2	2x	2x	NUM
ejpam-841	510	3	yw+	yw+	PROPN
ejpam-841	510	4	18xzw−	18xzw−	NUM
ejpam-841	510	5	yzw	yzw	PROPN
ejpam-841	510	6	−	−	PROPN
ejpam-841	510	7	4z2w	4z2w	NOUN
ejpam-841	510	8	,	,	PUNCT
ejpam-841	510	9	x	x	PUNCT
ejpam-841	510	10	r	r	NOUN
ejpam-841	510	11	′′1	′′1	NOUN
ejpam-841	510	12	(	(	PUNCT
ejpam-841	510	13	y	y	NOUN
ejpam-841	510	14	x	x	INTJ
ejpam-841	510	15	,	,	PUNCT
ejpam-841	510	16	z	z	NOUN
ejpam-841	510	17	x	x	SYM
ejpam-841	510	18	)	)	PUNCT
ejpam-841	511	1	=	=	PUNCT
ejpam-841	511	2	x	x	X
ejpam-841	511	3	{	{	PUNCT
ejpam-841	511	4	[	[	X
ejpam-841	511	5	(	(	PUNCT
ejpam-841	511	6	s+	s+	X
ejpam-841	511	7	t	t	PROPN
ejpam-841	511	8	−	−	PROPN
ejpam-841	511	9	2	2	NUM
ejpam-841	511	10	t	t	PROPN
ejpam-841	511	11	(	(	PUNCT
ejpam-841	511	12	y	y	NOUN
ejpam-841	511	13	x	x	PROPN
ejpam-841	511	14	)	)	PUNCT
ejpam-841	511	15	]	]	X
ejpam-841	511	16	w	w	ADP
ejpam-841	511	17	−	−	PROPN
ejpam-841	511	18	t	t	PROPN
ejpam-841	511	19	(	(	PUNCT
ejpam-841	511	20	z	z	NOUN
ejpam-841	511	21	x	x	SYM
ejpam-841	511	22	)	)	PUNCT
ejpam-841	512	1	z	z	X
ejpam-841	512	2	}	}	PUNCT
ejpam-841	512	3	=	=	PUNCT
ejpam-841	513	1	−8xws+	−8xws+	NOUN
ejpam-841	513	2	yws+	yws+	PROPN
ejpam-841	513	3	4zws−	4zws−	ADJ
ejpam-841	514	1	x2	x2	PROPN
ejpam-841	514	2	t	t	NOUN
ejpam-841	514	3	+	+	CCONJ
ejpam-841	514	4	2xzt	2xzt	NUM
ejpam-841	514	5	−	−	PROPN
ejpam-841	514	6	z2	z2	PROPN
ejpam-841	514	7	t	t	PROPN
ejpam-841	514	8	−	−	PROPN
ejpam-841	514	9	10xwt	10xwt	PRON
ejpam-841	515	1	+	+	CCONJ
ejpam-841	515	2	ywt	ywt	NOUN
ejpam-841	515	3	+	+	X
ejpam-841	515	4	4zwt	4zwt	NOUN
ejpam-841	515	5	.	.	PUNCT
ejpam-841	515	6	example	example	NOUN
ejpam-841	515	7	2	2	NUM
ejpam-841	515	8	.	.	X
ejpam-841	515	9	consider	consider	VERB
ejpam-841	515	10	the	the	DET
ejpam-841	515	11	non	non	ADJ
ejpam-841	515	12	-	-	ADJ
ejpam-841	515	13	singular	singular	ADJ
ejpam-841	515	14	rational	rational	ADJ
ejpam-841	515	15	degree	degree	NOUN
ejpam-841	515	16	7	7	NUM
ejpam-841	515	17	space	space	NOUN
ejpam-841	515	18	curve	curve	NOUN
ejpam-841	515	19	given	give	VERB
ejpam-841	515	20	by	by	ADP
ejpam-841	515	21	f(s	f(	NOUN
ejpam-841	515	22	,	,	PUNCT
ejpam-841	515	23	t	t	PROPN
ejpam-841	515	24	)	)	PUNCT
ejpam-841	515	25	=	=	PUNCT
ejpam-841	516	1	(	(	PUNCT
ejpam-841	516	2	s7	s7	PROPN
ejpam-841	516	3	,	,	PUNCT
ejpam-841	516	4	s6	s6	PROPN
ejpam-841	516	5	t	t	PROPN
ejpam-841	516	6	,	,	PUNCT
ejpam-841	516	7	st6	st6	PROPN
ejpam-841	516	8	,	,	PUNCT
ejpam-841	516	9	t7	t7	PROPN
ejpam-841	516	10	)	)	PUNCT
ejpam-841	516	11	.	.	PUNCT
ejpam-841	517	1	compute	compute	VERB
ejpam-841	517	2	a	a	DET
ejpam-841	517	3	µ-basis	µ-basis	NOUN
ejpam-841	517	4	for	for	ADP
ejpam-841	517	5	f(s	f(	NOUN
ejpam-841	517	6	,	,	PUNCT
ejpam-841	517	7	t	t	PROPN
ejpam-841	517	8	):	):	PUNCT
ejpam-841	517	9	p	p	X
ejpam-841	517	10	=	=	PUNCT
ejpam-841	517	11	ys−	ys−	PUNCT
ejpam-841	517	12	x	x	SYM
ejpam-841	517	13	t	t	PROPN
ejpam-841	517	14	,	,	PUNCT
ejpam-841	517	15	q	q	X
ejpam-841	517	16	=	=	SYM
ejpam-841	517	17	ws−	ws−	PUNCT
ejpam-841	517	18	zt	zt	PROPN
ejpam-841	517	19	,	,	PUNCT
ejpam-841	517	20	r	r	NOUN
ejpam-841	517	21	=	=	SYM
ejpam-841	517	22	zs5	zs5	NOUN
ejpam-841	517	23	−	−	PROPN
ejpam-841	517	24	y	y	PROPN
ejpam-841	517	25	t5	t5	PROPN
ejpam-841	517	26	.	.	PUNCT
ejpam-841	518	1	the	the	DET
ejpam-841	518	2	minimal	minimal	ADJ
ejpam-841	518	3	generators	generator	NOUN
ejpam-841	518	4	for	for	ADP
ejpam-841	518	5	the	the	DET
ejpam-841	518	6	rees	rees	PROPN
ejpam-841	518	7	algebra	algebra	NOUN
ejpam-841	518	8	associated	associate	VERB
ejpam-841	518	9	to	to	ADP
ejpam-841	518	10	this	this	DET
ejpam-841	518	11	curve	curve	NOUN
ejpam-841	518	12	consist	consist	NOUN
ejpam-841	518	13	of	of	ADP
ejpam-841	518	14	p	p	X
ejpam-841	518	15	,	,	PUNCT
ejpam-841	518	16	q	q	ADJ
ejpam-841	518	17	,	,	PUNCT
ejpam-841	518	18	r	r	NOUN
ejpam-841	518	19	,	,	PUNCT
ejpam-841	518	20	the	the	DET
ejpam-841	518	21	following	follow	VERB
ejpam-841	518	22	quadric	quadric	ADJ
ejpam-841	518	23	and	and	CCONJ
ejpam-841	518	24	sextic	sextic	ADJ
ejpam-841	518	25	implicit	implicit	ADJ
ejpam-841	518	26	equations	equation	NOUN
ejpam-841	518	27	of	of	ADP
ejpam-841	518	28	the	the	DET
ejpam-841	518	29	space	space	NOUN
ejpam-841	518	30	curve	curve	NOUN
ejpam-841	518	31	:	:	PUNCT
ejpam-841	518	32	sylvs	sylvs	NOUN
ejpam-841	518	33	,	,	PUNCT
ejpam-841	518	34	t(p	t(p	PROPN
ejpam-841	518	35	,	,	PUNCT
ejpam-841	518	36	q	q	NOUN
ejpam-841	518	37	)	)	PUNCT
ejpam-841	518	38	=	=	SYM
ejpam-841	518	39	xw−	xw−	PUNCT
ejpam-841	518	40	yz	yz	PROPN
ejpam-841	518	41	;	;	PUNCT
ejpam-841	518	42	z5r	z5r	NOUN
ejpam-841	518	43	′i	′i	NOUN
ejpam-841	518	44	(	(	PUNCT
ejpam-841	518	45	w	w	NOUN
ejpam-841	518	46	z	z	NOUN
ejpam-841	518	47	,	,	PUNCT
ejpam-841	518	48	y	y	PROPN
ejpam-841	518	49	x	x	PUNCT
ejpam-841	518	50	)	)	PUNCT
ejpam-841	518	51	=	=	PUNCT
ejpam-841	519	1	z5[z	z5[z	NOUN
ejpam-841	519	2	−	−	PROPN
ejpam-841	519	3	y	y	PROPN
ejpam-841	519	4	(	(	PUNCT
ejpam-841	519	5	w	w	PROPN
ejpam-841	519	6	z	z	NOUN
ejpam-841	519	7	)	)	PUNCT
ejpam-841	519	8	5	5	NUM
ejpam-841	519	9	]	]	PUNCT
ejpam-841	519	10	=	=	SYM
ejpam-841	519	11	z6	z6	PROPN
ejpam-841	520	1	−	−	PROPN
ejpam-841	521	1	yw5	yw5	NOUN
ejpam-841	521	2	;	;	PUNCT
ejpam-841	521	3	xz4r	xz4r	PROPN
ejpam-841	521	4	′i	′i	NOUN
ejpam-841	521	5	(	(	PUNCT
ejpam-841	521	6	w	w	PROPN
ejpam-841	521	7	z	z	NOUN
ejpam-841	521	8	,	,	PUNCT
ejpam-841	521	9	y	y	PROPN
ejpam-841	521	10	x	x	PUNCT
ejpam-841	521	11	)	)	PUNCT
ejpam-841	521	12	=	=	PUNCT
ejpam-841	522	1	xz4[z	xz4[z	PROPN
ejpam-841	522	2	−	−	PROPN
ejpam-841	523	1	y	y	PROPN
ejpam-841	523	2	(	(	PUNCT
ejpam-841	523	3	w	w	PROPN
ejpam-841	523	4	z	z	NOUN
ejpam-841	523	5	)	)	PUNCT
ejpam-841	523	6	4	4	NUM
ejpam-841	523	7	(	(	PUNCT
ejpam-841	523	8	y	y	NOUN
ejpam-841	523	9	x	x	PROPN
ejpam-841	523	10	)	)	PUNCT
ejpam-841	523	11	]	]	PUNCT
ejpam-841	524	1	=	=	PUNCT
ejpam-841	524	2	xz5	xz5	NOUN
ejpam-841	524	3	−	−	PROPN
ejpam-841	524	4	y2w4	y2w4	NOUN
ejpam-841	524	5	;	;	PUNCT
ejpam-841	524	6	x2z3r	x2z3r	PUNCT
ejpam-841	525	1	′i	′i	NOUN
ejpam-841	525	2	(	(	PUNCT
ejpam-841	525	3	w	w	PROPN
ejpam-841	525	4	z	z	NOUN
ejpam-841	525	5	,	,	PUNCT
ejpam-841	525	6	y	y	PROPN
ejpam-841	525	7	x	x	PUNCT
ejpam-841	525	8	)	)	PUNCT
ejpam-841	525	9	=	=	PUNCT
ejpam-841	526	1	x2z3[z	x2z3[z	PROPN
ejpam-841	527	1	−	−	PROPN
ejpam-841	527	2	y	y	PROPN
ejpam-841	527	3	(	(	PUNCT
ejpam-841	527	4	w	w	PROPN
ejpam-841	527	5	z	z	NOUN
ejpam-841	527	6	)	)	PUNCT
ejpam-841	527	7	3	3	NUM
ejpam-841	527	8	(	(	PUNCT
ejpam-841	527	9	y	y	NOUN
ejpam-841	527	10	x	x	PROPN
ejpam-841	527	11	)	)	PUNCT
ejpam-841	527	12	2	2	NUM
ejpam-841	527	13	]	]	PUNCT
ejpam-841	527	14	=	=	PUNCT
ejpam-841	527	15	x2z4	x2z4	PROPN
ejpam-841	528	1	−	−	PROPN
ejpam-841	528	2	y3w3	y3w3	NOUN
ejpam-841	528	3	;	;	PUNCT
ejpam-841	528	4	x3z2r	x3z2r	PUNCT
ejpam-841	529	1	′i	′i	NOUN
ejpam-841	529	2	(	(	PUNCT
ejpam-841	529	3	w	w	NOUN
ejpam-841	529	4	z	z	NOUN
ejpam-841	529	5	,	,	PUNCT
ejpam-841	529	6	y	y	PROPN
ejpam-841	529	7	x	x	PUNCT
ejpam-841	529	8	)	)	PUNCT
ejpam-841	529	9	=	=	PUNCT
ejpam-841	530	1	x3z2[z	x3z2[z	PROPN
ejpam-841	531	1	−	−	PROPN
ejpam-841	531	2	y	y	PROPN
ejpam-841	531	3	(	(	PUNCT
ejpam-841	531	4	w	w	PROPN
ejpam-841	531	5	z	z	NOUN
ejpam-841	531	6	)	)	PUNCT
ejpam-841	531	7	2	2	NUM
ejpam-841	531	8	(	(	PUNCT
ejpam-841	531	9	y	y	NOUN
ejpam-841	531	10	x	x	PROPN
ejpam-841	531	11	)	)	PUNCT
ejpam-841	531	12	3	3	X
ejpam-841	531	13	]	]	PUNCT
ejpam-841	531	14	=	=	PUNCT
ejpam-841	531	15	x3z3	x3z3	PUNCT
ejpam-841	531	16	−	−	PROPN
ejpam-841	531	17	y4w2	y4w2	NOUN
ejpam-841	531	18	;	;	PUNCT
ejpam-841	531	19	x4zr	x4zr	X
ejpam-841	532	1	′i	′i	NOUN
ejpam-841	532	2	(	(	PUNCT
ejpam-841	532	3	w	w	PROPN
ejpam-841	532	4	z	z	NOUN
ejpam-841	532	5	,	,	PUNCT
ejpam-841	532	6	y	y	PROPN
ejpam-841	532	7	x	x	PUNCT
ejpam-841	532	8	)	)	PUNCT
ejpam-841	532	9	=	=	SYM
ejpam-841	532	10	x4z[z	x4z[z	NOUN
ejpam-841	532	11	−	−	PROPN
ejpam-841	532	12	y	y	PROPN
ejpam-841	532	13	(	(	PUNCT
ejpam-841	532	14	w	w	PROPN
ejpam-841	532	15	z	z	NOUN
ejpam-841	532	16	)	)	PUNCT
ejpam-841	532	17	(	(	PUNCT
ejpam-841	532	18	y	y	NOUN
ejpam-841	532	19	x	x	PROPN
ejpam-841	532	20	)	)	PUNCT
ejpam-841	532	21	4	4	NUM
ejpam-841	532	22	]	]	PUNCT
ejpam-841	532	23	=	=	SYM
ejpam-841	532	24	x4z2	x4z2	X
ejpam-841	532	25	−	−	PROPN
ejpam-841	532	26	y5w	y5w	NOUN
ejpam-841	532	27	;	;	PUNCT
ejpam-841	532	28	x5r	x5r	X
ejpam-841	532	29	′i	′i	NOUN
ejpam-841	532	30	(	(	PUNCT
ejpam-841	532	31	w	w	PROPN
ejpam-841	532	32	z	z	NOUN
ejpam-841	532	33	,	,	PUNCT
ejpam-841	532	34	y	y	PROPN
ejpam-841	532	35	x	x	PUNCT
ejpam-841	532	36	)	)	PUNCT
ejpam-841	532	37	=	=	SYM
ejpam-841	532	38	x5[z	x5[z	PROPN
ejpam-841	533	1	−	−	PROPN
ejpam-841	533	2	y	y	PROPN
ejpam-841	533	3	(	(	PUNCT
ejpam-841	533	4	y	y	NOUN
ejpam-841	533	5	x	x	PROPN
ejpam-841	533	6	)	)	PUNCT
ejpam-841	533	7	5	5	NUM
ejpam-841	533	8	]	]	PUNCT
ejpam-841	533	9	=	=	PUNCT
ejpam-841	533	10	x5z	x5z	PUNCT
ejpam-841	534	1	−	−	NOUN
ejpam-841	534	2	y6	y6	ADJ
ejpam-841	534	3	,	,	PUNCT
ejpam-841	534	4	together	together	ADV
ejpam-841	534	5	with	with	ADP
ejpam-841	534	6	d(d	d(d	PROPN
ejpam-841	534	7	−	−	PROPN
ejpam-841	534	8	3	3	NUM
ejpam-841	534	9	)	)	SYM
ejpam-841	534	10	2	2	NUM
ejpam-841	534	11	=	=	SYM
ejpam-841	534	12	7(4	7(4	NUM
ejpam-841	534	13	)	)	PUNCT
ejpam-841	534	14	2	2	NUM
ejpam-841	534	15	=	=	SYM
ejpam-841	534	16	14	14	NUM
ejpam-841	534	17	moving	move	VERB
ejpam-841	534	18	planes	plane	NOUN
ejpam-841	534	19	of	of	ADP
ejpam-841	534	20	degree	degree	NOUN
ejpam-841	534	21	(	(	PUNCT
ejpam-841	534	22	d	d	NOUN
ejpam-841	534	23	−	−	PROPN
ejpam-841	534	24	2−	2−	NUM
ejpam-841	535	1	i	i	PRON
ejpam-841	535	2	,	,	PUNCT
ejpam-841	535	3	i	i	PRON
ejpam-841	535	4	+	+	NOUN
ejpam-841	535	5	1	1	NUM
ejpam-841	535	6	)	)	PUNCT
ejpam-841	535	7	,	,	PUNCT
ejpam-841	535	8	x	x	PROPN
ejpam-841	535	9	jz	jz	PROPN
ejpam-841	535	10	i−	i−	PROPN
ejpam-841	535	11	j	j	PROPN
ejpam-841	535	12	r	r	NOUN
ejpam-841	535	13	′j	′j	NOUN
ejpam-841	535	14	,	,	PUNCT
ejpam-841	535	15	i−	i−	PROPN
ejpam-841	535	16	j	j	PROPN
ejpam-841	535	17	(	(	PUNCT
ejpam-841	535	18	w	w	PROPN
ejpam-841	535	19	z	z	PROPN
ejpam-841	535	20	,	,	PUNCT
ejpam-841	535	21	y	y	PROPN
ejpam-841	535	22	x	x	PROPN
ejpam-841	535	23	)	)	PUNCT
ejpam-841	535	24	,	,	PUNCT
ejpam-841	535	25	∀i	∀i	NOUN
ejpam-841	535	26	=	=	SYM
ejpam-841	535	27	1	1	NUM
ejpam-841	535	28	,	,	PUNCT
ejpam-841	535	29	.	.	PUNCT
ejpam-841	535	30	.	.	PUNCT
ejpam-841	535	31	.	.	PUNCT
ejpam-841	536	1	,	,	PUNCT
ejpam-841	537	1	d	d	X
ejpam-841	537	2	−	−	X
ejpam-841	537	3	3=	3=	NUM
ejpam-841	537	4	4	4	NUM
ejpam-841	537	5	,	,	PUNCT
ejpam-841	537	6	0≤	0≤	NUM
ejpam-841	538	1	j	j	PROPN
ejpam-841	538	2	≤	≤	PROPN
ejpam-841	538	3	i	i	PRON
ejpam-841	538	4	,	,	PUNCT
ejpam-841	538	5	i.e.	i.e.	X
ejpam-841	538	6	,	,	PUNCT
ejpam-841	538	7	z[zs4	z[zs4	PROPN
ejpam-841	538	8	−	−	PROPN
ejpam-841	538	9	y	y	PROPN
ejpam-841	538	10	t4	t4	PROPN
ejpam-841	538	11	(	(	PUNCT
ejpam-841	538	12	w	w	PROPN
ejpam-841	538	13	z	z	NOUN
ejpam-841	538	14	)	)	PUNCT
ejpam-841	538	15	]	]	PUNCT
ejpam-841	539	1	=	=	PUNCT
ejpam-841	540	1	z2s4	z2s4	X
ejpam-841	540	2	−	−	X
ejpam-841	540	3	ywt4	ywt4	NOUN
ejpam-841	540	4	,	,	PUNCT
ejpam-841	540	5	x[zs4	x[zs4	PROPN
ejpam-841	541	1	−	−	PROPN
ejpam-841	541	2	y	y	PROPN
ejpam-841	541	3	t4	t4	PROPN
ejpam-841	541	4	(	(	PUNCT
ejpam-841	541	5	y	y	PROPN
ejpam-841	541	6	x	x	PROPN
ejpam-841	541	7	)	)	PUNCT
ejpam-841	541	8	]	]	PUNCT
ejpam-841	542	1	=	=	PUNCT
ejpam-841	542	2	xzs4	xzs4	PROPN
ejpam-841	542	3	−	−	PROPN
ejpam-841	542	4	y2	y2	PROPN
ejpam-841	542	5	t4	t4	PROPN
ejpam-841	542	6	,	,	PUNCT
ejpam-841	542	7	z2[zs3	z2[zs3	PROPN
ejpam-841	542	8	−	−	PROPN
ejpam-841	542	9	y	y	PROPN
ejpam-841	542	10	t3	t3	PROPN
ejpam-841	542	11	(	(	PUNCT
ejpam-841	542	12	w	w	PROPN
ejpam-841	542	13	z	z	NOUN
ejpam-841	542	14	)	)	PUNCT
ejpam-841	542	15	2	2	NUM
ejpam-841	542	16	]	]	PUNCT
ejpam-841	542	17	=	=	PUNCT
ejpam-841	542	18	z3s3	z3s3	NOUN
ejpam-841	542	19	−	−	PROPN
ejpam-841	542	20	yw2	yw2	PROPN
ejpam-841	542	21	t3	t3	PROPN
ejpam-841	542	22	,	,	PUNCT
ejpam-841	542	23	xz[zs3	xz[zs3	PROPN
ejpam-841	542	24	−	−	PROPN
ejpam-841	542	25	y	y	PROPN
ejpam-841	542	26	t3	t3	PROPN
ejpam-841	542	27	(	(	PUNCT
ejpam-841	542	28	y	y	PROPN
ejpam-841	542	29	x	x	PROPN
ejpam-841	542	30	)	)	PUNCT
ejpam-841	542	31	(	(	PUNCT
ejpam-841	542	32	w	w	NOUN
ejpam-841	542	33	z	z	NOUN
ejpam-841	542	34	)	)	PUNCT
ejpam-841	542	35	]	]	PUNCT
ejpam-841	543	1	=	=	PUNCT
ejpam-841	543	2	xz2s3	xz2s3	PROPN
ejpam-841	543	3	−	−	PROPN
ejpam-841	543	4	y2wt3	y2wt3	NOUN
ejpam-841	543	5	,	,	PUNCT
ejpam-841	543	6	x2[zs3	x2[zs3	PROPN
ejpam-841	543	7	−	−	PROPN
ejpam-841	543	8	y	y	PROPN
ejpam-841	543	9	t3	t3	PROPN
ejpam-841	543	10	(	(	PUNCT
ejpam-841	543	11	y	y	PROPN
ejpam-841	543	12	x	x	PROPN
ejpam-841	543	13	)	)	PUNCT
ejpam-841	543	14	2	2	NUM
ejpam-841	543	15	]	]	PUNCT
ejpam-841	543	16	=	=	PUNCT
ejpam-841	543	17	x2zs3	x2zs3	PROPN
ejpam-841	544	1	−	−	PROPN
ejpam-841	544	2	y3	y3	PROPN
ejpam-841	544	3	t3	t3	PROPN
ejpam-841	544	4	,	,	PUNCT
ejpam-841	544	5	z3[zs2	z3[zs2	PROPN
ejpam-841	544	6	−	−	PROPN
ejpam-841	544	7	y	y	PROPN
ejpam-841	544	8	t2	t2	PROPN
ejpam-841	544	9	(	(	PUNCT
ejpam-841	544	10	w	w	PROPN
ejpam-841	544	11	z	z	NOUN
ejpam-841	544	12	)	)	PUNCT
ejpam-841	544	13	3	3	X
ejpam-841	544	14	]	]	PUNCT
ejpam-841	544	15	=	=	PUNCT
ejpam-841	544	16	z4s2	z4s2	NUM
ejpam-841	544	17	−	−	NOUN
ejpam-841	544	18	yw3	yw3	PROPN
ejpam-841	544	19	t2	t2	NOUN
ejpam-841	544	20	,	,	PUNCT
ejpam-841	544	21	xz2[zs2	xz2[zs2	NUM
ejpam-841	545	1	−	−	PROPN
ejpam-841	545	2	y	y	PROPN
ejpam-841	545	3	t2	t2	PROPN
ejpam-841	545	4	(	(	PUNCT
ejpam-841	545	5	y	y	NOUN
ejpam-841	545	6	x	x	PROPN
ejpam-841	545	7	)	)	PUNCT
ejpam-841	545	8	(	(	PUNCT
ejpam-841	545	9	w	w	NOUN
ejpam-841	545	10	z	z	NOUN
ejpam-841	545	11	)	)	PUNCT
ejpam-841	545	12	2	2	NUM
ejpam-841	545	13	]	]	PUNCT
ejpam-841	545	14	=	=	PUNCT
ejpam-841	546	1	xz3s2	xz3s2	PROPN
ejpam-841	546	2	−	−	NOUN
ejpam-841	546	3	y2w2	y2w2	PROPN
ejpam-841	546	4	t2	t2	PROPN
ejpam-841	546	5	,	,	PUNCT
ejpam-841	546	6	x2z[zs2	x2z[zs2	NOUN
ejpam-841	547	1	−	−	PROPN
ejpam-841	547	2	y	y	PROPN
ejpam-841	547	3	t2	t2	PROPN
ejpam-841	547	4	(	(	PUNCT
ejpam-841	547	5	y	y	NOUN
ejpam-841	547	6	x	x	PROPN
ejpam-841	547	7	)	)	PUNCT
ejpam-841	547	8	2	2	NUM
ejpam-841	547	9	(	(	PUNCT
ejpam-841	547	10	w	w	NOUN
ejpam-841	547	11	z	z	NOUN
ejpam-841	547	12	)	)	PUNCT
ejpam-841	547	13	]	]	PUNCT
ejpam-841	548	1	=	=	PUNCT
ejpam-841	548	2	x2z2s2	x2z2s2	PROPN
ejpam-841	548	3	−	−	PROPN
ejpam-841	548	4	y3wt2	y3wt2	NOUN
ejpam-841	548	5	,	,	PUNCT
ejpam-841	548	6	j.	j.	PROPN
ejpam-841	548	7	hoffman	hoffman	PROPN
ejpam-841	548	8	,	,	PUNCT
ejpam-841	548	9	h.	h.	PROPN
ejpam-841	548	10	wang	wang	PROPN
ejpam-841	548	11	,	,	PUNCT
ejpam-841	548	12	x.	x.	PROPN
ejpam-841	548	13	jia	jia	PROPN
ejpam-841	548	14	,	,	PUNCT
ejpam-841	548	15	r.	r.	PROPN
ejpam-841	548	16	goldman	goldman	PROPN
ejpam-841	548	17	/	/	SYM
ejpam-841	548	18	eur	eur	PROPN
ejpam-841	548	19	.	.	PUNCT
ejpam-841	549	1	j.	j.	PROPN
ejpam-841	549	2	pure	pure	PROPN
ejpam-841	549	3	appl	appl	PROPN
ejpam-841	549	4	.	.	PROPN
ejpam-841	549	5	math	math	PROPN
ejpam-841	549	6	,	,	PUNCT
ejpam-841	549	7	3	3	NUM
ejpam-841	549	8	(	(	PUNCT
ejpam-841	549	9	2010	2010	NUM
ejpam-841	549	10	)	)	PUNCT
ejpam-841	549	11	,	,	PUNCT
ejpam-841	549	12	602	602	NUM
ejpam-841	549	13	-	-	SYM
ejpam-841	549	14	632	632	NUM
ejpam-841	549	15	620	620	NUM
ejpam-841	549	16	x3[zs2	x3[zs2	NUM
ejpam-841	549	17	−	−	PROPN
ejpam-841	550	1	y	y	PROPN
ejpam-841	550	2	t2	t2	PROPN
ejpam-841	550	3	(	(	PUNCT
ejpam-841	550	4	y	y	NOUN
ejpam-841	550	5	x	x	PROPN
ejpam-841	550	6	)	)	PUNCT
ejpam-841	550	7	3	3	X
ejpam-841	550	8	]	]	PUNCT
ejpam-841	550	9	=	=	PUNCT
ejpam-841	550	10	x3zs2	x3zs2	PROPN
ejpam-841	550	11	−	−	PROPN
ejpam-841	550	12	y4	y4	PROPN
ejpam-841	550	13	t2	t2	PROPN
ejpam-841	550	14	,	,	PUNCT
ejpam-841	550	15	z4[zs−	z4[zs−	PROPN
ejpam-841	550	16	y	y	PROPN
ejpam-841	550	17	t	t	PROPN
ejpam-841	550	18	(	(	PUNCT
ejpam-841	550	19	w	w	PROPN
ejpam-841	550	20	z	z	NOUN
ejpam-841	550	21	)	)	PUNCT
ejpam-841	550	22	4	4	NUM
ejpam-841	550	23	]	]	PUNCT
ejpam-841	550	24	=	=	PUNCT
ejpam-841	550	25	z5s−	z5s−	NUM
ejpam-841	550	26	yw4	yw4	PROPN
ejpam-841	550	27	t	t	PROPN
ejpam-841	550	28	,	,	PUNCT
ejpam-841	550	29	xz3[zs−	xz3[zs−	NUM
ejpam-841	550	30	y	y	PROPN
ejpam-841	550	31	t	t	PROPN
ejpam-841	550	32	(	(	PUNCT
ejpam-841	550	33	y	y	NOUN
ejpam-841	550	34	x	x	PROPN
ejpam-841	550	35	)	)	PUNCT
ejpam-841	550	36	(	(	PUNCT
ejpam-841	550	37	w	w	NOUN
ejpam-841	550	38	z	z	NOUN
ejpam-841	550	39	)	)	PUNCT
ejpam-841	550	40	3	3	X
ejpam-841	550	41	]	]	PUNCT
ejpam-841	550	42	=	=	PUNCT
ejpam-841	551	1	xz4s−	xz4s−	NUM
ejpam-841	551	2	y2w3	y2w3	PROPN
ejpam-841	551	3	t	t	PROPN
ejpam-841	551	4	,	,	PUNCT
ejpam-841	551	5	x2z2[zs−	x2z2[zs−	PROPN
ejpam-841	551	6	y	y	PROPN
ejpam-841	551	7	t	t	PROPN
ejpam-841	551	8	(	(	PUNCT
ejpam-841	551	9	y	y	PROPN
ejpam-841	551	10	x	x	PROPN
ejpam-841	551	11	)	)	PUNCT
ejpam-841	551	12	2	2	NUM
ejpam-841	551	13	(	(	PUNCT
ejpam-841	551	14	w	w	NOUN
ejpam-841	551	15	z	z	NOUN
ejpam-841	551	16	)	)	PUNCT
ejpam-841	551	17	2	2	NUM
ejpam-841	551	18	]	]	PUNCT
ejpam-841	551	19	=	=	SYM
ejpam-841	551	20	x2z3s−	x2z3s−	PUNCT
ejpam-841	551	21	y3w2	y3w2	PROPN
ejpam-841	551	22	t	t	PROPN
ejpam-841	551	23	,	,	PUNCT
ejpam-841	551	24	x3z[zs	x3z[z	VERB
ejpam-841	551	25	−	−	PROPN
ejpam-841	551	26	y	y	PROPN
ejpam-841	551	27	t	t	PROPN
ejpam-841	551	28	(	(	PUNCT
ejpam-841	551	29	y	y	PROPN
ejpam-841	551	30	x	x	PROPN
ejpam-841	551	31	)	)	PUNCT
ejpam-841	551	32	3	3	NUM
ejpam-841	551	33	(	(	PUNCT
ejpam-841	551	34	w	w	NOUN
ejpam-841	551	35	z	z	NOUN
ejpam-841	551	36	)	)	PUNCT
ejpam-841	551	37	]	]	PUNCT
ejpam-841	552	1	=	=	SYM
ejpam-841	552	2	x3z2s−	x3z2s−	PUNCT
ejpam-841	552	3	y4wt	y4wt	PUNCT
ejpam-841	552	4	,	,	PUNCT
ejpam-841	552	5	x4[zs−	x4[zs−	PROPN
ejpam-841	552	6	y	y	PROPN
ejpam-841	552	7	t	t	PROPN
ejpam-841	552	8	(	(	PUNCT
ejpam-841	552	9	y	y	NOUN
ejpam-841	552	10	x	x	PROPN
ejpam-841	552	11	)	)	PUNCT
ejpam-841	552	12	4	4	NUM
ejpam-841	552	13	]	]	PUNCT
ejpam-841	552	14	=	=	PUNCT
ejpam-841	552	15	x4zs−	x4zs−	NUM
ejpam-841	552	16	y5	y5	NOUN
ejpam-841	552	17	t.	t.	NOUN
ejpam-841	552	18	4	4	NUM
ejpam-841	552	19	.	.	PUNCT
ejpam-841	552	20	rational	rational	ADJ
ejpam-841	552	21	quartic	quartic	ADJ
ejpam-841	552	22	space	space	NOUN
ejpam-841	552	23	curves	curve	NOUN
ejpam-841	552	24	in	in	ADP
ejpam-841	552	25	this	this	DET
ejpam-841	552	26	section	section	NOUN
ejpam-841	552	27	,	,	PUNCT
ejpam-841	552	28	we	we	PRON
ejpam-841	552	29	are	be	AUX
ejpam-841	552	30	going	go	VERB
ejpam-841	552	31	to	to	PART
ejpam-841	552	32	discuss	discuss	VERB
ejpam-841	552	33	some	some	PRON
ejpam-841	552	34	of	of	ADP
ejpam-841	552	35	the	the	DET
ejpam-841	552	36	geometry	geometry	NOUN
ejpam-841	552	37	behind	behind	ADP
ejpam-841	552	38	the	the	DET
ejpam-841	552	39	generators	generator	NOUN
ejpam-841	552	40	for	for	ADP
ejpam-841	552	41	the	the	DET
ejpam-841	552	42	rees	rees	PROPN
ejpam-841	552	43	algebra	algebra	NOUN
ejpam-841	552	44	by	by	ADP
ejpam-841	552	45	studying	study	VERB
ejpam-841	552	46	rational	rational	ADJ
ejpam-841	552	47	quartic	quartic	ADJ
ejpam-841	552	48	space	space	NOUN
ejpam-841	552	49	curves	curve	NOUN
ejpam-841	552	50	.	.	PUNCT
ejpam-841	553	1	first	first	ADV
ejpam-841	553	2	,	,	PUNCT
ejpam-841	553	3	we	we	PRON
ejpam-841	553	4	would	would	AUX
ejpam-841	553	5	like	like	VERB
ejpam-841	553	6	to	to	PART
ejpam-841	553	7	find	find	VERB
ejpam-841	553	8	implicit	implicit	ADJ
ejpam-841	553	9	equations	equation	NOUN
ejpam-841	553	10	for	for	ADP
ejpam-841	553	11	rational	rational	ADJ
ejpam-841	553	12	quartic	quartic	ADJ
ejpam-841	553	13	space	space	NOUN
ejpam-841	553	14	curves	curve	NOUN
ejpam-841	553	15	.	.	PUNCT
ejpam-841	554	1	we	we	PRON
ejpam-841	554	2	need	need	VERB
ejpam-841	554	3	to	to	PART
ejpam-841	554	4	consider	consider	VERB
ejpam-841	554	5	both	both	CCONJ
ejpam-841	554	6	singular	singular	ADJ
ejpam-841	554	7	and	and	CCONJ
ejpam-841	554	8	nonsingular	nonsingular	ADJ
ejpam-841	554	9	curves	curve	NOUN
ejpam-841	554	10	.	.	PUNCT
ejpam-841	555	1	we	we	PRON
ejpam-841	555	2	begin	begin	VERB
ejpam-841	555	3	with	with	ADP
ejpam-841	555	4	some	some	DET
ejpam-841	555	5	generic	generic	ADJ
ejpam-841	555	6	results	result	NOUN
ejpam-841	555	7	.	.	PUNCT
ejpam-841	556	1	lemma	lemma	PROPN
ejpam-841	556	2	4	4	X
ejpam-841	556	3	.	.	PUNCT
ejpam-841	557	1	if	if	SCONJ
ejpam-841	557	2	a	a	DET
ejpam-841	557	3	rational	rational	ADJ
ejpam-841	557	4	quartic	quartic	ADJ
ejpam-841	557	5	space	space	NOUN
ejpam-841	557	6	curve	curve	NOUN
ejpam-841	557	7	c	c	PROPN
ejpam-841	557	8	is	be	AUX
ejpam-841	557	9	singular	singular	ADJ
ejpam-841	557	10	,	,	PUNCT
ejpam-841	557	11	then	then	ADV
ejpam-841	557	12	the	the	DET
ejpam-841	557	13	curve	curve	NOUN
ejpam-841	557	14	c	c	PROPN
ejpam-841	557	15	is	be	AUX
ejpam-841	557	16	a	a	DET
ejpam-841	557	17	complete	complete	ADJ
ejpam-841	557	18	intersection	intersection	NOUN
ejpam-841	557	19	of	of	ADP
ejpam-841	557	20	two	two	NUM
ejpam-841	557	21	quadric	quadric	ADJ
ejpam-841	557	22	surfaces	surface	NOUN
ejpam-841	557	23	.	.	PUNCT
ejpam-841	558	1	if	if	SCONJ
ejpam-841	558	2	a	a	DET
ejpam-841	558	3	rational	rational	ADJ
ejpam-841	558	4	quartic	quartic	ADJ
ejpam-841	558	5	space	space	NOUN
ejpam-841	558	6	curve	curve	NOUN
ejpam-841	558	7	c	c	PROPN
ejpam-841	558	8	is	be	AUX
ejpam-841	558	9	non	non	ADJ
ejpam-841	558	10	-	-	ADJ
ejpam-841	558	11	singular	singular	ADJ
ejpam-841	558	12	,	,	PUNCT
ejpam-841	558	13	then	then	ADV
ejpam-841	558	14	the	the	DET
ejpam-841	558	15	curve	curve	NOUN
ejpam-841	558	16	c	c	PROPN
ejpam-841	558	17	is	be	AUX
ejpam-841	558	18	the	the	DET
ejpam-841	558	19	projection	projection	NOUN
ejpam-841	558	20	of	of	ADP
ejpam-841	558	21	a	a	DET
ejpam-841	558	22	rational	rational	ADJ
ejpam-841	558	23	normal	normal	ADJ
ejpam-841	558	24	quartic	quartic	ADJ
ejpam-841	558	25	curve	curve	NOUN
ejpam-841	558	26	s(4	s(4	NOUN
ejpam-841	558	27	)	)	PUNCT
ejpam-841	558	28	∈	∈	PROPN
ejpam-841	558	29	p4	p4	NOUN
ejpam-841	558	30	from	from	ADP
ejpam-841	558	31	a	a	DET
ejpam-841	558	32	point	point	NOUN
ejpam-841	558	33	q	q	NOUN
ejpam-841	558	34	/∈	/∈	PUNCT
ejpam-841	558	35	s(4	s(4	NOUN
ejpam-841	558	36	)	)	PUNCT
ejpam-841	558	37	.	.	PUNCT
ejpam-841	559	1	proof	proof	NOUN
ejpam-841	559	2	.	.	PUNCT
ejpam-841	560	1	it	it	PRON
ejpam-841	560	2	is	be	AUX
ejpam-841	560	3	known	know	VERB
ejpam-841	560	4	[	[	X
ejpam-841	560	5	14	14	NUM
ejpam-841	560	6	,	,	PUNCT
ejpam-841	560	7	page	page	NOUN
ejpam-841	560	8	353	353	NUM
ejpam-841	560	9	]	]	PUNCT
ejpam-841	560	10	that	that	SCONJ
ejpam-841	560	11	the	the	DET
ejpam-841	560	12	geometric	geometric	ADJ
ejpam-841	560	13	genus	genus	NOUN
ejpam-841	560	14	of	of	ADP
ejpam-841	560	15	every	every	DET
ejpam-841	560	16	rational	rational	ADJ
ejpam-841	560	17	quartic	quartic	ADJ
ejpam-841	560	18	space	space	NOUN
ejpam-841	560	19	curve	curve	NOUN
ejpam-841	560	20	c	c	PROPN
ejpam-841	560	21	is	be	AUX
ejpam-841	560	22	zero	zero	NUM
ejpam-841	560	23	,	,	PUNCT
ejpam-841	560	24	although	although	SCONJ
ejpam-841	560	25	the	the	DET
ejpam-841	560	26	arithmetic	arithmetic	ADJ
ejpam-841	560	27	genus	genus	NOUN
ejpam-841	560	28	of	of	ADP
ejpam-841	560	29	c	c	NOUN
ejpam-841	560	30	may	may	AUX
ejpam-841	560	31	be	be	AUX
ejpam-841	560	32	either	either	CCONJ
ejpam-841	560	33	zero	zero	NUM
ejpam-841	560	34	or	or	CCONJ
ejpam-841	560	35	one	one	NUM
ejpam-841	560	36	.	.	PUNCT
ejpam-841	561	1	if	if	SCONJ
ejpam-841	561	2	c	c	PROPN
ejpam-841	561	3	is	be	AUX
ejpam-841	561	4	singular	singular	ADJ
ejpam-841	561	5	,	,	PUNCT
ejpam-841	561	6	then	then	ADV
ejpam-841	561	7	the	the	DET
ejpam-841	561	8	arithmetic	arithmetic	ADJ
ejpam-841	561	9	genus	genus	NOUN
ejpam-841	561	10	of	of	ADP
ejpam-841	561	11	c	c	PROPN
ejpam-841	561	12	is	be	AUX
ejpam-841	561	13	one	one	NUM
ejpam-841	561	14	,	,	PUNCT
ejpam-841	561	15	and	and	CCONJ
ejpam-841	561	16	c	c	NOUN
ejpam-841	561	17	is	be	AUX
ejpam-841	561	18	a	a	DET
ejpam-841	561	19	complete	complete	ADJ
ejpam-841	561	20	intersection	intersection	NOUN
ejpam-841	561	21	of	of	ADP
ejpam-841	561	22	two	two	NUM
ejpam-841	561	23	quadric	quadric	ADJ
ejpam-841	561	24	surfaces	surface	NOUN
ejpam-841	561	25	[	[	X
ejpam-841	561	26	13	13	NUM
ejpam-841	561	27	,	,	PUNCT
ejpam-841	561	28	chapter	chapter	NOUN
ejpam-841	561	29	1	1	NUM
ejpam-841	561	30	,	,	PUNCT
ejpam-841	561	31	page	page	NOUN
ejpam-841	561	32	44	44	NUM
ejpam-841	561	33	]	]	PUNCT
ejpam-841	561	34	.	.	PUNCT
ejpam-841	562	1	ifc	ifc	NOUN
ejpam-841	562	2	is	be	AUX
ejpam-841	562	3	non	non	ADJ
ejpam-841	562	4	-	-	ADJ
ejpam-841	562	5	singular	singular	ADJ
ejpam-841	562	6	,	,	PUNCT
ejpam-841	562	7	then	then	ADV
ejpam-841	562	8	the	the	DET
ejpam-841	562	9	arithmetic	arithmetic	ADJ
ejpam-841	562	10	genus	genus	NOUN
ejpam-841	562	11	ofc	ofc	NOUN
ejpam-841	562	12	is	be	AUX
ejpam-841	562	13	zero	zero	NUM
ejpam-841	562	14	,	,	PUNCT
ejpam-841	562	15	andc	andc	PROPN
ejpam-841	562	16	is	be	AUX
ejpam-841	562	17	contained	contain	VERB
ejpam-841	562	18	in	in	ADP
ejpam-841	562	19	a	a	DET
ejpam-841	562	20	unique	unique	ADJ
ejpam-841	562	21	non	non	ADJ
ejpam-841	562	22	-	-	ADJ
ejpam-841	562	23	singular	singular	ADJ
ejpam-841	562	24	quadric	quadric	ADJ
ejpam-841	562	25	surface	surface	NOUN
ejpam-841	562	26	.	.	PUNCT
ejpam-841	563	1	hence	hence	ADV
ejpam-841	563	2	,	,	PUNCT
ejpam-841	563	3	by	by	ADP
ejpam-841	563	4	theorem	theorem	NOUN
ejpam-841	563	5	6	6	NUM
ejpam-841	563	6	[	[	SYM
ejpam-841	563	7	27	27	NUM
ejpam-841	563	8	]	]	X
ejpam-841	563	9	,	,	PUNCT
ejpam-841	563	10	a	a	DET
ejpam-841	563	11	rational	rational	ADJ
ejpam-841	563	12	non	non	ADJ
ejpam-841	563	13	-	-	ADJ
ejpam-841	563	14	singular	singular	ADJ
ejpam-841	563	15	quartic	quartic	ADJ
ejpam-841	563	16	space	space	NOUN
ejpam-841	563	17	curve	curve	NOUN
ejpam-841	563	18	is	be	AUX
ejpam-841	563	19	the	the	DET
ejpam-841	563	20	projection	projection	NOUN
ejpam-841	563	21	of	of	ADP
ejpam-841	563	22	a	a	DET
ejpam-841	563	23	rational	rational	ADJ
ejpam-841	563	24	normal	normal	ADJ
ejpam-841	563	25	curve	curve	NOUN
ejpam-841	563	26	s(4	s(4	NOUN
ejpam-841	563	27	)	)	PUNCT
ejpam-841	563	28	∈	∈	PROPN
ejpam-841	563	29	p4	p4	NOUN
ejpam-841	563	30	from	from	ADP
ejpam-841	563	31	a	a	DET
ejpam-841	563	32	point	point	NOUN
ejpam-841	563	33	q	q	NOUN
ejpam-841	563	34	/∈	/∈	PUNCT
ejpam-841	563	35	s(4	s(4	NOUN
ejpam-841	563	36	)	)	PUNCT
ejpam-841	563	37	.	.	PUNCT
ejpam-841	564	1	by	by	ADP
ejpam-841	564	2	lemma	lemma	PROPN
ejpam-841	564	3	1	1	NUM
ejpam-841	564	4	,	,	PUNCT
ejpam-841	564	5	we	we	PRON
ejpam-841	564	6	know	know	VERB
ejpam-841	564	7	that	that	SCONJ
ejpam-841	564	8	a	a	DET
ejpam-841	564	9	rational	rational	ADJ
ejpam-841	564	10	quartic	quartic	ADJ
ejpam-841	564	11	space	space	NOUN
ejpam-841	564	12	curve	curve	NOUN
ejpam-841	564	13	is	be	AUX
ejpam-841	564	14	contained	contain	VERB
ejpam-841	564	15	in	in	ADP
ejpam-841	564	16	the	the	DET
ejpam-841	564	17	quadric	quadric	ADJ
ejpam-841	564	18	surface	surface	NOUN
ejpam-841	564	19	given	give	VERB
ejpam-841	564	20	sylvs	sylvs	PROPN
ejpam-841	564	21	,	,	PUNCT
ejpam-841	564	22	t(p	t(p	PROPN
ejpam-841	564	23	,	,	PUNCT
ejpam-841	564	24	q	q	NOUN
ejpam-841	564	25	)	)	PUNCT
ejpam-841	564	26	=	=	SYM
ejpam-841	565	1	0	0	X
ejpam-841	565	2	.	.	PUNCT
ejpam-841	566	1	next	next	ADV
ejpam-841	566	2	we	we	PRON
ejpam-841	566	3	provide	provide	VERB
ejpam-841	566	4	methods	method	NOUN
ejpam-841	566	5	for	for	ADP
ejpam-841	566	6	finding	find	VERB
ejpam-841	566	7	implicit	implicit	ADJ
ejpam-841	566	8	equations	equation	NOUN
ejpam-841	566	9	for	for	ADP
ejpam-841	566	10	both	both	CCONJ
ejpam-841	566	11	singular	singular	ADJ
ejpam-841	566	12	and	and	CCONJ
ejpam-841	566	13	non	non	ADJ
ejpam-841	566	14	-	-	ADJ
ejpam-841	566	15	singular	singular	ADJ
ejpam-841	566	16	rational	rational	ADJ
ejpam-841	566	17	quartic	quartic	ADJ
ejpam-841	566	18	space	space	NOUN
ejpam-841	566	19	curves	curve	NOUN
ejpam-841	566	20	using	use	VERB
ejpam-841	566	21	moving	move	VERB
ejpam-841	566	22	planes	plane	NOUN
ejpam-841	566	23	and	and	CCONJ
ejpam-841	566	24	µ-bases	µ-base	NOUN
ejpam-841	566	25	.	.	PUNCT
ejpam-841	567	1	4.1	4.1	NUM
ejpam-841	567	2	.	.	PUNCT
ejpam-841	568	1	implicit	implicit	ADJ
ejpam-841	568	2	equations	equation	NOUN
ejpam-841	568	3	of	of	ADP
ejpam-841	568	4	singular	singular	ADJ
ejpam-841	568	5	rational	rational	ADJ
ejpam-841	568	6	quartic	quartic	ADJ
ejpam-841	568	7	space	space	NOUN
ejpam-841	568	8	curves	curve	NOUN
ejpam-841	568	9	by	by	ADP
ejpam-841	568	10	proposition	proposition	NOUN
ejpam-841	568	11	2	2	NUM
ejpam-841	568	12	,	,	PUNCT
ejpam-841	568	13	for	for	ADP
ejpam-841	568	14	a	a	DET
ejpam-841	568	15	singular	singular	ADJ
ejpam-841	568	16	rational	rational	ADJ
ejpam-841	568	17	quartic	quartic	ADJ
ejpam-841	568	18	space	space	NOUN
ejpam-841	568	19	curve	curve	NOUN
ejpam-841	568	20	c	c	PROPN
ejpam-841	568	21	with	with	ADP
ejpam-841	568	22	a	a	DET
ejpam-841	568	23	µ-basis	µ-basis	NOUN
ejpam-841	568	24	p	p	NOUN
ejpam-841	568	25	,	,	PUNCT
ejpam-841	568	26	q	q	ADJ
ejpam-841	568	27	,	,	PUNCT
ejpam-841	568	28	r	r	NOUN
ejpam-841	568	29	,	,	PUNCT
ejpam-841	568	30	the	the	DET
ejpam-841	568	31	only	only	ADJ
ejpam-841	568	32	singular	singular	ADJ
ejpam-841	568	33	point	point	NOUN
ejpam-841	568	34	p	p	NOUN
ejpam-841	568	35	is	be	AUX
ejpam-841	568	36	the	the	DET
ejpam-841	568	37	double	double	ADJ
ejpam-841	568	38	point	point	NOUN
ejpam-841	568	39	at	at	ADP
ejpam-841	568	40	the	the	DET
ejpam-841	568	41	intersection	intersection	NOUN
ejpam-841	568	42	of	of	ADP
ejpam-841	568	43	the	the	DET
ejpam-841	568	44	axes	axis	NOUN
ejpam-841	568	45	of	of	ADP
ejpam-841	568	46	p	p	PROPN
ejpam-841	568	47	and	and	CCONJ
ejpam-841	568	48	q.	q.	PROPN
ejpam-841	568	49	now	now	ADV
ejpam-841	568	50	suppose	suppose	VERB
ejpam-841	568	51	that	that	SCONJ
ejpam-841	568	52	p	p	PROPN
ejpam-841	568	53	is	be	AUX
ejpam-841	568	54	a	a	DET
ejpam-841	568	55	singular	singular	ADJ
ejpam-841	568	56	point	point	NOUN
ejpam-841	568	57	on	on	ADP
ejpam-841	568	58	a	a	DET
ejpam-841	568	59	rational	rational	ADJ
ejpam-841	568	60	quartic	quartic	ADJ
ejpam-841	568	61	space	space	NOUN
ejpam-841	568	62	curve	curve	NOUN
ejpam-841	568	63	c	c	PROPN
ejpam-841	568	64	.	.	PUNCT
ejpam-841	569	1	if	if	SCONJ
ejpam-841	569	2	we	we	PRON
ejpam-841	569	3	take	take	VERB
ejpam-841	569	4	two	two	NUM
ejpam-841	569	5	distinct	distinct	ADJ
ejpam-841	569	6	points	point	NOUN
ejpam-841	569	7	f(s1	f(s1	NOUN
ejpam-841	569	8	,	,	PUNCT
ejpam-841	569	9	t1	t1	NOUN
ejpam-841	569	10	)	)	PUNCT
ejpam-841	569	11	,	,	PUNCT
ejpam-841	569	12	f(s2	f(s2	NOUN
ejpam-841	569	13	,	,	PUNCT
ejpam-841	569	14	t2	t2	NOUN
ejpam-841	569	15	)	)	PUNCT
ejpam-841	569	16	different	different	ADJ
ejpam-841	569	17	from	from	ADP
ejpam-841	569	18	the	the	DET
ejpam-841	569	19	singular	singular	ADJ
ejpam-841	569	20	point	point	NOUN
ejpam-841	569	21	p	p	NOUN
ejpam-841	569	22	on	on	ADP
ejpam-841	569	23	the	the	DET
ejpam-841	569	24	space	space	NOUN
ejpam-841	569	25	curvec	curvec	NOUN
ejpam-841	569	26	,	,	PUNCT
ejpam-841	569	27	then	then	ADV
ejpam-841	569	28	by	by	ADP
ejpam-841	569	29	theorem	theorem	NOUN
ejpam-841	569	30	3.3	3.3	NUM
ejpam-841	569	31	[	[	SYM
ejpam-841	569	32	26	26	NUM
ejpam-841	569	33	]	]	PUNCT
ejpam-841	569	34	there	there	PRON
ejpam-841	569	35	are	be	VERB
ejpam-841	569	36	two	two	NUM
ejpam-841	569	37	moving	move	VERB
ejpam-841	569	38	planes	plane	NOUN
ejpam-841	569	39	l1,l2	l1,l2	PROPN
ejpam-841	569	40	of	of	ADP
ejpam-841	569	41	degree	degree	NOUN
ejpam-841	569	42	one	one	NOUN
ejpam-841	569	43	that	that	PRON
ejpam-841	569	44	follow	follow	VERB
ejpam-841	569	45	the	the	DET
ejpam-841	569	46	space	space	NOUN
ejpam-841	569	47	curve	curve	NOUN
ejpam-841	569	48	c	c	PROPN
ejpam-841	569	49	with	with	ADP
ejpam-841	569	50	axes	axis	NOUN
ejpam-841	569	51	pf(s1	pf(s1	NOUN
ejpam-841	569	52	,	,	PUNCT
ejpam-841	569	53	t1	t1	NOUN
ejpam-841	569	54	)	)	PUNCT
ejpam-841	569	55	and	and	CCONJ
ejpam-841	569	56	pf(s2	pf(s2	NOUN
ejpam-841	569	57	,	,	PUNCT
ejpam-841	569	58	t2	t2	PROPN
ejpam-841	569	59	)	)	PUNCT
ejpam-841	569	60	.	.	PUNCT
ejpam-841	570	1	moreover	moreover	ADV
ejpam-841	570	2	we	we	PRON
ejpam-841	570	3	can	can	AUX
ejpam-841	570	4	easily	easily	ADV
ejpam-841	570	5	choose	choose	VERB
ejpam-841	570	6	f(s1	f(s1	NOUN
ejpam-841	570	7	,	,	PUNCT
ejpam-841	570	8	t1),f(s2	t1),f(s2	PROPN
ejpam-841	570	9	,	,	PUNCT
ejpam-841	570	10	t2	t2	NOUN
ejpam-841	570	11	)	)	PUNCT
ejpam-841	570	12	so	so	SCONJ
ejpam-841	570	13	that	that	SCONJ
ejpam-841	570	14	these	these	DET
ejpam-841	570	15	axes	axis	NOUN
ejpam-841	570	16	are	be	AUX
ejpam-841	570	17	distinct	distinct	ADJ
ejpam-841	570	18	.	.	PUNCT
ejpam-841	571	1	since	since	SCONJ
ejpam-841	571	2	the	the	DET
ejpam-841	571	3	axes	axis	NOUN
ejpam-841	571	4	of	of	ADP
ejpam-841	571	5	l1,l2	l1,l2	PROPN
ejpam-841	571	6	are	be	AUX
ejpam-841	571	7	distinct	distinct	ADJ
ejpam-841	571	8	,	,	PUNCT
ejpam-841	571	9	the	the	DET
ejpam-841	571	10	moving	move	VERB
ejpam-841	571	11	planes	plane	NOUN
ejpam-841	571	12	l1,l2	l1,l2	PROPN
ejpam-841	571	13	are	be	AUX
ejpam-841	571	14	linearly	linearly	ADV
ejpam-841	571	15	independent	independent	ADJ
ejpam-841	571	16	.	.	PUNCT
ejpam-841	572	1	therefore	therefore	ADV
ejpam-841	572	2	,	,	PUNCT
ejpam-841	572	3	l1,l2	l1,l2	PROPN
ejpam-841	572	4	,	,	PUNCT
ejpam-841	572	5	r	r	NOUN
ejpam-841	572	6	form	form	NOUN
ejpam-841	572	7	another	another	DET
ejpam-841	572	8	µ-basis	µ-basis	NOUN
ejpam-841	572	9	for	for	ADP
ejpam-841	572	10	the	the	DET
ejpam-841	572	11	curve	curve	NOUN
ejpam-841	572	12	c	c	PROPN
ejpam-841	572	13	.	.	PUNCT
ejpam-841	573	1	hence	hence	ADV
ejpam-841	573	2	,	,	PUNCT
ejpam-841	573	3	without	without	ADP
ejpam-841	573	4	loss	loss	NOUN
ejpam-841	573	5	of	of	ADP
ejpam-841	573	6	generality	generality	NOUN
ejpam-841	573	7	,	,	PUNCT
ejpam-841	573	8	we	we	PRON
ejpam-841	573	9	can	can	AUX
ejpam-841	573	10	assume	assume	VERB
ejpam-841	573	11	that	that	SCONJ
ejpam-841	573	12	the	the	DET
ejpam-841	573	13	axes	axis	NOUN
ejpam-841	573	14	of	of	ADP
ejpam-841	573	15	p	p	NOUN
ejpam-841	573	16	and	and	CCONJ
ejpam-841	573	17	q	q	PROPN
ejpam-841	573	18	intersect	intersect	ADJ
ejpam-841	573	19	the	the	DET
ejpam-841	573	20	space	space	NOUN
ejpam-841	573	21	curve	curve	NOUN
ejpam-841	573	22	c	c	PROPN
ejpam-841	573	23	at	at	ADP
ejpam-841	573	24	two	two	NUM
ejpam-841	573	25	distinct	distinct	ADJ
ejpam-841	573	26	points	point	NOUN
ejpam-841	573	27	f(s1	f(s1	NOUN
ejpam-841	573	28	,	,	PUNCT
ejpam-841	573	29	t1	t1	NOUN
ejpam-841	573	30	)	)	PUNCT
ejpam-841	573	31	and	and	CCONJ
ejpam-841	573	32	f(s2	f(s2	NOUN
ejpam-841	573	33	,	,	PUNCT
ejpam-841	573	34	t2	t2	NOUN
ejpam-841	573	35	)	)	PUNCT
ejpam-841	573	36	other	other	ADJ
ejpam-841	573	37	than	than	ADP
ejpam-841	573	38	the	the	DET
ejpam-841	573	39	singular	singular	NOUN
ejpam-841	573	40	point	point	NOUN
ejpam-841	573	41	p.	p.	NOUN
ejpam-841	573	42	in	in	ADP
ejpam-841	573	43	this	this	DET
ejpam-841	573	44	case	case	NOUN
ejpam-841	573	45	,	,	PUNCT
ejpam-841	573	46	the	the	DET
ejpam-841	573	47	plane	plane	NOUN
ejpam-841	573	48	that	that	PRON
ejpam-841	573	49	contains	contain	VERB
ejpam-841	573	50	the	the	DET
ejpam-841	573	51	axes	axis	NOUN
ejpam-841	573	52	of	of	ADP
ejpam-841	573	53	p	p	PROPN
ejpam-841	573	54	and	and	CCONJ
ejpam-841	573	55	q	q	NOUN
ejpam-841	573	56	has	have	VERB
ejpam-841	573	57	the	the	DET
ejpam-841	573	58	implicit	implicit	ADJ
ejpam-841	573	59	equation	equation	NOUN
ejpam-841	573	60	a(x	a(x	NOUN
ejpam-841	573	61	)	)	PUNCT
ejpam-841	573	62	=	=	PUNCT
ejpam-841	574	1	[	[	X
ejpam-841	574	2	f(s1	f(s1	NOUN
ejpam-841	574	3	,	,	PUNCT
ejpam-841	574	4	t1),f(s2	t1),f(s2	PROPN
ejpam-841	574	5	,	,	PUNCT
ejpam-841	574	6	t2	t2	NOUN
ejpam-841	574	7	)	)	PUNCT
ejpam-841	574	8	,	,	PUNCT
ejpam-841	574	9	p	p	X
ejpam-841	574	10	]	]	X
ejpam-841	574	11	·	·	PUNCT
ejpam-841	574	12	x=	x=	X
ejpam-841	574	13	0	0	NUM
ejpam-841	574	14	,	,	PUNCT
ejpam-841	574	15	(	(	PUNCT
ejpam-841	574	16	10	10	NUM
ejpam-841	574	17	)	)	PUNCT
ejpam-841	574	18	j.	j.	PROPN
ejpam-841	574	19	hoffman	hoffman	PROPN
ejpam-841	574	20	,	,	PUNCT
ejpam-841	574	21	h.	h.	PROPN
ejpam-841	574	22	wang	wang	PROPN
ejpam-841	574	23	,	,	PUNCT
ejpam-841	574	24	x.	x.	PROPN
ejpam-841	574	25	jia	jia	PROPN
ejpam-841	574	26	,	,	PUNCT
ejpam-841	574	27	r.	r.	PROPN
ejpam-841	574	28	goldman	goldman	PROPN
ejpam-841	574	29	/	/	SYM
ejpam-841	574	30	eur	eur	PROPN
ejpam-841	574	31	.	.	PUNCT
ejpam-841	575	1	j.	j.	PROPN
ejpam-841	575	2	pure	pure	PROPN
ejpam-841	575	3	appl	appl	PROPN
ejpam-841	575	4	.	.	PROPN
ejpam-841	575	5	math	math	PROPN
ejpam-841	575	6	,	,	PUNCT
ejpam-841	575	7	3	3	NUM
ejpam-841	575	8	(	(	PUNCT
ejpam-841	575	9	2010	2010	NUM
ejpam-841	575	10	)	)	PUNCT
ejpam-841	575	11	,	,	PUNCT
ejpam-841	575	12	602	602	NUM
ejpam-841	575	13	-	-	SYM
ejpam-841	575	14	632	632	NUM
ejpam-841	575	15	621	621	NUM
ejpam-841	576	1	where	where	SCONJ
ejpam-841	576	2	x=	x=	PUNCT
ejpam-841	576	3	(	(	PUNCT
ejpam-841	576	4	x	x	X
ejpam-841	576	5	,	,	PUNCT
ejpam-841	576	6	y	y	PROPN
ejpam-841	576	7	,	,	PUNCT
ejpam-841	576	8	z	z	PROPN
ejpam-841	576	9	,	,	PUNCT
ejpam-841	576	10	w	w	PROPN
ejpam-841	576	11	)	)	PUNCT
ejpam-841	576	12	and	and	CCONJ
ejpam-841	576	13	[	[	X
ejpam-841	576	14	·	·	PUNCT
ejpam-841	576	15	]	]	X
ejpam-841	576	16	denotes	denote	VERB
ejpam-841	576	17	the	the	DET
ejpam-841	576	18	outer	outer	ADJ
ejpam-841	576	19	product	product	NOUN
ejpam-841	576	20	.	.	PUNCT
ejpam-841	577	1	next	next	ADV
ejpam-841	577	2	we	we	PRON
ejpam-841	577	3	are	be	AUX
ejpam-841	577	4	going	go	VERB
ejpam-841	577	5	to	to	PART
ejpam-841	577	6	find	find	VERB
ejpam-841	577	7	two	two	NUM
ejpam-841	577	8	quadric	quadric	ADJ
ejpam-841	577	9	surfaces	surface	NOUN
ejpam-841	577	10	that	that	PRON
ejpam-841	577	11	contain	contain	VERB
ejpam-841	577	12	the	the	DET
ejpam-841	577	13	singular	singular	ADJ
ejpam-841	577	14	rational	rational	ADJ
ejpam-841	577	15	quartic	quartic	ADJ
ejpam-841	577	16	space	space	NOUN
ejpam-841	577	17	curve	curve	NOUN
ejpam-841	577	18	c	c	PROPN
ejpam-841	577	19	.	.	PUNCT
ejpam-841	578	1	to	to	PART
ejpam-841	578	2	compute	compute	VERB
ejpam-841	578	3	the	the	DET
ejpam-841	578	4	implicit	implicit	ADJ
ejpam-841	578	5	equations	equation	NOUN
ejpam-841	578	6	of	of	ADP
ejpam-841	578	7	a	a	DET
ejpam-841	578	8	singular	singular	ADJ
ejpam-841	578	9	rational	rational	ADJ
ejpam-841	578	10	quartic	quartic	ADJ
ejpam-841	578	11	space	space	NOUN
ejpam-841	578	12	curve	curve	NOUN
ejpam-841	578	13	,	,	PUNCT
ejpam-841	578	14	we	we	PRON
ejpam-841	578	15	write	write	VERB
ejpam-841	578	16	the	the	DET
ejpam-841	578	17	µ-basis	µ-basis	NOUN
ejpam-841	578	18	in	in	ADP
ejpam-841	578	19	the	the	DET
ejpam-841	578	20	form	form	NOUN
ejpam-841	578	21	:	:	PUNCT
ejpam-841	578	22	p	p	X
ejpam-841	578	23	=	=	PUNCT
ejpam-841	578	24	p1s+	p1s+	PROPN
ejpam-841	578	25	p0	p0	NOUN
ejpam-841	578	26	t	t	PROPN
ejpam-841	578	27	,	,	PUNCT
ejpam-841	578	28	q	q	PROPN
ejpam-841	579	1	=	=	PUNCT
ejpam-841	579	2	q1s+	q1s+	NOUN
ejpam-841	579	3	q0	q0	PROPN
ejpam-841	579	4	t	t	PROPN
ejpam-841	579	5	,	,	PUNCT
ejpam-841	579	6	r	r	NOUN
ejpam-841	579	7	=	=	PUNCT
ejpam-841	579	8	r2s2	r2s2	PROPN
ejpam-841	579	9	+	+	NOUN
ejpam-841	579	10	r1st	r1st	PUNCT
ejpam-841	580	1	+	+	CCONJ
ejpam-841	580	2	r0	r0	NOUN
ejpam-841	580	3	t2	t2	NOUN
ejpam-841	580	4	,	,	PUNCT
ejpam-841	580	5	where	where	SCONJ
ejpam-841	580	6	p1	p1	NOUN
ejpam-841	580	7	,	,	PUNCT
ejpam-841	580	8	p0,q1,q0	p0,q1,q0	ADJ
ejpam-841	580	9	,	,	PUNCT
ejpam-841	580	10	r2	r2	NOUN
ejpam-841	580	11	,	,	PUNCT
ejpam-841	580	12	r1	r1	NOUN
ejpam-841	580	13	,	,	PUNCT
ejpam-841	580	14	r0	r0	NOUN
ejpam-841	580	15	are	be	AUX
ejpam-841	580	16	homogeneous	homogeneous	ADJ
ejpam-841	580	17	polynomials	polynomial	NOUN
ejpam-841	580	18	of	of	ADP
ejpam-841	580	19	degree	degree	NOUN
ejpam-841	580	20	1	1	NUM
ejpam-841	580	21	in	in	ADP
ejpam-841	580	22	x	x	SYM
ejpam-841	580	23	,	,	PUNCT
ejpam-841	580	24	y	y	PROPN
ejpam-841	580	25	,	,	PUNCT
ejpam-841	580	26	z	z	PROPN
ejpam-841	580	27	,	,	PUNCT
ejpam-841	580	28	w.	w.	PROPN
ejpam-841	580	29	let	let	VERB
ejpam-841	580	30	m(p	m(p	PROPN
ejpam-841	580	31	,	,	PUNCT
ejpam-841	580	32	q	q	NOUN
ejpam-841	580	33	,	,	PUNCT
ejpam-841	580	34	r	r	NOUN
ejpam-841	580	35	)	)	PUNCT
ejpam-841	580	36	be	be	AUX
ejpam-841	580	37	the	the	DET
ejpam-841	580	38	3	3	NUM
ejpam-841	580	39	×	×	NOUN
ejpam-841	580	40	3	3	NUM
ejpam-841	580	41	coefficient	coefficient	NOUN
ejpam-841	580	42	matrix	matrix	NOUN
ejpam-841	580	43	of	of	ADP
ejpam-841	580	44	the	the	DET
ejpam-841	580	45	moving	move	VERB
ejpam-841	580	46	planes	plane	NOUN
ejpam-841	580	47	(	(	PUNCT
ejpam-841	580	48	t1s	t1	NOUN
ejpam-841	580	49	−	−	PROPN
ejpam-841	580	50	s1	s1	PROPN
ejpam-841	580	51	t)p	t)p	NOUN
ejpam-841	580	52	,	,	PUNCT
ejpam-841	580	53	(	(	PUNCT
ejpam-841	580	54	t2s	t2s	NOUN
ejpam-841	580	55	−	−	PROPN
ejpam-841	580	56	s2	s2	PROPN
ejpam-841	580	57	t)q	t)q	VERB
ejpam-841	580	58	,	,	PUNCT
ejpam-841	580	59	r.	r.	PROPN
ejpam-841	580	60	since	since	SCONJ
ejpam-841	580	61	(	(	PUNCT
ejpam-841	580	62	t1s−	t1s−	PRON
ejpam-841	580	63	s1	s1	PROPN
ejpam-841	580	64	t)p	t)p	X
ejpam-841	580	65	=	=	PUNCT
ejpam-841	580	66	t1p1s2	t1p1s2	PROPN
ejpam-841	580	67	+	+	CCONJ
ejpam-841	580	68	(	(	PUNCT
ejpam-841	580	69	t1p0	t1p0	X
ejpam-841	580	70	−	−	PROPN
ejpam-841	580	71	s1p1)st	s1p1)st	ADJ
ejpam-841	580	72	−	−	NOUN
ejpam-841	581	1	s1p0	s1p0	PROPN
ejpam-841	581	2	t2	t2	NOUN
ejpam-841	581	3	,	,	PUNCT
ejpam-841	581	4	(	(	PUNCT
ejpam-841	581	5	t2s−	t2s−	ADP
ejpam-841	581	6	s2	s2	VERB
ejpam-841	581	7	t)q	t)q	PUNCT
ejpam-841	581	8	=	=	PUNCT
ejpam-841	582	1	t2q1s2	t2q1s2	PROPN
ejpam-841	582	2	+	+	CCONJ
ejpam-841	582	3	(	(	PUNCT
ejpam-841	582	4	t2q0−	t2q0−	NUM
ejpam-841	582	5	s2q1)st	s2q1)st	NOUN
ejpam-841	582	6	−	−	PROPN
ejpam-841	582	7	s2q0t2	s2q0t2	PROPN
ejpam-841	582	8	,	,	PUNCT
ejpam-841	582	9	it	it	PRON
ejpam-841	582	10	follows	follow	VERB
ejpam-841	582	11	that	that	SCONJ
ejpam-841	582	12	m(p	m(p	PROPN
ejpam-841	582	13	,	,	PUNCT
ejpam-841	582	14	q	q	NOUN
ejpam-841	582	15	,	,	PUNCT
ejpam-841	582	16	r	r	NOUN
ejpam-841	582	17	)	)	PUNCT
ejpam-841	582	18	=	=	NOUN
ejpam-841	582	19			X
ejpam-841	582	20			ADJ
ejpam-841	582	21			NOUN
ejpam-841	582	22	t1p1	t1p1	PUNCT
ejpam-841	582	23	t1p0	t1p0	X
ejpam-841	582	24	−	−	NOUN
ejpam-841	582	25	s1p1	s1p1	NOUN
ejpam-841	582	26	−s1p0	−s1p0	NOUN
ejpam-841	582	27	t2q1	t2q1	INTJ
ejpam-841	582	28	t2q0−	t2q0−	CCONJ
ejpam-841	582	29	s2q1	s2q1	PROPN
ejpam-841	582	30	−s2q0	−s2q0	PROPN
ejpam-841	582	31	r2	r2	PROPN
ejpam-841	582	32	r1	r1	PROPN
ejpam-841	582	33	r0	r0	NOUN
ejpam-841	582	34			PROPN
ejpam-841	582	35			PROPN
ejpam-841	582	36			PROPN
ejpam-841	582	37	.	.	PUNCT
ejpam-841	583	1	(	(	PUNCT
ejpam-841	583	2	11	11	NUM
ejpam-841	583	3	)	)	PUNCT
ejpam-841	583	4	now	now	ADV
ejpam-841	583	5	we	we	PRON
ejpam-841	583	6	quote	quote	VERB
ejpam-841	583	7	some	some	DET
ejpam-841	583	8	results	result	NOUN
ejpam-841	583	9	concerning	concern	VERB
ejpam-841	583	10	the	the	DET
ejpam-841	583	11	geometric	geometric	ADJ
ejpam-841	583	12	construction	construction	NOUN
ejpam-841	583	13	of	of	ADP
ejpam-841	583	14	the	the	DET
ejpam-841	583	15	implicit	implicit	ADJ
ejpam-841	583	16	equations	equation	NOUN
ejpam-841	583	17	of	of	ADP
ejpam-841	583	18	the	the	DET
ejpam-841	583	19	singular	singular	ADJ
ejpam-841	583	20	quartic	quartic	ADJ
ejpam-841	583	21	space	space	NOUN
ejpam-841	583	22	curve	curve	NOUN
ejpam-841	583	23	.	.	PUNCT
ejpam-841	584	1	detailed	detailed	ADJ
ejpam-841	584	2	proofs	proof	NOUN
ejpam-841	584	3	can	can	AUX
ejpam-841	584	4	be	be	AUX
ejpam-841	584	5	found	find	VERB
ejpam-841	584	6	in	in	ADP
ejpam-841	584	7	[	[	X
ejpam-841	584	8	17	17	NUM
ejpam-841	584	9	,	,	PUNCT
ejpam-841	584	10	theorem	theorem	VERB
ejpam-841	584	11	4.9	4.9	NUM
ejpam-841	584	12	and	and	CCONJ
ejpam-841	584	13	theorem	theorem	VERB
ejpam-841	584	14	4.11	4.11	NUM
ejpam-841	584	15	]	]	PUNCT
ejpam-841	584	16	.	.	PUNCT
ejpam-841	585	1	theorem	theorem	NOUN
ejpam-841	585	2	5	5	NUM
ejpam-841	585	3	.	.	PUNCT
ejpam-841	586	1	let	let	VERB
ejpam-841	586	2	c	c	PRON
ejpam-841	586	3	be	be	AUX
ejpam-841	586	4	a	a	DET
ejpam-841	586	5	singular	singular	ADJ
ejpam-841	586	6	rational	rational	ADJ
ejpam-841	586	7	quartic	quartic	ADJ
ejpam-841	586	8	space	space	NOUN
ejpam-841	586	9	curve	curve	NOUN
ejpam-841	586	10	,	,	PUNCT
ejpam-841	586	11	and	and	CCONJ
ejpam-841	586	12	let	let	VERB
ejpam-841	586	13	m	m	VERB
ejpam-841	586	14	=	=	PUNCT
ejpam-841	586	15	m(p	m(p	PROPN
ejpam-841	586	16	,	,	PUNCT
ejpam-841	586	17	q	q	NOUN
ejpam-841	586	18	,	,	PUNCT
ejpam-841	586	19	r	r	NOUN
ejpam-841	586	20	)	)	PUNCT
ejpam-841	586	21	be	be	AUX
ejpam-841	586	22	the	the	DET
ejpam-841	586	23	matrix	matrix	NOUN
ejpam-841	586	24	in	in	ADP
ejpam-841	586	25	equation	equation	NOUN
ejpam-841	586	26	(	(	PUNCT
ejpam-841	586	27	11	11	NUM
ejpam-841	586	28	)	)	PUNCT
ejpam-841	586	29	constructed	construct	VERB
ejpam-841	586	30	from	from	ADP
ejpam-841	586	31	a	a	DET
ejpam-841	586	32	µ-basis	µ-basis	NOUN
ejpam-841	586	33	for	for	ADP
ejpam-841	586	34	the	the	DET
ejpam-841	586	35	curve	curve	NOUN
ejpam-841	586	36	.	.	PUNCT
ejpam-841	587	1	then	then	ADV
ejpam-841	587	2	det(m	det(m	PROPN
ejpam-841	587	3	)	)	PUNCT
ejpam-841	587	4	=	=	PUNCT
ejpam-841	588	1	ah	ah	INTJ
ejpam-841	588	2	,	,	PUNCT
ejpam-841	588	3	where	where	SCONJ
ejpam-841	588	4	a(x	a(x	NOUN
ejpam-841	588	5	,	,	PUNCT
ejpam-841	588	6	y	y	PROPN
ejpam-841	588	7	,	,	PUNCT
ejpam-841	588	8	z	z	PROPN
ejpam-841	588	9	,	,	PUNCT
ejpam-841	588	10	w	w	PROPN
ejpam-841	588	11	)	)	PUNCT
ejpam-841	588	12	=	=	SYM
ejpam-841	588	13	0	0	NUM
ejpam-841	588	14	is	be	AUX
ejpam-841	588	15	the	the	DET
ejpam-841	588	16	implicit	implicit	ADJ
ejpam-841	588	17	equation	equation	NOUN
ejpam-841	588	18	of	of	ADP
ejpam-841	588	19	the	the	DET
ejpam-841	588	20	plane	plane	NOUN
ejpam-841	588	21	that	that	PRON
ejpam-841	588	22	contains	contain	VERB
ejpam-841	588	23	the	the	DET
ejpam-841	588	24	axes	axis	NOUN
ejpam-841	588	25	of	of	ADP
ejpam-841	588	26	p	p	PROPN
ejpam-841	588	27	and	and	CCONJ
ejpam-841	588	28	q	q	NOUN
ejpam-841	588	29	,	,	PUNCT
ejpam-841	588	30	and	and	CCONJ
ejpam-841	588	31	h(x	h(x	PROPN
ejpam-841	588	32	,	,	PUNCT
ejpam-841	588	33	y	y	PROPN
ejpam-841	588	34	,	,	PUNCT
ejpam-841	588	35	z	z	PROPN
ejpam-841	588	36	,	,	PUNCT
ejpam-841	588	37	w	w	PROPN
ejpam-841	588	38	)	)	PUNCT
ejpam-841	589	1	=	=	SYM
ejpam-841	589	2	0	0	NUM
ejpam-841	589	3	is	be	AUX
ejpam-841	589	4	a	a	DET
ejpam-841	589	5	quadric	quadric	ADJ
ejpam-841	589	6	surface	surface	NOUN
ejpam-841	589	7	that	that	PRON
ejpam-841	589	8	contains	contain	VERB
ejpam-841	589	9	the	the	DET
ejpam-841	589	10	singular	singular	ADJ
ejpam-841	589	11	rational	rational	ADJ
ejpam-841	589	12	quartic	quartic	ADJ
ejpam-841	589	13	space	space	NOUN
ejpam-841	589	14	curve	curve	NOUN
ejpam-841	589	15	c	c	PROPN
ejpam-841	589	16	.	.	PUNCT
ejpam-841	590	1	theorem	theorem	ADJ
ejpam-841	590	2	6	6	NUM
ejpam-841	590	3	.	.	PUNCT
ejpam-841	591	1	let	let	VERB
ejpam-841	591	2	c	c	PRON
ejpam-841	591	3	be	be	AUX
ejpam-841	591	4	a	a	DET
ejpam-841	591	5	singular	singular	ADJ
ejpam-841	591	6	rational	rational	ADJ
ejpam-841	591	7	quartic	quartic	ADJ
ejpam-841	591	8	space	space	NOUN
ejpam-841	591	9	curve	curve	NOUN
ejpam-841	591	10	,	,	PUNCT
ejpam-841	591	11	and	and	CCONJ
ejpam-841	591	12	let	let	VERB
ejpam-841	591	13	m	m	VERB
ejpam-841	591	14	=	=	PUNCT
ejpam-841	591	15	m(p	m(p	PROPN
ejpam-841	591	16	,	,	PUNCT
ejpam-841	591	17	q	q	NOUN
ejpam-841	591	18	,	,	PUNCT
ejpam-841	591	19	r	r	NOUN
ejpam-841	591	20	)	)	PUNCT
ejpam-841	591	21	be	be	AUX
ejpam-841	591	22	the	the	DET
ejpam-841	591	23	matrix	matrix	NOUN
ejpam-841	591	24	in	in	ADP
ejpam-841	591	25	equation	equation	NOUN
ejpam-841	591	26	(	(	PUNCT
ejpam-841	591	27	11	11	NUM
ejpam-841	591	28	)	)	PUNCT
ejpam-841	591	29	constructed	construct	VERB
ejpam-841	591	30	from	from	ADP
ejpam-841	591	31	a	a	DET
ejpam-841	591	32	µ-basis	µ-basis	NOUN
ejpam-841	591	33	p	p	NOUN
ejpam-841	591	34	,	,	PUNCT
ejpam-841	591	35	q	q	NOUN
ejpam-841	591	36	,	,	PUNCT
ejpam-841	591	37	r	r	NOUN
ejpam-841	591	38	for	for	ADP
ejpam-841	591	39	the	the	DET
ejpam-841	591	40	curve	curve	NOUN
ejpam-841	591	41	c	c	PROPN
ejpam-841	591	42	.	.	PUNCT
ejpam-841	592	1	suppose	suppose	VERB
ejpam-841	592	2	nthe	nthe	ADJ
ejpam-841	592	3	plane	plane	NOUN
ejpam-841	592	4	that	that	PRON
ejpam-841	592	5	contains	contain	VERB
ejpam-841	592	6	the	the	DET
ejpam-841	592	7	axes	axis	NOUN
ejpam-841	592	8	of	of	ADP
ejpam-841	592	9	p	p	NOUN
ejpam-841	592	10	,	,	PUNCT
ejpam-841	592	11	q	q	PROPN
ejpam-841	592	12	has	have	VERB
ejpam-841	592	13	implicit	implicit	ADJ
ejpam-841	592	14	equation	equation	NOUN
ejpam-841	592	15	a	a	DET
ejpam-841	592	16	=	=	NOUN
ejpam-841	592	17	0	0	PROPN
ejpam-841	592	18	.	.	PUNCT
ejpam-841	593	1	then	then	ADV
ejpam-841	593	2	the	the	DET
ejpam-841	593	3	two	two	NUM
ejpam-841	593	4	quadric	quadric	ADJ
ejpam-841	593	5	surfaces	surface	NOUN
ejpam-841	593	6	f	f	NOUN
ejpam-841	593	7	=	=	SYM
ejpam-841	593	8	sylvs	sylvs	PROPN
ejpam-841	593	9	,	,	PUNCT
ejpam-841	593	10	t(p	t(p	PROPN
ejpam-841	593	11	,	,	PUNCT
ejpam-841	593	12	q	q	NOUN
ejpam-841	593	13	)	)	PUNCT
ejpam-841	593	14	=	=	SYM
ejpam-841	593	15	0	0	NUM
ejpam-841	593	16	and	and	CCONJ
ejpam-841	593	17	g	g	NOUN
ejpam-841	593	18	=	=	SYM
ejpam-841	593	19	det(m	det(m	PROPN
ejpam-841	593	20	)	)	PUNCT
ejpam-841	593	21	a	a	DET
ejpam-841	593	22	=	=	SYM
ejpam-841	593	23	0	0	NUM
ejpam-841	593	24	form	form	NOUN
ejpam-841	593	25	set	set	NOUN
ejpam-841	593	26	-	-	PUNCT
ejpam-841	593	27	theoretic	theoretic	ADJ
ejpam-841	593	28	complete	complete	ADJ
ejpam-841	593	29	intersection	intersection	NOUN
ejpam-841	593	30	generators	generator	NOUN
ejpam-841	593	31	for	for	ADP
ejpam-841	593	32	the	the	DET
ejpam-841	593	33	curve	curve	NOUN
ejpam-841	593	34	c	c	PROPN
ejpam-841	593	35	.	.	PUNCT
ejpam-841	594	1	next	next	ADV
ejpam-841	594	2	we	we	PRON
ejpam-841	594	3	will	will	AUX
ejpam-841	594	4	use	use	VERB
ejpam-841	594	5	a	a	DET
ejpam-841	594	6	very	very	ADV
ejpam-841	594	7	simple	simple	ADJ
ejpam-841	594	8	example	example	NOUN
ejpam-841	594	9	to	to	PART
ejpam-841	594	10	illustrate	illustrate	VERB
ejpam-841	594	11	our	our	PRON
ejpam-841	594	12	method	method	NOUN
ejpam-841	594	13	for	for	ADP
ejpam-841	594	14	finding	find	VERB
ejpam-841	594	15	both	both	CCONJ
ejpam-841	594	16	the	the	DET
ejpam-841	594	17	singular	singular	ADJ
ejpam-841	594	18	point	point	NOUN
ejpam-841	594	19	and	and	CCONJ
ejpam-841	594	20	the	the	DET
ejpam-841	594	21	implicit	implicit	ADJ
ejpam-841	594	22	equations	equation	NOUN
ejpam-841	594	23	for	for	ADP
ejpam-841	594	24	a	a	DET
ejpam-841	594	25	singular	singular	ADJ
ejpam-841	594	26	rational	rational	ADJ
ejpam-841	594	27	quartic	quartic	ADJ
ejpam-841	594	28	space	space	NOUN
ejpam-841	594	29	curve	curve	NOUN
ejpam-841	594	30	.	.	PUNCT
ejpam-841	594	31	example	example	NOUN
ejpam-841	595	1	3	3	X
ejpam-841	595	2	.	.	PUNCT
ejpam-841	595	3	let	let	VERB
ejpam-841	595	4	the	the	DET
ejpam-841	595	5	singular	singular	ADJ
ejpam-841	595	6	rational	rational	ADJ
ejpam-841	595	7	quartic	quartic	ADJ
ejpam-841	595	8	space	space	NOUN
ejpam-841	595	9	curve	curve	NOUN
ejpam-841	595	10	c	c	AUX
ejpam-841	595	11	be	be	AUX
ejpam-841	595	12	given	give	VERB
ejpam-841	595	13	as	as	ADP
ejpam-841	595	14	the	the	DET
ejpam-841	595	15	image	image	NOUN
ejpam-841	595	16	of	of	ADP
ejpam-841	595	17	the	the	DET
ejpam-841	595	18	parameterization	parameterization	NOUN
ejpam-841	595	19	:	:	PUNCT
ejpam-841	595	20	(	(	PUNCT
ejpam-841	595	21	x	x	X
ejpam-841	595	22	,	,	PUNCT
ejpam-841	595	23	y	y	PROPN
ejpam-841	595	24	,	,	PUNCT
ejpam-841	595	25	z	z	PROPN
ejpam-841	595	26	,	,	PUNCT
ejpam-841	595	27	w	w	NOUN
ejpam-841	595	28	)	)	PUNCT
ejpam-841	595	29	=	=	SYM
ejpam-841	595	30	(	(	PUNCT
ejpam-841	595	31	s4	s4	PROPN
ejpam-841	595	32	,	,	PUNCT
ejpam-841	595	33	s3	s3	PROPN
ejpam-841	595	34	t	t	PROPN
ejpam-841	595	35	,	,	PUNCT
ejpam-841	595	36	s2	s2	NOUN
ejpam-841	595	37	t2	t2	PROPN
ejpam-841	595	38	,	,	PUNCT
ejpam-841	595	39	t4	t4	PROPN
ejpam-841	595	40	)	)	PUNCT
ejpam-841	595	41	.	.	PUNCT
ejpam-841	596	1	j.	j.	PROPN
ejpam-841	596	2	hoffman	hoffman	PROPN
ejpam-841	596	3	,	,	PUNCT
ejpam-841	596	4	h.	h.	PROPN
ejpam-841	596	5	wang	wang	PROPN
ejpam-841	596	6	,	,	PUNCT
ejpam-841	596	7	x.	x.	PROPN
ejpam-841	596	8	jia	jia	PROPN
ejpam-841	596	9	,	,	PUNCT
ejpam-841	596	10	r.	r.	PROPN
ejpam-841	596	11	goldman	goldman	PROPN
ejpam-841	596	12	/	/	SYM
ejpam-841	596	13	eur	eur	PROPN
ejpam-841	596	14	.	.	PUNCT
ejpam-841	597	1	j.	j.	PROPN
ejpam-841	597	2	pure	pure	PROPN
ejpam-841	597	3	appl	appl	PROPN
ejpam-841	597	4	.	.	PROPN
ejpam-841	597	5	math	math	PROPN
ejpam-841	597	6	,	,	PUNCT
ejpam-841	597	7	3	3	NUM
ejpam-841	597	8	(	(	PUNCT
ejpam-841	597	9	2010	2010	NUM
ejpam-841	597	10	)	)	PUNCT
ejpam-841	597	11	,	,	PUNCT
ejpam-841	597	12	602	602	NUM
ejpam-841	597	13	-	-	SYM
ejpam-841	597	14	632	632	NUM
ejpam-841	597	15	622	622	NUM
ejpam-841	597	16	compute	compute	NOUN
ejpam-841	597	17	a	a	DET
ejpam-841	597	18	µ-basis	µ-basis	NOUN
ejpam-841	597	19	using	use	VERB
ejpam-841	597	20	the	the	DET
ejpam-841	597	21	algorithm	algorithm	NOUN
ejpam-841	597	22	in	in	ADP
ejpam-841	597	23	[	[	X
ejpam-841	597	24	24	24	NUM
ejpam-841	597	25	]	]	X
ejpam-841	597	26	:	:	PUNCT
ejpam-841	597	27	p	p	X
ejpam-841	597	28	=	=	SYM
ejpam-841	597	29	(	(	PUNCT
ejpam-841	597	30	y	y	PROPN
ejpam-841	597	31	−	−	PROPN
ejpam-841	597	32	z)s+	z)s+	NUM
ejpam-841	597	33	(	(	PUNCT
ejpam-841	597	34	y	y	PROPN
ejpam-841	597	35	−	−	PROPN
ejpam-841	597	36	x)t	x)t	NOUN
ejpam-841	597	37	,	,	PUNCT
ejpam-841	597	38	q	q	NOUN
ejpam-841	597	39	=	=	SYM
ejpam-841	597	40	zs−	zs−	NUM
ejpam-841	597	41	y	y	PROPN
ejpam-841	597	42	t	t	PROPN
ejpam-841	597	43	,	,	PUNCT
ejpam-841	597	44	r	r	NOUN
ejpam-841	597	45	=	=	SYM
ejpam-841	597	46	ws2	ws2	NOUN
ejpam-841	597	47	−	−	PROPN
ejpam-841	597	48	zt2	zt2	PROPN
ejpam-841	597	49	.	.	PUNCT
ejpam-841	598	1	the	the	DET
ejpam-841	598	2	axis	axis	NOUN
ejpam-841	598	3	of	of	ADP
ejpam-841	598	4	p	p	NOUN
ejpam-841	598	5	is	be	AUX
ejpam-841	598	6	the	the	DET
ejpam-841	598	7	line	line	NOUN
ejpam-841	598	8	through	through	ADP
ejpam-841	598	9	the	the	DET
ejpam-841	598	10	points	point	NOUN
ejpam-841	598	11	(	(	PUNCT
ejpam-841	598	12	0,0,0,1	0,0,0,1	NOUN
ejpam-841	598	13	)	)	PUNCT
ejpam-841	598	14	and	and	CCONJ
ejpam-841	598	15	(	(	PUNCT
ejpam-841	598	16	1,1,1,1	1,1,1,1	NUM
ejpam-841	598	17	)	)	PUNCT
ejpam-841	598	18	,	,	PUNCT
ejpam-841	598	19	and	and	CCONJ
ejpam-841	598	20	the	the	DET
ejpam-841	598	21	axis	axis	NOUN
ejpam-841	598	22	of	of	ADP
ejpam-841	598	23	q	q	NOUN
ejpam-841	598	24	is	be	AUX
ejpam-841	598	25	the	the	DET
ejpam-841	598	26	line	line	NOUN
ejpam-841	598	27	through	through	ADP
ejpam-841	598	28	the	the	DET
ejpam-841	598	29	points	point	NOUN
ejpam-841	598	30	(	(	PUNCT
ejpam-841	598	31	1,0,0,0	1,0,0,0	NUM
ejpam-841	598	32	)	)	PUNCT
ejpam-841	598	33	and	and	CCONJ
ejpam-841	598	34	(	(	PUNCT
ejpam-841	598	35	0,0,0,1	0,0,0,1	NOUN
ejpam-841	598	36	)	)	PUNCT
ejpam-841	598	37	.	.	PUNCT
ejpam-841	599	1	these	these	DET
ejpam-841	599	2	axes	axis	NOUN
ejpam-841	599	3	intersect	intersect	ADJ
ejpam-841	599	4	at	at	ADP
ejpam-841	599	5	the	the	DET
ejpam-841	599	6	point	point	NOUN
ejpam-841	599	7	(	(	PUNCT
ejpam-841	599	8	0,0,0,1)—the	0,0,0,1)—the	DET
ejpam-841	599	9	only	only	ADJ
ejpam-841	599	10	singular	singular	ADJ
ejpam-841	599	11	point	point	NOUN
ejpam-841	599	12	,	,	PUNCT
ejpam-841	599	13	a	a	DET
ejpam-841	599	14	point	point	NOUN
ejpam-841	599	15	of	of	ADP
ejpam-841	599	16	order	order	NOUN
ejpam-841	599	17	2	2	NUM
ejpam-841	599	18	—	—	PUNCT
ejpam-841	599	19	which	which	PRON
ejpam-841	599	20	corresponds	correspond	VERB
ejpam-841	599	21	to	to	ADP
ejpam-841	599	22	the	the	DET
ejpam-841	599	23	parameters	parameter	NOUN
ejpam-841	599	24	(	(	PUNCT
ejpam-841	599	25	s0	s0	PROPN
ejpam-841	599	26	,	,	PUNCT
ejpam-841	599	27	t0	t0	PROPN
ejpam-841	599	28	)	)	PUNCT
ejpam-841	599	29	=	=	SYM
ejpam-841	599	30	(	(	PUNCT
ejpam-841	599	31	0,1	0,1	NUM
ejpam-841	599	32	)	)	PUNCT
ejpam-841	599	33	.	.	PUNCT
ejpam-841	600	1	the	the	DET
ejpam-841	600	2	curve	curve	NOUN
ejpam-841	600	3	c	c	PROPN
ejpam-841	600	4	intersects	intersect	VERB
ejpam-841	600	5	the	the	DET
ejpam-841	600	6	axis	axis	NOUN
ejpam-841	600	7	of	of	ADP
ejpam-841	600	8	p	p	NOUN
ejpam-841	600	9	at	at	ADP
ejpam-841	600	10	the	the	DET
ejpam-841	600	11	point	point	NOUN
ejpam-841	600	12	(	(	PUNCT
ejpam-841	600	13	1,1,1,1	1,1,1,1	NUM
ejpam-841	600	14	)	)	PUNCT
ejpam-841	600	15	with	with	ADP
ejpam-841	600	16	parameters	parameter	NOUN
ejpam-841	600	17	(	(	PUNCT
ejpam-841	600	18	s1	s1	NOUN
ejpam-841	600	19	,	,	PUNCT
ejpam-841	600	20	t1	t1	NOUN
ejpam-841	600	21	)	)	PUNCT
ejpam-841	601	1	=	=	SYM
ejpam-841	601	2	(	(	PUNCT
ejpam-841	601	3	1,1	1,1	NUM
ejpam-841	601	4	)	)	PUNCT
ejpam-841	601	5	,	,	PUNCT
ejpam-841	601	6	and	and	CCONJ
ejpam-841	601	7	intersects	intersect	VERB
ejpam-841	601	8	the	the	DET
ejpam-841	601	9	axis	axis	NOUN
ejpam-841	601	10	of	of	ADP
ejpam-841	601	11	q	q	NOUN
ejpam-841	601	12	at	at	ADP
ejpam-841	601	13	the	the	DET
ejpam-841	601	14	point	point	NOUN
ejpam-841	601	15	(	(	PUNCT
ejpam-841	601	16	1,0,0,0	1,0,0,0	NUM
ejpam-841	601	17	)	)	PUNCT
ejpam-841	601	18	with	with	ADP
ejpam-841	601	19	parameters	parameter	NOUN
ejpam-841	601	20	(	(	PUNCT
ejpam-841	601	21	s2	s2	PROPN
ejpam-841	601	22	,	,	PUNCT
ejpam-841	601	23	t2	t2	NOUN
ejpam-841	601	24	)	)	PUNCT
ejpam-841	601	25	=	=	PUNCT
ejpam-841	601	26	(	(	PUNCT
ejpam-841	601	27	1,0	1,0	NUM
ejpam-841	601	28	)	)	PUNCT
ejpam-841	601	29	.	.	PUNCT
ejpam-841	602	1	now	now	ADV
ejpam-841	602	2	f	f	X
ejpam-841	602	3	=	=	SYM
ejpam-841	602	4	sylvs	sylvs	PROPN
ejpam-841	602	5	,	,	PUNCT
ejpam-841	602	6	t(p	t(p	PROPN
ejpam-841	602	7	,	,	PUNCT
ejpam-841	602	8	q	q	NOUN
ejpam-841	602	9	)	)	PUNCT
ejpam-841	602	10	=	=	SYM
ejpam-841	602	11	det	det	PROPN
ejpam-841	602	12	�	�	PROPN
ejpam-841	602	13	y	y	PROPN
ejpam-841	602	14	−	−	PROPN
ejpam-841	603	1	z	z	NOUN
ejpam-841	603	2	y	y	NOUN
ejpam-841	604	1	−	−	NOUN
ejpam-841	604	2	x	x	SYM
ejpam-841	604	3	z	z	NOUN
ejpam-841	604	4	−y	−y	VERB
ejpam-841	604	5	�	�	PROPN
ejpam-841	604	6	=	=	SYM
ejpam-841	604	7	xz−	xz−	PROPN
ejpam-841	604	8	y2	y2	PROPN
ejpam-841	604	9	,	,	PUNCT
ejpam-841	604	10	g	g	PROPN
ejpam-841	604	11	=	=	PUNCT
ejpam-841	604	12	det(m(p	det(m(p	PROPN
ejpam-841	604	13	,	,	PUNCT
ejpam-841	604	14	q	q	NOUN
ejpam-841	604	15	,	,	PUNCT
ejpam-841	604	16	r	r	NOUN
ejpam-841	604	17	)	)	PUNCT
ejpam-841	604	18	)	)	PUNCT
ejpam-841	605	1	=	=	SYM
ejpam-841	605	2	det	det	PROPN
ejpam-841	605	3			PROPN
ejpam-841	605	4			NOUN
ejpam-841	605	5			NOUN
ejpam-841	605	6	t1p1	t1p1	PUNCT
ejpam-841	605	7	t1p0	t1p0	X
ejpam-841	605	8	−	−	PRON
ejpam-841	605	9	s1p1	s1p1	NOUN
ejpam-841	605	10	−s1p0	−s1p0	NOUN
ejpam-841	605	11	t2q1	t2q1	INTJ
ejpam-841	605	12	t2q0−	t2q0−	CCONJ
ejpam-841	605	13	s2q1	s2q1	PROPN
ejpam-841	605	14	−s2q0	−s2q0	PROPN
ejpam-841	605	15	r2	r2	PROPN
ejpam-841	605	16	r1	r1	PROPN
ejpam-841	605	17	r0	r0	PROPN
ejpam-841	605	18			PROPN
ejpam-841	605	19			VERB
ejpam-841	605	20			PUNCT
ejpam-841	606	1	=	=	PUNCT
ejpam-841	606	2	det	det	PROPN
ejpam-841	606	3			PROPN
ejpam-841	606	4			NOUN
ejpam-841	606	5			NOUN
ejpam-841	606	6	y	y	NOUN
ejpam-841	606	7	−	−	PROPN
ejpam-841	607	1	z	z	NOUN
ejpam-841	607	2	z	z	NOUN
ejpam-841	608	1	−	−	NOUN
ejpam-841	608	2	x	x	SYM
ejpam-841	609	1	x	x	PUNCT
ejpam-841	609	2	−	−	PUNCT
ejpam-841	609	3	y	y	PROPN
ejpam-841	609	4	0	0	NUM
ejpam-841	609	5	−z	−z	NOUN
ejpam-841	609	6	y	y	PROPN
ejpam-841	609	7	w	w	PROPN
ejpam-841	609	8	0	0	NUM
ejpam-841	609	9	−z	−z	NOUN
ejpam-841	609	10			NOUN
ejpam-841	609	11			VERB
ejpam-841	609	12			PUNCT
ejpam-841	610	1	=	=	NOUN
ejpam-841	610	2	z2	z2	NUM
ejpam-841	610	3	y	y	NOUN
ejpam-841	610	4	−	−	NOUN
ejpam-841	610	5	x	x	SYM
ejpam-841	610	6	yw−	yw−	NUM
ejpam-841	610	7	z3	z3	PROPN
ejpam-841	610	8	+	+	CCONJ
ejpam-841	610	9	xzw	xzw	PROPN
ejpam-841	610	10	.	.	PUNCT
ejpam-841	611	1	the	the	DET
ejpam-841	611	2	implicit	implicit	ADJ
ejpam-841	611	3	equation	equation	NOUN
ejpam-841	611	4	of	of	ADP
ejpam-841	611	5	the	the	DET
ejpam-841	611	6	plane	plane	NOUN
ejpam-841	611	7	containing	contain	VERB
ejpam-841	611	8	the	the	DET
ejpam-841	611	9	axes	axis	NOUN
ejpam-841	611	10	of	of	ADP
ejpam-841	611	11	p(s	p(s	PROPN
ejpam-841	611	12	,	,	PUNCT
ejpam-841	611	13	t	t	PROPN
ejpam-841	611	14	)	)	PUNCT
ejpam-841	611	15	and	and	CCONJ
ejpam-841	611	16	q(s	q(s	PROPN
ejpam-841	611	17	,	,	PUNCT
ejpam-841	611	18	t	t	PROPN
ejpam-841	611	19	)	)	PUNCT
ejpam-841	611	20	is	be	AUX
ejpam-841	611	21	:	:	PUNCT
ejpam-841	611	22	a	a	DET
ejpam-841	611	23	=	=	PUNCT
ejpam-841	611	24	�	�	PROPN
ejpam-841	611	25	�	�	PROPN
ejpam-841	611	26	�	�	PROPN
ejpam-841	611	27	�	�	PROPN
ejpam-841	611	28	�	�	PROPN
ejpam-841	611	29	�	�	PROPN
ejpam-841	611	30	�	�	PROPN
ejpam-841	611	31	�	�	PROPN
ejpam-841	611	32	x	x	PUNCT
ejpam-841	611	33	y	y	PROPN
ejpam-841	611	34	z	z	PROPN
ejpam-841	611	35	w	w	NOUN
ejpam-841	611	36	1	1	NUM
ejpam-841	611	37	1	1	NUM
ejpam-841	611	38	1	1	NUM
ejpam-841	611	39	1	1	NUM
ejpam-841	611	40	1	1	NUM
ejpam-841	611	41	0	0	NUM
ejpam-841	611	42	0	0	NUM
ejpam-841	611	43	0	0	NUM
ejpam-841	611	44	0	0	NUM
ejpam-841	611	45	0	0	NUM
ejpam-841	611	46	0	0	NUM
ejpam-841	611	47	1	1	NUM
ejpam-841	611	48	�	�	PROPN
ejpam-841	611	49	�	�	PROPN
ejpam-841	611	50	�	�	PROPN
ejpam-841	611	51	�	�	PROPN
ejpam-841	611	52	�	�	PROPN
ejpam-841	611	53	�	�	PROPN
ejpam-841	611	54	�	�	PROPN
ejpam-841	611	55	�	�	PROPN
ejpam-841	611	56	=	=	PUNCT
ejpam-841	611	57	y	y	PROPN
ejpam-841	611	58	−	−	NOUN
ejpam-841	611	59	z	z	NOUN
ejpam-841	612	1	=	=	NOUN
ejpam-841	613	1	0	0	X
ejpam-841	613	2	.	.	PUNCT
ejpam-841	614	1	therefore	therefore	ADV
ejpam-841	614	2	the	the	DET
ejpam-841	614	3	other	other	ADJ
ejpam-841	614	4	quadric	quadric	ADJ
ejpam-841	614	5	surface	surface	NOUN
ejpam-841	614	6	that	that	PRON
ejpam-841	614	7	contains	contain	VERB
ejpam-841	614	8	the	the	DET
ejpam-841	614	9	quartic	quartic	ADJ
ejpam-841	614	10	curve	curve	NOUN
ejpam-841	614	11	is	be	AUX
ejpam-841	614	12	h=	h=	NOUN
ejpam-841	614	13	g	g	PROPN
ejpam-841	614	14	a	a	DET
ejpam-841	614	15	=	=	PROPN
ejpam-841	614	16	z2	z2	PROPN
ejpam-841	614	17	−	−	PROPN
ejpam-841	614	18	xw	xw	PROPN
ejpam-841	614	19	.	.	PUNCT
ejpam-841	615	1	hence	hence	ADV
ejpam-841	615	2	the	the	DET
ejpam-841	615	3	implicit	implicit	ADJ
ejpam-841	615	4	equations	equation	NOUN
ejpam-841	615	5	of	of	ADP
ejpam-841	615	6	the	the	DET
ejpam-841	615	7	singular	singular	ADJ
ejpam-841	615	8	rational	rational	ADJ
ejpam-841	615	9	quartic	quartic	ADJ
ejpam-841	615	10	curve	curve	NOUN
ejpam-841	615	11	c	c	PROPN
ejpam-841	615	12	are	be	AUX
ejpam-841	615	13	(	(	PUNCT
ejpam-841	615	14	see	see	VERB
ejpam-841	615	15	figure	figure	NOUN
ejpam-841	615	16	1	1	NUM
ejpam-841	615	17	)	)	PUNCT
ejpam-841	615	18	xz	xz	NOUN
ejpam-841	616	1	−	−	PROPN
ejpam-841	616	2	y2	y2	NOUN
ejpam-841	616	3	=	=	SYM
ejpam-841	616	4	0	0	NUM
ejpam-841	616	5	,	,	PUNCT
ejpam-841	616	6	z2	z2	PROPN
ejpam-841	616	7	−	−	PROPN
ejpam-841	616	8	xw	xw	PROPN
ejpam-841	616	9	=	=	SYM
ejpam-841	616	10	0	0	PROPN
ejpam-841	616	11	.	.	NOUN
ejpam-841	616	12	4.2	4.2	NUM
ejpam-841	616	13	.	.	PUNCT
ejpam-841	617	1	implicit	implicit	ADJ
ejpam-841	617	2	equations	equation	NOUN
ejpam-841	617	3	of	of	ADP
ejpam-841	617	4	non	non	ADJ
ejpam-841	617	5	-	-	ADJ
ejpam-841	617	6	singular	singular	ADJ
ejpam-841	617	7	rational	rational	ADJ
ejpam-841	617	8	quartic	quartic	ADJ
ejpam-841	617	9	space	space	NOUN
ejpam-841	617	10	curves	curve	NOUN
ejpam-841	617	11	next	next	ADV
ejpam-841	617	12	we	we	PRON
ejpam-841	617	13	consider	consider	VERB
ejpam-841	617	14	the	the	DET
ejpam-841	617	15	case	case	NOUN
ejpam-841	617	16	where	where	SCONJ
ejpam-841	617	17	the	the	DET
ejpam-841	617	18	rational	rational	ADJ
ejpam-841	617	19	quartic	quartic	ADJ
ejpam-841	617	20	space	space	NOUN
ejpam-841	617	21	curvec	curvec	NOUN
ejpam-841	617	22	given	give	VERB
ejpam-841	617	23	by	by	ADP
ejpam-841	617	24	the	the	DET
ejpam-841	617	25	parametrization	parametrization	NOUN
ejpam-841	617	26	f0	f0	PROPN
ejpam-841	617	27	,	,	PUNCT
ejpam-841	617	28	f1	f1	NOUN
ejpam-841	617	29	,	,	PUNCT
ejpam-841	617	30	f2	f2	PROPN
ejpam-841	617	31	,	,	PUNCT
ejpam-841	617	32	f3	f3	PROPN
ejpam-841	617	33	is	be	AUX
ejpam-841	617	34	smooth	smooth	ADJ
ejpam-841	617	35	.	.	PUNCT
ejpam-841	618	1	by	by	ADP
ejpam-841	618	2	lemma	lemma	PROPN
ejpam-841	618	3	4	4	NUM
ejpam-841	618	4	,	,	PUNCT
ejpam-841	618	5	we	we	PRON
ejpam-841	618	6	know	know	VERB
ejpam-841	618	7	that	that	SCONJ
ejpam-841	618	8	the	the	DET
ejpam-841	618	9	curve	curve	NOUN
ejpam-841	618	10	c	c	PROPN
ejpam-841	618	11	is	be	AUX
ejpam-841	618	12	the	the	DET
ejpam-841	618	13	image	image	NOUN
ejpam-841	618	14	of	of	ADP
ejpam-841	618	15	the	the	DET
ejpam-841	618	16	projection	projection	NOUN
ejpam-841	618	17	of	of	ADP
ejpam-841	618	18	the	the	DET
ejpam-841	618	19	rational	rational	ADJ
ejpam-841	618	20	normal	normal	ADJ
ejpam-841	618	21	scroll	scroll	NOUN
ejpam-841	618	22	s(4	s(4	NOUN
ejpam-841	618	23	)	)	PUNCT
ejpam-841	618	24	from	from	ADP
ejpam-841	618	25	a	a	DET
ejpam-841	618	26	point	point	NOUN
ejpam-841	618	27	p	p	X
ejpam-841	618	28	/∈	/∈	PROPN
ejpam-841	618	29	s(4)⊂	s(4)⊂	NOUN
ejpam-841	618	30	p4	p4	ADJ
ejpam-841	618	31	to	to	ADP
ejpam-841	618	32	a	a	DET
ejpam-841	618	33	hyperplane	hyperplane	NOUN
ejpam-841	618	34	in	in	ADP
ejpam-841	618	35	p4	p4	ADJ
ejpam-841	618	36	.	.	PUNCT
ejpam-841	619	1	lemma	lemma	PROPN
ejpam-841	619	2	5	5	NUM
ejpam-841	619	3	.	.	PUNCT
ejpam-841	620	1	the	the	DET
ejpam-841	620	2	implicit	implicit	ADJ
ejpam-841	620	3	equations	equation	NOUN
ejpam-841	620	4	of	of	ADP
ejpam-841	620	5	a	a	DET
ejpam-841	620	6	non	non	ADJ
ejpam-841	620	7	-	-	ADJ
ejpam-841	620	8	singular	singular	ADJ
ejpam-841	620	9	rational	rational	ADJ
ejpam-841	620	10	quartic	quartic	ADJ
ejpam-841	620	11	space	space	NOUN
ejpam-841	620	12	curve	curve	NOUN
ejpam-841	620	13	c	c	PROPN
ejpam-841	620	14	are	be	AUX
ejpam-841	620	15	given	give	VERB
ejpam-841	620	16	by	by	ADP
ejpam-841	620	17	4	4	NUM
ejpam-841	620	18	equations	equation	NOUN
ejpam-841	620	19	f1	f1	NOUN
ejpam-841	620	20	=	=	SYM
ejpam-841	620	21	0	0	NUM
ejpam-841	620	22	,	,	PUNCT
ejpam-841	620	23	f2	f2	NOUN
ejpam-841	620	24	=	=	SYM
ejpam-841	620	25	0	0	NUM
ejpam-841	620	26	,	,	PUNCT
ejpam-841	620	27	f3	f3	NOUN
ejpam-841	620	28	=	=	SYM
ejpam-841	620	29	0	0	NUM
ejpam-841	620	30	,	,	PUNCT
ejpam-841	620	31	f4	f4	NOUN
ejpam-841	620	32	=	=	SYM
ejpam-841	620	33	0	0	NUM
ejpam-841	620	34	,	,	PUNCT
ejpam-841	620	35	one	one	NUM
ejpam-841	620	36	quadric	quadric	ADJ
ejpam-841	620	37	surface	surface	NOUN
ejpam-841	620	38	and	and	CCONJ
ejpam-841	620	39	three	three	NUM
ejpam-841	620	40	cubic	cubic	ADJ
ejpam-841	620	41	surfaces	surface	NOUN
ejpam-841	620	42	.	.	PUNCT
ejpam-841	621	1	moreover	moreover	ADV
ejpam-841	621	2	,	,	PUNCT
ejpam-841	621	3	the	the	DET
ejpam-841	621	4	quadric	quadric	ADJ
ejpam-841	621	5	surface	surface	NOUN
ejpam-841	621	6	is	be	AUX
ejpam-841	621	7	given	give	VERB
ejpam-841	621	8	by	by	ADP
ejpam-841	621	9	f1	f1	PROPN
ejpam-841	621	10	=	=	SYM
ejpam-841	621	11	sylvs	sylvs	PROPN
ejpam-841	621	12	,	,	PUNCT
ejpam-841	621	13	t(p	t(p	PROPN
ejpam-841	621	14	,	,	PUNCT
ejpam-841	621	15	q	q	NOUN
ejpam-841	621	16	)	)	PUNCT
ejpam-841	621	17	=	=	SYM
ejpam-841	621	18	0	0	NUM
ejpam-841	621	19	,	,	PUNCT
ejpam-841	621	20	where	where	SCONJ
ejpam-841	621	21	p	p	X
ejpam-841	621	22	,	,	PUNCT
ejpam-841	621	23	q	q	X
ejpam-841	621	24	are	be	AUX
ejpam-841	621	25	the	the	DET
ejpam-841	621	26	degree	degree	NOUN
ejpam-841	621	27	1	1	NUM
ejpam-841	621	28	elements	element	NOUN
ejpam-841	621	29	of	of	ADP
ejpam-841	621	30	a	a	DET
ejpam-841	621	31	µ-basis	µ-basis	NOUN
ejpam-841	621	32	for	for	ADP
ejpam-841	621	33	c	c	PROPN
ejpam-841	621	34	.	.	PUNCT
ejpam-841	622	1	j.	j.	PROPN
ejpam-841	622	2	hoffman	hoffman	PROPN
ejpam-841	622	3	,	,	PUNCT
ejpam-841	622	4	h.	h.	PROPN
ejpam-841	622	5	wang	wang	PROPN
ejpam-841	622	6	,	,	PUNCT
ejpam-841	622	7	x.	x.	PROPN
ejpam-841	622	8	jia	jia	PROPN
ejpam-841	622	9	,	,	PUNCT
ejpam-841	622	10	r.	r.	PROPN
ejpam-841	622	11	goldman	goldman	PROPN
ejpam-841	622	12	/	/	SYM
ejpam-841	622	13	eur	eur	PROPN
ejpam-841	622	14	.	.	PUNCT
ejpam-841	623	1	j.	j.	PROPN
ejpam-841	623	2	pure	pure	PROPN
ejpam-841	623	3	appl	appl	PROPN
ejpam-841	623	4	.	.	PROPN
ejpam-841	623	5	math	math	PROPN
ejpam-841	623	6	,	,	PUNCT
ejpam-841	623	7	3	3	NUM
ejpam-841	623	8	(	(	PUNCT
ejpam-841	623	9	2010	2010	NUM
ejpam-841	623	10	)	)	PUNCT
ejpam-841	623	11	,	,	PUNCT
ejpam-841	623	12	602	602	NUM
ejpam-841	623	13	-	-	SYM
ejpam-841	623	14	632	632	NUM
ejpam-841	623	15	623	623	NUM
ejpam-841	623	16	figure	figure	NOUN
ejpam-841	623	17	1	1	NUM
ejpam-841	623	18	:	:	PUNCT
ejpam-841	623	19	set	set	VERB
ejpam-841	623	20	-	-	PUNCT
ejpam-841	623	21	theoreti	theoreti	NOUN
ejpam-841	623	22	omplete	omplete	ADJ
ejpam-841	623	23	interse	interse	ADJ
ejpam-841	623	24	tion	tion	NOUN
ejpam-841	623	25	of	of	ADP
ejpam-841	623	26	a	a	DET
ejpam-841	623	27	singular	singular	ADJ
ejpam-841	623	28	rational	rational	ADJ
ejpam-841	623	29	quarti	quarti	PROPN
ejpam-841	623	30	spa	spa	NOUN
ejpam-841	623	31	e	e	NOUN
ejpam-841	623	32	urve	urve	NOUN
ejpam-841	623	33	.	.	PUNCT
ejpam-841	624	1	proof	proof	NOUN
ejpam-841	624	2	.	.	PUNCT
ejpam-841	625	1	let	let	VERB
ejpam-841	625	2	p	p	NOUN
ejpam-841	625	3	=	=	X
ejpam-841	625	4	(	(	PUNCT
ejpam-841	625	5	a	a	PRON
ejpam-841	625	6	,	,	PUNCT
ejpam-841	625	7	b	b	NOUN
ejpam-841	625	8	,	,	PUNCT
ejpam-841	625	9	c	c	NOUN
ejpam-841	625	10	,	,	PUNCT
ejpam-841	625	11	d	d	NOUN
ejpam-841	625	12	,	,	PUNCT
ejpam-841	625	13	1	1	X
ejpam-841	625	14	)	)	PUNCT
ejpam-841	625	15	∈	∈	PROPN
ejpam-841	625	16	p4	p4	NOUN
ejpam-841	625	17	be	be	AUX
ejpam-841	625	18	a	a	DET
ejpam-841	625	19	point	point	NOUN
ejpam-841	625	20	not	not	PART
ejpam-841	625	21	contained	contain	VERB
ejpam-841	625	22	in	in	ADP
ejpam-841	625	23	the	the	DET
ejpam-841	625	24	rational	rational	ADJ
ejpam-841	625	25	normal	normal	ADJ
ejpam-841	625	26	curve	curve	NOUN
ejpam-841	625	27	s(4	s(4	NOUN
ejpam-841	625	28	)	)	PUNCT
ejpam-841	625	29	given	give	VERB
ejpam-841	625	30	by	by	ADP
ejpam-841	625	31	the	the	DET
ejpam-841	625	32	parametric	parametric	ADJ
ejpam-841	625	33	representation	representation	NOUN
ejpam-841	625	34	x	x	X
ejpam-841	625	35	=	=	SYM
ejpam-841	625	36	s4	s4	PROPN
ejpam-841	625	37	,	,	PUNCT
ejpam-841	625	38	y	y	PROPN
ejpam-841	625	39	=	=	PROPN
ejpam-841	625	40	s3	s3	PROPN
ejpam-841	625	41	t	t	PROPN
ejpam-841	625	42	,	,	PUNCT
ejpam-841	625	43	z	z	NOUN
ejpam-841	625	44	=	=	SYM
ejpam-841	625	45	s2	s2	NOUN
ejpam-841	625	46	t2	t2	NOUN
ejpam-841	625	47	,	,	PUNCT
ejpam-841	625	48	w	w	PROPN
ejpam-841	625	49	=	=	SYM
ejpam-841	625	50	st3	st3	PROPN
ejpam-841	625	51	,	,	PUNCT
ejpam-841	625	52	u	u	PROPN
ejpam-841	625	53	=	=	PROPN
ejpam-841	625	54	t4	t4	PROPN
ejpam-841	625	55	.	.	PROPN
ejpam-841	626	1	without	without	ADP
ejpam-841	626	2	loss	loss	NOUN
ejpam-841	626	3	of	of	ADP
ejpam-841	626	4	generality	generality	NOUN
ejpam-841	626	5	,	,	PUNCT
ejpam-841	626	6	let	let	VERB
ejpam-841	626	7	the	the	DET
ejpam-841	626	8	non	non	ADJ
ejpam-841	626	9	-	-	ADJ
ejpam-841	626	10	singular	singular	ADJ
ejpam-841	626	11	rational	rational	ADJ
ejpam-841	626	12	quartic	quartic	ADJ
ejpam-841	626	13	space	space	NOUN
ejpam-841	626	14	curve	curve	NOUN
ejpam-841	626	15	c	c	AUX
ejpam-841	626	16	be	be	AUX
ejpam-841	626	17	the	the	DET
ejpam-841	626	18	image	image	NOUN
ejpam-841	626	19	of	of	ADP
ejpam-841	626	20	the	the	DET
ejpam-841	626	21	projection	projection	NOUN
ejpam-841	626	22	of	of	ADP
ejpam-841	626	23	the	the	DET
ejpam-841	626	24	rational	rational	ADJ
ejpam-841	626	25	normal	normal	ADJ
ejpam-841	626	26	curve	curve	NOUN
ejpam-841	626	27	s(4	s(4	NOUN
ejpam-841	626	28	)	)	PUNCT
ejpam-841	626	29	from	from	ADP
ejpam-841	626	30	the	the	DET
ejpam-841	626	31	point	point	NOUN
ejpam-841	626	32	p	p	NOUN
ejpam-841	626	33	to	to	ADP
ejpam-841	626	34	the	the	DET
ejpam-841	626	35	hyperplane	hyperplane	NOUN
ejpam-841	626	36	u	u	NOUN
ejpam-841	626	37	=	=	PROPN
ejpam-841	626	38	0	0	X
ejpam-841	626	39	.	.	PUNCT
ejpam-841	626	40	observe	observe	VERB
ejpam-841	626	41	that	that	SCONJ
ejpam-841	626	42	the	the	DET
ejpam-841	626	43	parametrization	parametrization	NOUN
ejpam-841	626	44	of	of	ADP
ejpam-841	626	45	the	the	DET
ejpam-841	626	46	image	image	NOUN
ejpam-841	626	47	can	can	AUX
ejpam-841	626	48	be	be	AUX
ejpam-841	626	49	described	describe	VERB
ejpam-841	626	50	as	as	ADP
ejpam-841	626	51	f(s	f(	NOUN
ejpam-841	626	52	,	,	PUNCT
ejpam-841	626	53	t	t	PROPN
ejpam-841	626	54	)	)	PUNCT
ejpam-841	627	1	=	=	SYM
ejpam-841	627	2	(	(	PUNCT
ejpam-841	627	3	s4	s4	PROPN
ejpam-841	627	4	−	−	PROPN
ejpam-841	627	5	at4	at4	PROPN
ejpam-841	627	6	,	,	PUNCT
ejpam-841	627	7	s3	s3	PROPN
ejpam-841	627	8	t	t	PROPN
ejpam-841	627	9	−	−	PROPN
ejpam-841	627	10	bt4	bt4	PROPN
ejpam-841	627	11	,	,	PUNCT
ejpam-841	627	12	s2	s2	NOUN
ejpam-841	627	13	t2	t2	NOUN
ejpam-841	627	14	−	−	PROPN
ejpam-841	627	15	ct4	ct4	PROPN
ejpam-841	627	16	,	,	PUNCT
ejpam-841	627	17	st3	st3	PROPN
ejpam-841	627	18	−	−	PROPN
ejpam-841	627	19	d	d	PROPN
ejpam-841	627	20	t4	t4	PROPN
ejpam-841	627	21	)	)	PUNCT
ejpam-841	627	22	.	.	PUNCT
ejpam-841	628	1	therefore	therefore	ADV
ejpam-841	628	2	the	the	DET
ejpam-841	628	3	implicit	implicit	ADJ
ejpam-841	628	4	equations	equation	NOUN
ejpam-841	628	5	of	of	ADP
ejpam-841	628	6	c	c	PROPN
ejpam-841	628	7	are	be	AUX
ejpam-841	628	8	the	the	DET
ejpam-841	628	9	generators	generator	NOUN
ejpam-841	628	10	of	of	ADP
ejpam-841	628	11	the	the	DET
ejpam-841	628	12	following	follow	VERB
ejpam-841	628	13	ideal	ideal	NOUN
ejpam-841	628	14	:	:	PUNCT
ejpam-841	628	15	〈	〈	PROPN
ejpam-841	628	16	x	x	SYM
ejpam-841	628	17	−	−	PROPN
ejpam-841	628	18	(	(	PUNCT
ejpam-841	628	19	s4	s4	PROPN
ejpam-841	628	20	−	−	PROPN
ejpam-841	628	21	at4	at4	PROPN
ejpam-841	628	22	)	)	PUNCT
ejpam-841	628	23	,	,	PUNCT
ejpam-841	628	24	y	y	PROPN
ejpam-841	628	25	−	−	PROPN
ejpam-841	628	26	(	(	PUNCT
ejpam-841	628	27	s3	s3	PROPN
ejpam-841	628	28	t	t	PROPN
ejpam-841	628	29	−	−	PROPN
ejpam-841	628	30	bt4	bt4	PROPN
ejpam-841	628	31	)	)	PUNCT
ejpam-841	628	32	,	,	PUNCT
ejpam-841	628	33	z	z	NOUN
ejpam-841	628	34	−	−	PROPN
ejpam-841	629	1	(	(	PUNCT
ejpam-841	629	2	s2t2	s2t2	ADP
ejpam-841	629	3	−	−	NOUN
ejpam-841	629	4	ct4	ct4	PROPN
ejpam-841	629	5	)	)	PUNCT
ejpam-841	629	6	,	,	PUNCT
ejpam-841	629	7	w	w	ADP
ejpam-841	629	8	−	−	PROPN
ejpam-841	629	9	(	(	PUNCT
ejpam-841	629	10	st3	st3	PROPN
ejpam-841	629	11	−	−	PROPN
ejpam-841	629	12	d	d	PROPN
ejpam-841	629	13	t4	t4	PROPN
ejpam-841	629	14	)	)	PUNCT
ejpam-841	629	15	〉	〉	NOUN
ejpam-841	629	16	⋂	⋂	PROPN
ejpam-841	629	17	k(a	k(a	PROPN
ejpam-841	629	18	,	,	PUNCT
ejpam-841	629	19	b	b	NOUN
ejpam-841	629	20	,	,	PUNCT
ejpam-841	629	21	c	c	NOUN
ejpam-841	629	22	,	,	PUNCT
ejpam-841	629	23	d)[x	d)[x	PROPN
ejpam-841	629	24	,	,	PUNCT
ejpam-841	629	25	y	y	PROPN
ejpam-841	629	26	,	,	PUNCT
ejpam-841	629	27	z	z	PROPN
ejpam-841	629	28	,	,	PUNCT
ejpam-841	629	29	w	w	NOUN
ejpam-841	629	30	]	]	PUNCT
ejpam-841	629	31	.	.	PUNCT
ejpam-841	630	1	with	with	ADP
ejpam-841	630	2	the	the	DET
ejpam-841	630	3	aid	aid	NOUN
ejpam-841	630	4	of	of	ADP
ejpam-841	630	5	a	a	DET
ejpam-841	630	6	computer	computer	NOUN
ejpam-841	630	7	algebra	algebra	NOUN
ejpam-841	630	8	system	system	NOUN
ejpam-841	630	9	,	,	PUNCT
ejpam-841	630	10	we	we	PRON
ejpam-841	630	11	compute	compute	VERB
ejpam-841	630	12	four	four	NUM
ejpam-841	630	13	polynomial	polynomial	ADJ
ejpam-841	630	14	generators	generator	NOUN
ejpam-841	630	15	:	:	PUNCT
ejpam-841	630	16	f1	f1	NOUN
ejpam-841	630	17	,	,	PUNCT
ejpam-841	630	18	f2	f2	PROPN
ejpam-841	630	19	,	,	PUNCT
ejpam-841	630	20	f3	f3	PROPN
ejpam-841	630	21	,	,	PUNCT
ejpam-841	630	22	f4	f4	PROPN
ejpam-841	630	23	;	;	PUNCT
ejpam-841	630	24	with	with	ADP
ejpam-841	630	25	one	one	NUM
ejpam-841	630	26	quadric	quadric	ADJ
ejpam-841	630	27	f1	f1	NOUN
ejpam-841	630	28	and	and	CCONJ
ejpam-841	630	29	three	three	NUM
ejpam-841	630	30	cubics	cubic	NOUN
ejpam-841	630	31	f2	f2	PROPN
ejpam-841	630	32	,	,	PUNCT
ejpam-841	630	33	f3	f3	PROPN
ejpam-841	630	34	,	,	PUNCT
ejpam-841	630	35	f4	f4	PROPN
ejpam-841	630	36	–	–	PUNCT
ejpam-841	630	37	see	see	NOUN
ejpam-841	630	38	below	below	ADV
ejpam-841	630	39	.	.	PUNCT
ejpam-841	631	1	moreover	moreover	ADV
ejpam-841	631	2	,	,	PUNCT
ejpam-841	631	3	f1	f1	PROPN
ejpam-841	631	4	=	=	SYM
ejpam-841	631	5	sylvs	sylvs	PROPN
ejpam-841	631	6	,	,	PUNCT
ejpam-841	631	7	t(p	t(p	PROPN
ejpam-841	631	8	,	,	PUNCT
ejpam-841	631	9	q	q	NOUN
ejpam-841	631	10	)	)	PUNCT
ejpam-841	631	11	,	,	PUNCT
ejpam-841	631	12	since	since	SCONJ
ejpam-841	631	13	by	by	ADP
ejpam-841	631	14	lemma	lemma	PROPN
ejpam-841	631	15	4	4	NUM
ejpam-841	631	16	any	any	DET
ejpam-841	631	17	non	non	ADJ
ejpam-841	631	18	-	-	ADJ
ejpam-841	631	19	singular	singular	ADJ
ejpam-841	631	20	quartic	quartic	ADJ
ejpam-841	631	21	space	space	NOUN
ejpam-841	631	22	curve	curve	NOUN
ejpam-841	631	23	is	be	AUX
ejpam-841	631	24	contained	contain	VERB
ejpam-841	631	25	in	in	ADP
ejpam-841	631	26	exactly	exactly	ADV
ejpam-841	631	27	one	one	NUM
ejpam-841	631	28	quadric	quadric	ADJ
ejpam-841	631	29	.	.	PUNCT
ejpam-841	632	1	f1	f1	PROPN
ejpam-841	632	2	=(	=(	NOUN
ejpam-841	632	3	−c	−c	NOUN
ejpam-841	632	4	+	+	CCONJ
ejpam-841	632	5	d2)xz+	d2)xz+	X
ejpam-841	632	6	(	(	PUNCT
ejpam-841	632	7	b−	b−	NOUN
ejpam-841	632	8	cd)xw+	cd)xw+	NOUN
ejpam-841	632	9	(	(	PUNCT
ejpam-841	632	10	c	c	NOUN
ejpam-841	632	11	−	−	PROPN
ejpam-841	632	12	d2)y2	d2)y2	PROPN
ejpam-841	632	13	+	+	CCONJ
ejpam-841	632	14	(	(	PUNCT
ejpam-841	632	15	−b+	−b+	NOUN
ejpam-841	632	16	cd)yz	cd)yz	X
ejpam-841	633	1	+	+	CCONJ
ejpam-841	633	2	(	(	PUNCT
ejpam-841	633	3	−a+	−a+	X
ejpam-841	633	4	2bd	2bd	NOUN
ejpam-841	633	5	−	−	PROPN
ejpam-841	633	6	c2)yw+	c2)yw+	NOUN
ejpam-841	633	7	(	(	PUNCT
ejpam-841	633	8	a−	a−	PROPN
ejpam-841	633	9	bd)z2	bd)z2	NOUN
ejpam-841	633	10	+	+	CCONJ
ejpam-841	633	11	(	(	PUNCT
ejpam-841	633	12	−ad	−ad	X
ejpam-841	633	13	+	+	NOUN
ejpam-841	633	14	bc)zw	bc)zw	NOUN
ejpam-841	633	15	+	+	CCONJ
ejpam-841	633	16	(	(	PUNCT
ejpam-841	633	17	ac	ac	PROPN
ejpam-841	633	18	−	−	PROPN
ejpam-841	633	19	b2)w2	b2)w2	PROPN
ejpam-841	633	20	;	;	PUNCT
ejpam-841	633	21	f2	f2	PROPN
ejpam-841	633	22	=(	=(	NOUN
ejpam-841	633	23	c	c	NOUN
ejpam-841	633	24	−	−	PROPN
ejpam-841	633	25	d2)y2w	d2)y2w	NOUN
ejpam-841	634	1	+	+	CCONJ
ejpam-841	634	2	(	(	PUNCT
ejpam-841	634	3	−c+	−c+	NOUN
ejpam-841	634	4	d2)yz2	d2)yz2	VERB
ejpam-841	634	5	+	+	X
ejpam-841	634	6	(	(	PUNCT
ejpam-841	634	7	−b+	−b+	NOUN
ejpam-841	634	8	cd)yzw	cd)yzw	PROPN
ejpam-841	635	1	+	+	CCONJ
ejpam-841	635	2	(	(	PUNCT
ejpam-841	635	3	2bd	2bd	ADJ
ejpam-841	635	4	−	−	PROPN
ejpam-841	635	5	2c2)yw2	2c2)yw2	NUM
ejpam-841	635	6	+	+	CCONJ
ejpam-841	635	7	(	(	PUNCT
ejpam-841	635	8	b−	b−	PROPN
ejpam-841	635	9	d3)z3	d3)z3	PROPN
ejpam-841	635	10	+	+	CCONJ
ejpam-841	636	1	(	(	PUNCT
ejpam-841	636	2	−3bd	−3bd	PROPN
ejpam-841	636	3	+	+	CCONJ
ejpam-841	636	4	3cd2)z2w	3cd2)z2w	PROPN
ejpam-841	636	5	+	+	CCONJ
ejpam-841	636	6	(	(	PUNCT
ejpam-841	636	7	3bc−	3bc−	NUM
ejpam-841	636	8	3c2d)zw2	3c2d)zw2	NUM
ejpam-841	636	9	+	+	CCONJ
ejpam-841	636	10	(	(	PUNCT
ejpam-841	636	11	−b2	−b2	PROPN
ejpam-841	636	12	+	+	CCONJ
ejpam-841	636	13	c3)w3	c3)w3	NOUN
ejpam-841	636	14	;	;	PUNCT
ejpam-841	636	15	f3	f3	PROPN
ejpam-841	636	16	=(	=(	PROPN
ejpam-841	636	17	−c2	−c2	PROPN
ejpam-841	636	18	+	+	CCONJ
ejpam-841	636	19	2cd2	2cd2	NUM
ejpam-841	636	20	−	−	PROPN
ejpam-841	636	21	d4)x	d4)x	NOUN
ejpam-841	636	22	yw+	yw+	PROPN
ejpam-841	636	23	(	(	PUNCT
ejpam-841	636	24	b2	b2	NOUN
ejpam-841	636	25	−	−	PROPN
ejpam-841	636	26	3bcd	3bcd	PROPN
ejpam-841	636	27	+	+	NUM
ejpam-841	636	28	bd3	bd3	ADJ
ejpam-841	636	29	+	+	CCONJ
ejpam-841	636	30	c3)xw2	c3)xw2	NOUN
ejpam-841	636	31	+	+	CCONJ
ejpam-841	636	32	(	(	PUNCT
ejpam-841	636	33	c2	c2	PROPN
ejpam-841	636	34	−	−	PROPN
ejpam-841	636	35	2cd2	2cd2	PUNCT
ejpam-841	637	1	+	+	CCONJ
ejpam-841	637	2	d4)y2z+	d4)y2z+	PROPN
ejpam-841	637	3	(	(	PUNCT
ejpam-841	637	4	bc	bc	PROPN
ejpam-841	637	5	−	−	PROPN
ejpam-841	637	6	bd2−	bd2−	NOUN
ejpam-841	637	7	c2d	c2d	PROPN
ejpam-841	637	8	+	+	CCONJ
ejpam-841	637	9	cd3)y2w	cd3)y2w	NOUN
ejpam-841	637	10	+	+	CCONJ
ejpam-841	637	11	(	(	PUNCT
ejpam-841	637	12	−bc+	−bc+	PROPN
ejpam-841	637	13	bd2	bd2	NOUN
ejpam-841	637	14	+	+	PROPN
ejpam-841	637	15	cd3	cd3	PROPN
ejpam-841	637	16	−	−	PROPN
ejpam-841	637	17	d5)yz2	d5)yz2	VERB
ejpam-841	637	18	+	+	CCONJ
ejpam-841	637	19	(	(	PUNCT
ejpam-841	637	20	−b2	−b2	PROPN
ejpam-841	637	21	+	+	CCONJ
ejpam-841	638	1	4bcd	4bcd	NUM
ejpam-841	638	2	−	−	NUM
ejpam-841	638	3	2bd3−	2bd3−	NUM
ejpam-841	638	4	3c2d2	3c2d2	NUM
ejpam-841	638	5	+	+	NOUN
ejpam-841	638	6	2cd4)yzw	2cd4)yzw	NUM
ejpam-841	638	7	+	+	CCONJ
ejpam-841	638	8	(	(	PUNCT
ejpam-841	638	9	−ab+	−ab+	PROPN
ejpam-841	638	10	ad3	ad3	PROPN
ejpam-841	638	11	+	+	CCONJ
ejpam-841	638	12	2b2d	2b2d	NOUN
ejpam-841	638	13	−	−	NOUN
ejpam-841	638	14	bc2	bc2	NOUN
ejpam-841	639	1	−	−	PROPN
ejpam-841	639	2	2bcd2	2bcd2	NUM
ejpam-841	639	3	+	+	NUM
ejpam-841	639	4	2c3d	2c3d	NOUN
ejpam-841	639	5	−	−	ADP
ejpam-841	640	1	c2d3)yw2	c2d3)yw2	PRON
ejpam-841	640	2	+	+	NUM
ejpam-841	640	3	(	(	PUNCT
ejpam-841	640	4	ab−	ab−	NUM
ejpam-841	640	5	ad3−	ad3−	VERB
ejpam-841	640	6	b2d	b2d	NOUN
ejpam-841	640	7	+	+	CCONJ
ejpam-841	640	8	bd4)z2w	bd4)z2w	PRON
ejpam-841	640	9	+	+	CCONJ
ejpam-841	640	10	(	(	PUNCT
ejpam-841	640	11	−abd	−abd	NOUN
ejpam-841	640	12	−	−	PROPN
ejpam-841	640	13	ac2	ac2	PROPN
ejpam-841	640	14	+	+	CCONJ
ejpam-841	640	15	2acd2	2acd2	NUM
ejpam-841	640	16	+	+	CCONJ
ejpam-841	640	17	b2d2	b2d2	PROPN
ejpam-841	640	18	+	+	ADJ
ejpam-841	640	19	bc2d	bc2d	NOUN
ejpam-841	640	20	−	−	NOUN
ejpam-841	640	21	2bcd3)zw2	2bcd3)zw2	NUM
ejpam-841	640	22	+	+	CCONJ
ejpam-841	640	23	(	(	PUNCT
ejpam-841	640	24	2abc	2abc	PROPN
ejpam-841	640	25	−	−	PROPN
ejpam-841	640	26	abd2−	abd2−	NOUN
ejpam-841	640	27	ac2d	ac2d	PROPN
ejpam-841	640	28	−	−	PROPN
ejpam-841	640	29	b3	b3	PROPN
ejpam-841	640	30	+	+	CCONJ
ejpam-841	640	31	b2cd	b2cd	PUNCT
ejpam-841	640	32	−	−	NOUN
ejpam-841	640	33	bc3	bc3	NOUN
ejpam-841	640	34	+	+	CCONJ
ejpam-841	640	35	bc2d2)w3	bc2d2)w3	NOUN
ejpam-841	640	36	;	;	PUNCT
ejpam-841	640	37	f4	f4	NUM
ejpam-841	640	38	=(	=(	NOUN
ejpam-841	640	39	−c2	−c2	NOUN
ejpam-841	640	40	+	+	CCONJ
ejpam-841	640	41	2cd2	2cd2	NUM
ejpam-841	640	42	−	−	PROPN
ejpam-841	640	43	d4)x2w	d4)x2w	VERB
ejpam-841	640	44	+	+	CCONJ
ejpam-841	640	45	(	(	PUNCT
ejpam-841	640	46	c2	c2	PROPN
ejpam-841	640	47	−	−	PROPN
ejpam-841	640	48	2cd2	2cd2	SYM
ejpam-841	640	49	+	+	NUM
ejpam-841	640	50	d4)x	d4)x	NOUN
ejpam-841	640	51	yz	yz	X
ejpam-841	640	52	+	+	CCONJ
ejpam-841	640	53	(	(	PUNCT
ejpam-841	640	54	bc	bc	PROPN
ejpam-841	640	55	−	−	PROPN
ejpam-841	640	56	bd2	bd2	NOUN
ejpam-841	640	57	−	−	NOUN
ejpam-841	640	58	c2d	c2d	NOUN
ejpam-841	641	1	+	+	CCONJ
ejpam-841	641	2	cd3)x	cd3)x	PROPN
ejpam-841	641	3	yw+	yw+	INTJ
ejpam-841	641	4	(	(	PUNCT
ejpam-841	641	5	−bc	−bc	NOUN
ejpam-841	641	6	+	+	NOUN
ejpam-841	641	7	bd2	bd2	NOUN
ejpam-841	641	8	+	+	NOUN
ejpam-841	641	9	cd3	cd3	NOUN
ejpam-841	641	10	−	−	PROPN
ejpam-841	641	11	d5)xz2	d5)xz2	X
ejpam-841	641	12	+	+	CCONJ
ejpam-841	641	13	(	(	PUNCT
ejpam-841	641	14	bcd	bcd	PROPN
ejpam-841	641	15	−	−	PROPN
ejpam-841	641	16	bd3	bd3	PROPN
ejpam-841	641	17	+	+	PROPN
ejpam-841	641	18	c3	c3	PROPN
ejpam-841	641	19	−	−	PROPN
ejpam-841	641	20	3c2d2	3c2d2	NUM
ejpam-841	641	21	+	+	NUM
ejpam-841	641	22	2cd4)xzw+	2cd4)xzw+	NUM
ejpam-841	641	23	(	(	PUNCT
ejpam-841	641	24	ab−	ab−	PROPN
ejpam-841	641	25	j.	j.	PROPN
ejpam-841	641	26	hoffman	hoffman	PROPN
ejpam-841	641	27	,	,	PUNCT
ejpam-841	641	28	h.	h.	PROPN
ejpam-841	641	29	wang	wang	PROPN
ejpam-841	641	30	,	,	PUNCT
ejpam-841	641	31	x.	x.	PROPN
ejpam-841	641	32	jia	jia	PROPN
ejpam-841	641	33	,	,	PUNCT
ejpam-841	641	34	r.	r.	PROPN
ejpam-841	641	35	goldman	goldman	PROPN
ejpam-841	641	36	/	/	SYM
ejpam-841	641	37	eur	eur	PROPN
ejpam-841	641	38	.	.	PUNCT
ejpam-841	642	1	j.	j.	PROPN
ejpam-841	642	2	pure	pure	PROPN
ejpam-841	642	3	appl	appl	PROPN
ejpam-841	642	4	.	.	PROPN
ejpam-841	642	5	math	math	PROPN
ejpam-841	642	6	,	,	PUNCT
ejpam-841	642	7	3	3	NUM
ejpam-841	642	8	(	(	PUNCT
ejpam-841	642	9	2010	2010	NUM
ejpam-841	642	10	)	)	PUNCT
ejpam-841	642	11	,	,	PUNCT
ejpam-841	642	12	602	602	NUM
ejpam-841	642	13	-	-	SYM
ejpam-841	642	14	632	632	NUM
ejpam-841	642	15	624	624	NUM
ejpam-841	642	16	3acd	3acd	NUM
ejpam-841	642	17	+	+	CCONJ
ejpam-841	642	18	2ad3−	2ad3−	NUM
ejpam-841	642	19	bcd2	bcd2	NOUN
ejpam-841	642	20	+	+	CCONJ
ejpam-841	642	21	2c3d	2c3d	NOUN
ejpam-841	642	22	−	−	ADP
ejpam-841	642	23	c2d3)xw2	c2d3)xw2	PROPN
ejpam-841	643	1	+	+	NUM
ejpam-841	643	2	(	(	PUNCT
ejpam-841	643	3	−ab+	−ab+	NOUN
ejpam-841	643	4	3acd	3acd	NUM
ejpam-841	643	5	−	−	PROPN
ejpam-841	643	6	2ad3−	2ad3−	NUM
ejpam-841	643	7	bc2	bc2	NOUN
ejpam-841	644	1	+	+	CCONJ
ejpam-841	644	2	2bcd2−	2bcd2−	X
ejpam-841	644	3	c3d)yzw	c3d)yzw	PROPN
ejpam-841	644	4	+	+	CCONJ
ejpam-841	644	5	(	(	PUNCT
ejpam-841	644	6	−a2	−a2	NOUN
ejpam-841	644	7	+	+	NOUN
ejpam-841	644	8	2abd	2abd	NOUN
ejpam-841	644	9	−	−	PROPN
ejpam-841	644	10	2bc2d	2bc2d	NUM
ejpam-841	644	11	+	+	CCONJ
ejpam-841	644	12	c4)yw2	c4)yw2	NOUN
ejpam-841	644	13	+	+	CCONJ
ejpam-841	644	14	(	(	PUNCT
ejpam-841	644	15	a2−	a2−	PROPN
ejpam-841	644	16	abd	abd	PROPN
ejpam-841	644	17	−	−	PROPN
ejpam-841	644	18	ac2	ac2	PROPN
ejpam-841	644	19	−	−	PROPN
ejpam-841	644	20	acd2	acd2	NOUN
ejpam-841	644	21	+	+	CCONJ
ejpam-841	644	22	ad4	ad4	PROPN
ejpam-841	644	23	+	+	PROPN
ejpam-841	644	24	b2c	b2c	NOUN
ejpam-841	644	25	−	−	PUNCT
ejpam-841	644	26	b2d2	b2d2	X
ejpam-841	644	27	+	+	NOUN
ejpam-841	644	28	bc2d)z2w	bc2d)z2w	PROPN
ejpam-841	644	29	+	+	CCONJ
ejpam-841	644	30	(	(	PUNCT
ejpam-841	644	31	−a2d	−a2d	NUM
ejpam-841	644	32	−	−	PROPN
ejpam-841	644	33	abc	abc	PROPN
ejpam-841	644	34	+	+	CCONJ
ejpam-841	644	35	2abd2	2abd2	NUM
ejpam-841	644	36	+	+	SYM
ejpam-841	644	37	3ac2d	3ac2d	NUM
ejpam-841	644	38	−	−	PROPN
ejpam-841	644	39	2acd3−	2acd3−	NUM
ejpam-841	644	40	bc3)zw2	bc3)zw2	X
ejpam-841	644	41	+	+	CCONJ
ejpam-841	644	42	(	(	PUNCT
ejpam-841	644	43	2a2c	2a2c	NUM
ejpam-841	644	44	−	−	NOUN
ejpam-841	644	45	a2d2	a2d2	ADP
ejpam-841	644	46	−	−	PROPN
ejpam-841	644	47	ab2	ab2	NOUN
ejpam-841	644	48	−	−	NOUN
ejpam-841	644	49	2ac3	2ac3	NUM
ejpam-841	644	50	+	+	CCONJ
ejpam-841	644	51	ac2d2	ac2d2	X
ejpam-841	644	52	+	+	NOUN
ejpam-841	644	53	b2c2)w3	b2c2)w3	NOUN
ejpam-841	644	54	.	.	PUNCT
ejpam-841	645	1	lemma	lemma	PROPN
ejpam-841	645	2	5	5	NUM
ejpam-841	645	3	is	be	AUX
ejpam-841	645	4	a	a	DET
ejpam-841	645	5	theoretical	theoretical	ADJ
ejpam-841	645	6	result	result	NOUN
ejpam-841	645	7	.	.	PUNCT
ejpam-841	646	1	our	our	PRON
ejpam-841	646	2	next	next	ADJ
ejpam-841	646	3	goal	goal	NOUN
ejpam-841	646	4	is	be	AUX
ejpam-841	646	5	to	to	PART
ejpam-841	646	6	find	find	VERB
ejpam-841	646	7	simple	simple	ADJ
ejpam-841	646	8	explicit	explicit	ADJ
ejpam-841	646	9	expressions	expression	NOUN
ejpam-841	646	10	for	for	ADP
ejpam-841	646	11	these	these	DET
ejpam-841	646	12	four	four	NUM
ejpam-841	646	13	implicit	implicit	ADJ
ejpam-841	646	14	equations	equation	NOUN
ejpam-841	646	15	.	.	PUNCT
ejpam-841	647	1	to	to	PART
ejpam-841	647	2	proceed	proceed	VERB
ejpam-841	647	3	,	,	PUNCT
ejpam-841	647	4	we	we	PRON
ejpam-841	647	5	first	first	ADV
ejpam-841	647	6	dehomogenize	dehomogenize	VERB
ejpam-841	647	7	the	the	DET
ejpam-841	647	8	µ-basis	µ-basis	NOUN
ejpam-841	647	9	elements	element	NOUN
ejpam-841	647	10	by	by	ADP
ejpam-841	647	11	setting	set	VERB
ejpam-841	647	12	t	t	NOUN
ejpam-841	647	13	=	=	SYM
ejpam-841	647	14	1	1	NUM
ejpam-841	647	15	,	,	PUNCT
ejpam-841	647	16	so	so	SCONJ
ejpam-841	647	17	that	that	SCONJ
ejpam-841	647	18	p	p	NOUN
ejpam-841	647	19	=	=	NOUN
ejpam-841	647	20	p1s+	p1s+	NOUN
ejpam-841	647	21	p0	p0	NOUN
ejpam-841	647	22	,	,	PUNCT
ejpam-841	647	23	q	q	NOUN
ejpam-841	647	24	=	=	PUNCT
ejpam-841	647	25	q1s+	q1s+	NOUN
ejpam-841	647	26	q0	q0	NOUN
ejpam-841	647	27	,	,	PUNCT
ejpam-841	647	28	r	r	NOUN
ejpam-841	647	29	=	=	SYM
ejpam-841	647	30	r2s2	r2s2	PROPN
ejpam-841	647	31	+	+	NOUN
ejpam-841	647	32	r1s+	r1s+	NOUN
ejpam-841	647	33	r0	r0	NOUN
ejpam-841	647	34	,	,	PUNCT
ejpam-841	647	35	where	where	SCONJ
ejpam-841	647	36	deg(p1	deg(p1	NOUN
ejpam-841	647	37	)	)	PUNCT
ejpam-841	647	38	=	=	SYM
ejpam-841	648	1	deg(p0	deg(p0	PROPN
ejpam-841	648	2	)	)	PUNCT
ejpam-841	648	3	=	=	SYM
ejpam-841	648	4	deg(q1	deg(q1	NOUN
ejpam-841	648	5	)	)	PUNCT
ejpam-841	648	6	=	=	SYM
ejpam-841	648	7	deg(q0	deg(q0	NOUN
ejpam-841	648	8	)	)	PUNCT
ejpam-841	648	9	=	=	SYM
ejpam-841	648	10	deg(r2	deg(r2	NOUN
ejpam-841	648	11	)	)	PUNCT
ejpam-841	648	12	=	=	SYM
ejpam-841	649	1	deg(r1	deg(r1	PROPN
ejpam-841	649	2	)	)	PUNCT
ejpam-841	649	3	=	=	SYM
ejpam-841	650	1	deg(r1	deg(r1	NOUN
ejpam-841	650	2	)	)	PUNCT
ejpam-841	650	3	=	=	SYM
ejpam-841	651	1	1	1	X
ejpam-841	651	2	.	.	PUNCT
ejpam-841	652	1	if	if	SCONJ
ejpam-841	652	2	we	we	PRON
ejpam-841	652	3	take	take	VERB
ejpam-841	652	4	the	the	DET
ejpam-841	652	5	resultants	resultant	NOUN
ejpam-841	652	6	with	with	ADP
ejpam-841	652	7	respect	respect	NOUN
ejpam-841	652	8	to	to	ADP
ejpam-841	652	9	the	the	DET
ejpam-841	652	10	variable	variable	NOUN
ejpam-841	652	11	s	s	NOUN
ejpam-841	652	12	,	,	PUNCT
ejpam-841	652	13	then	then	ADV
ejpam-841	652	14	we	we	PRON
ejpam-841	652	15	generate	generate	VERB
ejpam-841	652	16	the	the	DET
ejpam-841	652	17	following	follow	VERB
ejpam-841	652	18	three	three	NUM
ejpam-841	652	19	expressions	expression	NOUN
ejpam-841	652	20	:	:	PUNCT
ejpam-841	652	21	res(p	res(p	NOUN
ejpam-841	652	22	,	,	PUNCT
ejpam-841	652	23	q	q	NOUN
ejpam-841	652	24	)	)	PUNCT
ejpam-841	652	25	=	=	SYM
ejpam-841	652	26	det	det	PROPN
ejpam-841	652	27	�	�	PROPN
ejpam-841	652	28	p1	p1	PROPN
ejpam-841	652	29	p0	p0	PROPN
ejpam-841	652	30	q1	q1	PROPN
ejpam-841	652	31	q0	q0	PROPN
ejpam-841	652	32	�	�	PROPN
ejpam-841	652	33	=	=	SYM
ejpam-841	652	34	sylvs	sylvs	PROPN
ejpam-841	652	35	,	,	PUNCT
ejpam-841	652	36	t(p	t(p	PROPN
ejpam-841	652	37	,	,	PUNCT
ejpam-841	652	38	q	q	NOUN
ejpam-841	652	39	)	)	PUNCT
ejpam-841	652	40	=	=	SYM
ejpam-841	653	1	p1q0	p1q0	X
ejpam-841	653	2	−	−	X
ejpam-841	653	3	p0q1	p0q1	PROPN
ejpam-841	653	4	,	,	PUNCT
ejpam-841	653	5	res(p	res(p	PROPN
ejpam-841	653	6	,	,	PUNCT
ejpam-841	653	7	r	r	NOUN
ejpam-841	653	8	)	)	PUNCT
ejpam-841	653	9	=	=	SYM
ejpam-841	653	10	det	det	PROPN
ejpam-841	653	11			PROPN
ejpam-841	653	12			NOUN
ejpam-841	653	13			PROPN
ejpam-841	653	14	p1	p1	NOUN
ejpam-841	653	15	0	0	NUM
ejpam-841	653	16	r2	r2	PROPN
ejpam-841	653	17	p0	p0	PROPN
ejpam-841	653	18	p1	p1	PROPN
ejpam-841	653	19	r1	r1	PROPN
ejpam-841	653	20	0	0	NUM
ejpam-841	653	21	p0	p0	PROPN
ejpam-841	653	22	r0	r0	NOUN
ejpam-841	653	23			PROPN
ejpam-841	653	24			VERB
ejpam-841	653	25			PUNCT
ejpam-841	654	1	=	=	PUNCT
ejpam-841	654	2	r2p2	r2p2	ADJ
ejpam-841	654	3	0	0	NUM
ejpam-841	654	4	−	−	NOUN
ejpam-841	654	5	r1p1p0	r1p1p0	NOUN
ejpam-841	654	6	+	+	CCONJ
ejpam-841	654	7	r0p2	r0p2	NOUN
ejpam-841	654	8	1	1	NUM
ejpam-841	654	9	,	,	PUNCT
ejpam-841	654	10	res(q	res(q	NOUN
ejpam-841	654	11	,	,	PUNCT
ejpam-841	654	12	r	r	NOUN
ejpam-841	654	13	)	)	PUNCT
ejpam-841	654	14	=	=	SYM
ejpam-841	654	15	det	det	PROPN
ejpam-841	654	16			PROPN
ejpam-841	654	17			PROPN
ejpam-841	654	18			PROPN
ejpam-841	654	19	q1	q1	NOUN
ejpam-841	654	20	0	0	NUM
ejpam-841	654	21	r2	r2	PROPN
ejpam-841	654	22	q0	q0	PROPN
ejpam-841	654	23	q1	q1	PROPN
ejpam-841	654	24	r1	r1	PROPN
ejpam-841	654	25	0	0	NUM
ejpam-841	655	1	q0	q0	PROPN
ejpam-841	655	2	r0	r0	NOUN
ejpam-841	655	3			PROPN
ejpam-841	655	4			VERB
ejpam-841	655	5			PUNCT
ejpam-841	656	1	=	=	PUNCT
ejpam-841	656	2	r2q2	r2q2	NOUN
ejpam-841	656	3	0	0	NUM
ejpam-841	656	4	−	−	NOUN
ejpam-841	656	5	r1q1q0	r1q1q0	NOUN
ejpam-841	656	6	+	+	CCONJ
ejpam-841	656	7	r0q2	r0q2	PROPN
ejpam-841	656	8	1	1	NUM
ejpam-841	656	9	,	,	PUNCT
ejpam-841	656	10	where	where	SCONJ
ejpam-841	656	11	deg(res(p	deg(res(p	NOUN
ejpam-841	656	12	,	,	PUNCT
ejpam-841	656	13	q	q	NOUN
ejpam-841	656	14	)	)	PUNCT
ejpam-841	656	15	)	)	PUNCT
ejpam-841	656	16	=	=	SYM
ejpam-841	656	17	2	2	NUM
ejpam-841	656	18	,	,	PUNCT
ejpam-841	656	19	deg(res(p	deg(res(p	PROPN
ejpam-841	656	20	,	,	PUNCT
ejpam-841	656	21	r	r	NOUN
ejpam-841	656	22	)	)	PUNCT
ejpam-841	656	23	)	)	PUNCT
ejpam-841	656	24	=	=	SYM
ejpam-841	656	25	3	3	NUM
ejpam-841	656	26	,	,	PUNCT
ejpam-841	656	27	and	and	CCONJ
ejpam-841	656	28	deg(res(q	deg(res(q	NOUN
ejpam-841	656	29	,	,	PUNCT
ejpam-841	656	30	r	r	NOUN
ejpam-841	656	31	)	)	PUNCT
ejpam-841	656	32	)	)	PUNCT
ejpam-841	656	33	=	=	SYM
ejpam-841	657	1	3	3	NUM
ejpam-841	657	2	in	in	ADP
ejpam-841	657	3	x	x	X
ejpam-841	657	4	,	,	PUNCT
ejpam-841	657	5	y	y	PROPN
ejpam-841	657	6	,	,	PUNCT
ejpam-841	657	7	z	z	PROPN
ejpam-841	657	8	,	,	PUNCT
ejpam-841	657	9	w.	w.	PROPN
ejpam-841	657	10	remark	remark	PROPN
ejpam-841	657	11	4	4	NUM
ejpam-841	657	12	.	.	PUNCT
ejpam-841	658	1	if	if	SCONJ
ejpam-841	658	2	p	p	X
ejpam-841	658	3	,	,	PUNCT
ejpam-841	658	4	q	q	ADJ
ejpam-841	658	5	,	,	PUNCT
ejpam-841	658	6	r	r	NOUN
ejpam-841	658	7	are	be	AUX
ejpam-841	658	8	a	a	DET
ejpam-841	658	9	µ-basis	µ-basis	NOUN
ejpam-841	658	10	for	for	ADP
ejpam-841	658	11	the	the	DET
ejpam-841	658	12	space	space	NOUN
ejpam-841	658	13	curve	curve	NOUN
ejpam-841	658	14	c	c	PROPN
ejpam-841	658	15	,	,	PUNCT
ejpam-841	658	16	then	then	ADV
ejpam-841	658	17	c	c	PROPN
ejpam-841	658	18	is	be	AUX
ejpam-841	658	19	contained	contain	VERB
ejpam-841	658	20	in	in	ADP
ejpam-841	658	21	the	the	DET
ejpam-841	658	22	three	three	NUM
ejpam-841	658	23	surfaces	surface	NOUN
ejpam-841	658	24	res(p	res(p	ADP
ejpam-841	658	25	,	,	PUNCT
ejpam-841	658	26	q	q	NOUN
ejpam-841	658	27	)	)	PUNCT
ejpam-841	659	1	=	=	SYM
ejpam-841	659	2	0	0	NUM
ejpam-841	659	3	,	,	PUNCT
ejpam-841	659	4	res(p	res(p	PROPN
ejpam-841	659	5	,	,	PUNCT
ejpam-841	659	6	r	r	NOUN
ejpam-841	659	7	)	)	PUNCT
ejpam-841	659	8	=	=	SYM
ejpam-841	659	9	0	0	NUM
ejpam-841	659	10	and	and	CCONJ
ejpam-841	659	11	res(q	res(q	NOUN
ejpam-841	659	12	,	,	PUNCT
ejpam-841	659	13	r	r	NOUN
ejpam-841	659	14	)	)	PUNCT
ejpam-841	659	15	=	=	SYM
ejpam-841	659	16	0	0	PUNCT
ejpam-841	659	17	because	because	SCONJ
ejpam-841	659	18	each	each	DET
ejpam-841	659	19	element	element	NOUN
ejpam-841	659	20	of	of	ADP
ejpam-841	659	21	the	the	DET
ejpam-841	659	22	µ-basis	µ-basis	NOUN
ejpam-841	659	23	follows	follow	VERB
ejpam-841	659	24	the	the	DET
ejpam-841	659	25	curve	curve	NOUN
ejpam-841	659	26	.	.	PUNCT
ejpam-841	660	1	lemma	lemma	PROPN
ejpam-841	660	2	6	6	NUM
ejpam-841	660	3	.	.	PUNCT
ejpam-841	661	1	gcd(res(p	gcd(res(p	PROPN
ejpam-841	661	2	,	,	PUNCT
ejpam-841	661	3	q),res(p	q),res(p	PROPN
ejpam-841	661	4	,	,	PUNCT
ejpam-841	661	5	r	r	NOUN
ejpam-841	661	6	)	)	PUNCT
ejpam-841	661	7	)	)	PUNCT
ejpam-841	662	1	=	=	PUNCT
ejpam-841	662	2	gcd(res(p	gcd(res(p	PROPN
ejpam-841	662	3	,	,	PUNCT
ejpam-841	662	4	q),res(q	q),res(q	PROPN
ejpam-841	662	5	,	,	PUNCT
ejpam-841	662	6	r	r	NOUN
ejpam-841	662	7	)	)	PUNCT
ejpam-841	662	8	)	)	PUNCT
ejpam-841	663	1	=	=	SYM
ejpam-841	663	2	gcd(res(p	gcd(res(p	PROPN
ejpam-841	663	3	,	,	PUNCT
ejpam-841	663	4	r),res(q	r),res(q	NOUN
ejpam-841	663	5	,	,	PUNCT
ejpam-841	663	6	r	r	NOUN
ejpam-841	663	7	)	)	PUNCT
ejpam-841	663	8	)	)	PUNCT
ejpam-841	663	9	=	=	SYM
ejpam-841	664	1	1	1	X
ejpam-841	664	2	.	.	PUNCT
ejpam-841	664	3	proof	proof	NOUN
ejpam-841	664	4	.	.	PUNCT
ejpam-841	665	1	suppose	suppose	VERB
ejpam-841	665	2	that	that	SCONJ
ejpam-841	665	3	gcd(res(p	gcd(res(p	PROPN
ejpam-841	665	4	,	,	PUNCT
ejpam-841	665	5	q),res(p	q),res(p	PROPN
ejpam-841	665	6	,	,	PUNCT
ejpam-841	665	7	r	r	NOUN
ejpam-841	665	8	)	)	PUNCT
ejpam-841	665	9	)	)	PUNCT
ejpam-841	665	10	=	=	PUNCT
ejpam-841	665	11	g	g	PROPN
ejpam-841	665	12	6=	6=	PROPN
ejpam-841	665	13	1	1	NUM
ejpam-841	665	14	.	.	PUNCT
ejpam-841	665	15	then	then	ADV
ejpam-841	665	16	for	for	ADP
ejpam-841	665	17	any	any	DET
ejpam-841	665	18	point	point	NOUN
ejpam-841	665	19	a	a	DET
ejpam-841	665	20	satisfying	satisfying	NOUN
ejpam-841	665	21	g(a	g(a	PROPN
ejpam-841	665	22	)	)	PUNCT
ejpam-841	665	23	=	=	SYM
ejpam-841	665	24	0	0	NUM
ejpam-841	665	25	,	,	PUNCT
ejpam-841	665	26	we	we	PRON
ejpam-841	665	27	have	have	VERB
ejpam-841	665	28	deg(gcd(p(s	deg(gcd(p(s	PROPN
ejpam-841	665	29	,	,	PUNCT
ejpam-841	665	30	t	t	PROPN
ejpam-841	665	31	,	,	PUNCT
ejpam-841	665	32	a),q(s	a),q(s	PROPN
ejpam-841	665	33	,	,	PUNCT
ejpam-841	665	34	t	t	PROPN
ejpam-841	665	35	,	,	PUNCT
ejpam-841	665	36	a	a	PRON
ejpam-841	665	37	)	)	PUNCT
ejpam-841	665	38	)	)	PUNCT
ejpam-841	665	39	)	)	PUNCT
ejpam-841	665	40	≥	≥	NOUN
ejpam-841	665	41	1	1	NUM
ejpam-841	665	42	,	,	PUNCT
ejpam-841	665	43	and	and	CCONJ
ejpam-841	665	44	deg(gcd(p(s	deg(gcd(p(s	PROPN
ejpam-841	665	45	,	,	PUNCT
ejpam-841	665	46	t	t	PROPN
ejpam-841	665	47	,	,	PUNCT
ejpam-841	665	48	a	a	PRON
ejpam-841	665	49	)	)	PUNCT
ejpam-841	665	50	,	,	PUNCT
ejpam-841	665	51	r(s	r(s	PROPN
ejpam-841	665	52	,	,	PUNCT
ejpam-841	665	53	t	t	PROPN
ejpam-841	665	54	,	,	PUNCT
ejpam-841	665	55	a	a	PRON
ejpam-841	665	56	)	)	PUNCT
ejpam-841	665	57	)	)	PUNCT
ejpam-841	665	58	)	)	PUNCT
ejpam-841	665	59	≥	≥	NOUN
ejpam-841	666	1	1	1	NUM
ejpam-841	666	2	.	.	PUNCT
ejpam-841	667	1	since	since	SCONJ
ejpam-841	667	2	deg(p	deg(p	PROPN
ejpam-841	667	3	)	)	PUNCT
ejpam-841	667	4	=	=	SYM
ejpam-841	667	5	deg(q	deg(q	NOUN
ejpam-841	667	6	)	)	PUNCT
ejpam-841	667	7	=	=	SYM
ejpam-841	667	8	1	1	NUM
ejpam-841	667	9	and	and	CCONJ
ejpam-841	667	10	deg(r	deg(r	PROPN
ejpam-841	667	11	)	)	PUNCT
ejpam-841	667	12	=	=	SYM
ejpam-841	667	13	2	2	NUM
ejpam-841	667	14	in	in	ADP
ejpam-841	667	15	the	the	DET
ejpam-841	667	16	variables	variable	NOUN
ejpam-841	667	17	s	s	PROPN
ejpam-841	667	18	,	,	PUNCT
ejpam-841	667	19	t	t	PROPN
ejpam-841	667	20	,	,	PUNCT
ejpam-841	667	21	we	we	PRON
ejpam-841	667	22	have	have	VERB
ejpam-841	667	23	deg(gcd(p(s	deg(gcd(p(s	PROPN
ejpam-841	667	24	,	,	PUNCT
ejpam-841	667	25	t	t	PROPN
ejpam-841	667	26	,	,	PUNCT
ejpam-841	667	27	a),q(s	a),q(s	PROPN
ejpam-841	667	28	,	,	PUNCT
ejpam-841	667	29	t	t	PROPN
ejpam-841	667	30	,	,	PUNCT
ejpam-841	667	31	a	a	PRON
ejpam-841	667	32	)	)	PUNCT
ejpam-841	667	33	,	,	PUNCT
ejpam-841	667	34	r(s	r(s	PROPN
ejpam-841	667	35	,	,	PUNCT
ejpam-841	667	36	t	t	PROPN
ejpam-841	667	37	,	,	PUNCT
ejpam-841	667	38	a	a	PRON
ejpam-841	667	39	)	)	PUNCT
ejpam-841	667	40	)	)	PUNCT
ejpam-841	667	41	)	)	PUNCT
ejpam-841	667	42	≥	≥	NOUN
ejpam-841	668	1	1	1	X
ejpam-841	668	2	.	.	PUNCT
ejpam-841	668	3	j.	j.	PROPN
ejpam-841	668	4	hoffman	hoffman	PROPN
ejpam-841	668	5	,	,	PUNCT
ejpam-841	668	6	h.	h.	PROPN
ejpam-841	668	7	wang	wang	PROPN
ejpam-841	668	8	,	,	PUNCT
ejpam-841	668	9	x.	x.	PROPN
ejpam-841	668	10	jia	jia	PROPN
ejpam-841	668	11	,	,	PUNCT
ejpam-841	668	12	r.	r.	PROPN
ejpam-841	668	13	goldman	goldman	PROPN
ejpam-841	668	14	/	/	SYM
ejpam-841	668	15	eur	eur	PROPN
ejpam-841	668	16	.	.	PUNCT
ejpam-841	669	1	j.	j.	PROPN
ejpam-841	669	2	pure	pure	PROPN
ejpam-841	669	3	appl	appl	PROPN
ejpam-841	669	4	.	.	PROPN
ejpam-841	669	5	math	math	PROPN
ejpam-841	669	6	,	,	PUNCT
ejpam-841	669	7	3	3	NUM
ejpam-841	669	8	(	(	PUNCT
ejpam-841	669	9	2010	2010	NUM
ejpam-841	669	10	)	)	PUNCT
ejpam-841	669	11	,	,	PUNCT
ejpam-841	669	12	602	602	NUM
ejpam-841	669	13	-	-	SYM
ejpam-841	669	14	632	632	NUM
ejpam-841	669	15	625	625	NUM
ejpam-841	669	16	therefore	therefore	ADV
ejpam-841	669	17	by	by	ADP
ejpam-841	669	18	proposition	proposition	NOUN
ejpam-841	669	19	1	1	NUM
ejpam-841	669	20	the	the	DET
ejpam-841	669	21	point	point	NOUN
ejpam-841	669	22	a	a	PRON
ejpam-841	669	23	is	be	AUX
ejpam-841	669	24	on	on	ADP
ejpam-841	669	25	the	the	DET
ejpam-841	669	26	rational	rational	ADJ
ejpam-841	669	27	space	space	NOUN
ejpam-841	669	28	curve	curve	NOUN
ejpam-841	669	29	c	c	PROPN
ejpam-841	669	30	.	.	PUNCT
ejpam-841	670	1	hence	hence	ADV
ejpam-841	670	2	the	the	DET
ejpam-841	670	3	surface	surface	NOUN
ejpam-841	670	4	g	g	PROPN
ejpam-841	670	5	=	=	SYM
ejpam-841	670	6	0	0	NUM
ejpam-841	670	7	is	be	AUX
ejpam-841	670	8	contained	contain	VERB
ejpam-841	670	9	in	in	ADP
ejpam-841	670	10	the	the	DET
ejpam-841	670	11	space	space	NOUN
ejpam-841	670	12	curve	curve	NOUN
ejpam-841	670	13	c	c	PROPN
ejpam-841	670	14	.	.	PUNCT
ejpam-841	671	1	this	this	PRON
ejpam-841	671	2	is	be	AUX
ejpam-841	671	3	impossible	impossible	ADJ
ejpam-841	671	4	.	.	PUNCT
ejpam-841	672	1	therefore	therefore	ADV
ejpam-841	672	2	,	,	PUNCT
ejpam-841	672	3	gcd(res(p	gcd(res(p	PROPN
ejpam-841	672	4	,	,	PUNCT
ejpam-841	672	5	q),res(p	q),res(p	PROPN
ejpam-841	672	6	,	,	PUNCT
ejpam-841	672	7	r	r	NOUN
ejpam-841	672	8	)	)	PUNCT
ejpam-841	672	9	)	)	PUNCT
ejpam-841	673	1	=	=	PUNCT
ejpam-841	673	2	1	1	X
ejpam-841	673	3	.	.	PUNCT
ejpam-841	673	4	a	a	DET
ejpam-841	673	5	similar	similar	ADJ
ejpam-841	673	6	argument	argument	NOUN
ejpam-841	673	7	applies	apply	VERB
ejpam-841	673	8	to	to	PART
ejpam-841	673	9	show	show	VERB
ejpam-841	673	10	that	that	SCONJ
ejpam-841	673	11	gcd(res(p	gcd(res(p	NOUN
ejpam-841	673	12	,	,	PUNCT
ejpam-841	673	13	q	q	NOUN
ejpam-841	673	14	)	)	PUNCT
ejpam-841	673	15	,	,	PUNCT
ejpam-841	673	16	res(q	res(q	PROPN
ejpam-841	673	17	,	,	PUNCT
ejpam-841	673	18	r	r	NOUN
ejpam-841	673	19	)	)	PUNCT
ejpam-841	673	20	)	)	PUNCT
ejpam-841	674	1	=	=	SYM
ejpam-841	675	1	1	1	X
ejpam-841	675	2	.	.	PUNCT
ejpam-841	675	3	now	now	ADV
ejpam-841	675	4	suppose	suppose	VERB
ejpam-841	675	5	that	that	SCONJ
ejpam-841	675	6	gcd(res(p	gcd(res(p	PROPN
ejpam-841	675	7	,	,	PUNCT
ejpam-841	675	8	r),res(q	r),res(q	NOUN
ejpam-841	675	9	,	,	PUNCT
ejpam-841	675	10	r	r	NOUN
ejpam-841	675	11	)	)	PUNCT
ejpam-841	675	12	)	)	PUNCT
ejpam-841	675	13	=	=	PUNCT
ejpam-841	676	1	g	g	PROPN
ejpam-841	676	2	6=	6=	PROPN
ejpam-841	676	3	1	1	NUM
ejpam-841	676	4	.	.	PUNCT
ejpam-841	677	1	if	if	SCONJ
ejpam-841	677	2	deg(g	deg(g	PROPN
ejpam-841	677	3	)	)	PUNCT
ejpam-841	678	1	=	=	SYM
ejpam-841	678	2	3	3	NUM
ejpam-841	678	3	,	,	PUNCT
ejpam-841	678	4	then	then	ADV
ejpam-841	678	5	res(p	res(p	PROPN
ejpam-841	678	6	,	,	PUNCT
ejpam-841	678	7	r	r	NOUN
ejpam-841	678	8	)	)	PUNCT
ejpam-841	678	9	=	=	NOUN
ejpam-841	679	1	kres(q	kres(q	NOUN
ejpam-841	679	2	,	,	PUNCT
ejpam-841	679	3	r	r	NOUN
ejpam-841	679	4	)	)	PUNCT
ejpam-841	679	5	for	for	ADP
ejpam-841	679	6	some	some	DET
ejpam-841	679	7	nonzero	nonzero	NOUN
ejpam-841	679	8	constant	constant	ADJ
ejpam-841	679	9	k.	k.	PROPN
ejpam-841	679	10	hence	hence	ADV
ejpam-841	679	11	since	since	SCONJ
ejpam-841	679	12	the	the	DET
ejpam-841	679	13	axis	axis	NOUN
ejpam-841	679	14	of	of	ADP
ejpam-841	679	15	p	p	NOUN
ejpam-841	679	16	is	be	AUX
ejpam-841	679	17	contained	contain	VERB
ejpam-841	679	18	in	in	ADP
ejpam-841	679	19	the	the	DET
ejpam-841	679	20	cubic	cubic	ADJ
ejpam-841	679	21	surface	surface	PROPN
ejpam-841	679	22	res(p	res(p	PROPN
ejpam-841	679	23	,	,	PUNCT
ejpam-841	679	24	r	r	NOUN
ejpam-841	679	25	)	)	PUNCT
ejpam-841	679	26	=	=	SYM
ejpam-841	679	27	0	0	NUM
ejpam-841	679	28	,	,	PUNCT
ejpam-841	679	29	the	the	DET
ejpam-841	679	30	axis	axis	NOUN
ejpam-841	679	31	of	of	ADP
ejpam-841	679	32	p	p	NOUN
ejpam-841	679	33	is	be	AUX
ejpam-841	679	34	also	also	ADV
ejpam-841	679	35	contained	contain	VERB
ejpam-841	679	36	in	in	ADP
ejpam-841	679	37	the	the	DET
ejpam-841	679	38	surface	surface	NOUN
ejpam-841	679	39	res(q	res(q	NOUN
ejpam-841	679	40	,	,	PUNCT
ejpam-841	679	41	r	r	NOUN
ejpam-841	679	42	)	)	PUNCT
ejpam-841	679	43	=	=	SYM
ejpam-841	680	1	0	0	X
ejpam-841	680	2	.	.	PUNCT
ejpam-841	680	3	thus	thus	ADV
ejpam-841	680	4	for	for	ADP
ejpam-841	680	5	a	a	DET
ejpam-841	680	6	point	point	NOUN
ejpam-841	680	7	a	a	PRON
ejpam-841	680	8	on	on	ADP
ejpam-841	680	9	the	the	DET
ejpam-841	680	10	axis	axis	NOUN
ejpam-841	680	11	of	of	ADP
ejpam-841	680	12	p	p	X
ejpam-841	680	13	,	,	PUNCT
ejpam-841	680	14	we	we	PRON
ejpam-841	680	15	have	have	VERB
ejpam-841	680	16	p(s	p(s	PROPN
ejpam-841	680	17	,	,	PUNCT
ejpam-841	680	18	t	t	PROPN
ejpam-841	680	19	,	,	PUNCT
ejpam-841	680	20	a	a	PRON
ejpam-841	680	21	)	)	PUNCT
ejpam-841	680	22	=	=	SYM
ejpam-841	680	23	0	0	NUM
ejpam-841	680	24	and	and	CCONJ
ejpam-841	680	25	deg(gcd(q(s	deg(gcd(q(s	PROPN
ejpam-841	680	26	,	,	PUNCT
ejpam-841	680	27	t	t	PROPN
ejpam-841	680	28	,	,	PUNCT
ejpam-841	680	29	a	a	PRON
ejpam-841	680	30	)	)	PUNCT
ejpam-841	680	31	,	,	PUNCT
ejpam-841	680	32	r(s	r(s	PROPN
ejpam-841	680	33	,	,	PUNCT
ejpam-841	680	34	t	t	PROPN
ejpam-841	680	35	,	,	PUNCT
ejpam-841	680	36	a	a	PRON
ejpam-841	680	37	)	)	PUNCT
ejpam-841	680	38	)	)	PUNCT
ejpam-841	680	39	)	)	PUNCT
ejpam-841	680	40	≥	≥	NOUN
ejpam-841	681	1	1	1	NUM
ejpam-841	681	2	.	.	PUNCT
ejpam-841	681	3	hence	hence	ADV
ejpam-841	681	4	deg(gcd(p(s	deg(gcd(p(s	PROPN
ejpam-841	681	5	,	,	PUNCT
ejpam-841	681	6	t	t	PROPN
ejpam-841	681	7	,	,	PUNCT
ejpam-841	681	8	a),q(s	a),q(s	PROPN
ejpam-841	681	9	,	,	PUNCT
ejpam-841	681	10	t	t	PROPN
ejpam-841	681	11	,	,	PUNCT
ejpam-841	681	12	a	a	PRON
ejpam-841	681	13	)	)	PUNCT
ejpam-841	681	14	,	,	PUNCT
ejpam-841	681	15	r(s	r(s	PROPN
ejpam-841	681	16	,	,	PUNCT
ejpam-841	681	17	t	t	PROPN
ejpam-841	681	18	,	,	PUNCT
ejpam-841	681	19	a	a	PRON
ejpam-841	681	20	)	)	PUNCT
ejpam-841	681	21	)	)	PUNCT
ejpam-841	681	22	)	)	PUNCT
ejpam-841	681	23	≥	≥	NOUN
ejpam-841	681	24	1	1	NUM
ejpam-841	681	25	,	,	PUNCT
ejpam-841	681	26	so	so	ADV
ejpam-841	681	27	the	the	DET
ejpam-841	681	28	point	point	NOUN
ejpam-841	681	29	a	a	PRON
ejpam-841	681	30	is	be	AUX
ejpam-841	681	31	on	on	ADP
ejpam-841	681	32	the	the	DET
ejpam-841	681	33	curve	curve	NOUN
ejpam-841	681	34	c	c	PROPN
ejpam-841	681	35	.	.	PUNCT
ejpam-841	682	1	therefore	therefore	ADV
ejpam-841	682	2	,	,	PUNCT
ejpam-841	682	3	the	the	DET
ejpam-841	682	4	axis	axis	ADJ
ejpam-841	682	5	line	line	NOUN
ejpam-841	682	6	of	of	ADP
ejpam-841	682	7	p	p	NOUN
ejpam-841	682	8	is	be	AUX
ejpam-841	682	9	contained	contain	VERB
ejpam-841	682	10	in	in	ADP
ejpam-841	682	11	the	the	DET
ejpam-841	682	12	rational	rational	ADJ
ejpam-841	682	13	space	space	NOUN
ejpam-841	682	14	curve	curve	NOUN
ejpam-841	682	15	c	c	PROPN
ejpam-841	682	16	.	.	PUNCT
ejpam-841	683	1	this	this	PRON
ejpam-841	683	2	is	be	AUX
ejpam-841	683	3	impossible	impossible	ADJ
ejpam-841	683	4	.	.	PUNCT
ejpam-841	684	1	if	if	SCONJ
ejpam-841	684	2	deg(g	deg(g	PROPN
ejpam-841	684	3	)	)	PUNCT
ejpam-841	684	4	=	=	SYM
ejpam-841	684	5	2	2	NUM
ejpam-841	684	6	,	,	PUNCT
ejpam-841	684	7	then	then	ADV
ejpam-841	684	8	since	since	SCONJ
ejpam-841	684	9	a	a	DET
ejpam-841	684	10	rational	rational	ADJ
ejpam-841	684	11	non	non	ADJ
ejpam-841	684	12	-	-	ADJ
ejpam-841	684	13	singular	singular	ADJ
ejpam-841	684	14	quartic	quartic	ADJ
ejpam-841	684	15	curve	curve	NOUN
ejpam-841	684	16	can	can	AUX
ejpam-841	684	17	be	be	AUX
ejpam-841	684	18	contained	contain	VERB
ejpam-841	684	19	in	in	ADP
ejpam-841	684	20	only	only	ADV
ejpam-841	684	21	one	one	NUM
ejpam-841	684	22	quadric	quadric	ADJ
ejpam-841	684	23	surface	surface	NOUN
ejpam-841	684	24	,	,	PUNCT
ejpam-841	684	25	we	we	PRON
ejpam-841	684	26	would	would	AUX
ejpam-841	684	27	have	have	VERB
ejpam-841	684	28	g	g	PROPN
ejpam-841	684	29	=	=	SYM
ejpam-841	684	30	res(p	res(p	PROPN
ejpam-841	684	31	,	,	PUNCT
ejpam-841	684	32	q	q	NOUN
ejpam-841	684	33	)	)	PUNCT
ejpam-841	684	34	,	,	PUNCT
ejpam-841	684	35	again	again	ADV
ejpam-841	684	36	contradicting	contradict	VERB
ejpam-841	684	37	the	the	DET
ejpam-841	684	38	fact	fact	NOUN
ejpam-841	684	39	that	that	SCONJ
ejpam-841	684	40	gcd(res(p	gcd(res(p	PROPN
ejpam-841	684	41	,	,	PUNCT
ejpam-841	684	42	q),res(p	q),res(p	PROPN
ejpam-841	684	43	,	,	PUNCT
ejpam-841	684	44	r	r	NOUN
ejpam-841	684	45	)	)	PUNCT
ejpam-841	684	46	)	)	PUNCT
ejpam-841	684	47	=	=	PUNCT
ejpam-841	685	1	1	1	X
ejpam-841	685	2	.	.	X
ejpam-841	685	3	if	if	SCONJ
ejpam-841	685	4	deg(g	deg(g	PROPN
ejpam-841	685	5	)	)	PUNCT
ejpam-841	685	6	=	=	SYM
ejpam-841	685	7	1	1	NUM
ejpam-841	685	8	,	,	PUNCT
ejpam-841	685	9	then	then	ADV
ejpam-841	685	10	since	since	SCONJ
ejpam-841	685	11	c	c	PROPN
ejpam-841	685	12	is	be	AUX
ejpam-841	685	13	a	a	DET
ejpam-841	685	14	space	space	NOUN
ejpam-841	685	15	curve	curve	NOUN
ejpam-841	685	16	contained	contain	VERB
ejpam-841	685	17	in	in	ADP
ejpam-841	685	18	the	the	DET
ejpam-841	685	19	cubic	cubic	ADJ
ejpam-841	685	20	surface	surface	PROPN
ejpam-841	685	21	res(p	res(p	PROPN
ejpam-841	685	22	,	,	PUNCT
ejpam-841	685	23	r	r	NOUN
ejpam-841	685	24	)	)	PUNCT
ejpam-841	685	25	=	=	SYM
ejpam-841	685	26	0	0	NUM
ejpam-841	685	27	,	,	PUNCT
ejpam-841	685	28	we	we	PRON
ejpam-841	685	29	would	would	AUX
ejpam-841	685	30	have	have	VERB
ejpam-841	685	31	c	c	PROPN
ejpam-841	685	32	is	be	AUX
ejpam-841	685	33	contained	contain	VERB
ejpam-841	685	34	in	in	ADP
ejpam-841	685	35	the	the	DET
ejpam-841	685	36	quadric	quadric	ADJ
ejpam-841	685	37	surface	surface	NOUN
ejpam-841	685	38	res(p	res(p	PROPN
ejpam-841	685	39	,	,	PUNCT
ejpam-841	685	40	r	r	NOUN
ejpam-841	685	41	)	)	PUNCT
ejpam-841	685	42	g	g	NOUN
ejpam-841	685	43	=	=	SYM
ejpam-841	685	44	0	0	X
ejpam-841	685	45	.	.	PUNCT
ejpam-841	686	1	again	again	ADV
ejpam-841	686	2	since	since	SCONJ
ejpam-841	686	3	the	the	DET
ejpam-841	686	4	nonsingular	nonsingular	ADJ
ejpam-841	686	5	quartic	quartic	ADJ
ejpam-841	686	6	space	space	NOUN
ejpam-841	686	7	curve	curve	NOUN
ejpam-841	686	8	c	c	PROPN
ejpam-841	686	9	can	can	AUX
ejpam-841	686	10	be	be	AUX
ejpam-841	686	11	contained	contain	VERB
ejpam-841	686	12	in	in	ADP
ejpam-841	686	13	only	only	ADV
ejpam-841	686	14	one	one	NUM
ejpam-841	686	15	quadric	quadric	ADJ
ejpam-841	686	16	surface	surface	NOUN
ejpam-841	686	17	,	,	PUNCT
ejpam-841	686	18	we	we	PRON
ejpam-841	686	19	would	would	AUX
ejpam-841	686	20	have	have	VERB
ejpam-841	686	21	res(p	res(p	PROPN
ejpam-841	686	22	,	,	PUNCT
ejpam-841	686	23	r	r	NOUN
ejpam-841	686	24	)	)	PUNCT
ejpam-841	686	25	g	g	NOUN
ejpam-841	686	26	=	=	SYM
ejpam-841	686	27	res(p	res(p	PROPN
ejpam-841	686	28	,	,	PUNCT
ejpam-841	686	29	q	q	NOUN
ejpam-841	686	30	)	)	PUNCT
ejpam-841	686	31	,	,	PUNCT
ejpam-841	686	32	again	again	ADV
ejpam-841	686	33	contradicting	contradict	VERB
ejpam-841	686	34	the	the	DET
ejpam-841	686	35	fact	fact	NOUN
ejpam-841	686	36	that	that	SCONJ
ejpam-841	686	37	gcd(res(p	gcd(res(p	PROPN
ejpam-841	686	38	,	,	PUNCT
ejpam-841	686	39	q),res(p	q),res(p	PROPN
ejpam-841	686	40	,	,	PUNCT
ejpam-841	686	41	r	r	NOUN
ejpam-841	686	42	)	)	PUNCT
ejpam-841	686	43	)	)	PUNCT
ejpam-841	686	44	=	=	SYM
ejpam-841	687	1	1	1	X
ejpam-841	687	2	.	.	PUNCT
ejpam-841	688	1	therefore	therefore	ADV
ejpam-841	688	2	,	,	PUNCT
ejpam-841	688	3	gcd(res(p	gcd(res(p	PROPN
ejpam-841	688	4	,	,	PUNCT
ejpam-841	688	5	r),res(q	r),res(q	NOUN
ejpam-841	688	6	,	,	PUNCT
ejpam-841	688	7	r	r	NOUN
ejpam-841	688	8	)	)	PUNCT
ejpam-841	688	9	)	)	PUNCT
ejpam-841	689	1	=	=	SYM
ejpam-841	689	2	1	1	X
ejpam-841	689	3	.	.	PUNCT
ejpam-841	689	4	now	now	ADV
ejpam-841	689	5	let	let	VERB
ejpam-841	689	6	n(p	n(p	ADJ
ejpam-841	689	7	,	,	PUNCT
ejpam-841	689	8	q	q	NOUN
ejpam-841	689	9	,	,	PUNCT
ejpam-841	689	10	r	r	NOUN
ejpam-841	689	11	)	)	PUNCT
ejpam-841	689	12	be	be	AUX
ejpam-841	689	13	the	the	DET
ejpam-841	689	14	3×	3×	NUM
ejpam-841	689	15	3	3	NUM
ejpam-841	689	16	coefficient	coefficient	NOUN
ejpam-841	689	17	matrix	matrix	NOUN
ejpam-841	689	18	of	of	ADP
ejpam-841	689	19	the	the	DET
ejpam-841	689	20	moving	move	VERB
ejpam-841	689	21	planes	plane	NOUN
ejpam-841	689	22	sp	sp	NOUN
ejpam-841	689	23	,	,	PUNCT
ejpam-841	689	24	tq	tq	INTJ
ejpam-841	689	25	,	,	PUNCT
ejpam-841	689	26	r.	r.	NOUN
ejpam-841	689	27	since	since	SCONJ
ejpam-841	689	28	sp	sp	ADP
ejpam-841	689	29	=	=	PUNCT
ejpam-841	689	30	p1s2	p1s2	CCONJ
ejpam-841	689	31	+	+	PUNCT
ejpam-841	689	32	p0st	p0st	NOUN
ejpam-841	689	33	,	,	PUNCT
ejpam-841	689	34	tq	tq	ADP
ejpam-841	689	35	=	=	SYM
ejpam-841	689	36	q1st	q1st	X
ejpam-841	690	1	+	+	CCONJ
ejpam-841	690	2	q0	q0	ADJ
ejpam-841	690	3	t2	t2	NOUN
ejpam-841	690	4	,	,	PUNCT
ejpam-841	690	5	r	r	NOUN
ejpam-841	690	6	=	=	PUNCT
ejpam-841	690	7	r2s2	r2s2	PROPN
ejpam-841	690	8	+	+	NOUN
ejpam-841	690	9	r1st	r1st	PUNCT
ejpam-841	691	1	+	+	CCONJ
ejpam-841	691	2	r0t2	r0t2	AUX
ejpam-841	691	3	it	it	PRON
ejpam-841	691	4	follows	follow	VERB
ejpam-841	691	5	that	that	SCONJ
ejpam-841	691	6	n(p	n(p	PROPN
ejpam-841	691	7	,	,	PUNCT
ejpam-841	691	8	q	q	NOUN
ejpam-841	691	9	,	,	PUNCT
ejpam-841	691	10	r	r	NOUN
ejpam-841	691	11	)	)	PUNCT
ejpam-841	691	12	=	=	NOUN
ejpam-841	691	13			X
ejpam-841	691	14			ADJ
ejpam-841	691	15			NUM
ejpam-841	691	16	p1	p1	NOUN
ejpam-841	691	17	0	0	NUM
ejpam-841	691	18	r2	r2	PROPN
ejpam-841	691	19	p0	p0	PROPN
ejpam-841	691	20	q1	q1	PROPN
ejpam-841	691	21	r1	r1	PROPN
ejpam-841	691	22	0	0	NUM
ejpam-841	692	1	q0	q0	PROPN
ejpam-841	692	2	r0	r0	NOUN
ejpam-841	692	3			PROPN
ejpam-841	692	4			PROPN
ejpam-841	692	5			PROPN
ejpam-841	692	6	,	,	PUNCT
ejpam-841	692	7	and	and	CCONJ
ejpam-841	692	8	det(n	det(n	PROPN
ejpam-841	692	9	)	)	PUNCT
ejpam-841	692	10	=	=	SYM
ejpam-841	692	11	r2p0q0	r2p0q0	ADJ
ejpam-841	692	12	−	−	DET
ejpam-841	692	13	r1p1q0	r1p1q0	NOUN
ejpam-841	692	14	+	+	CCONJ
ejpam-841	692	15	r0p1q1	r0p1q1	NOUN
ejpam-841	692	16	.	.	PUNCT
ejpam-841	693	1	(	(	PUNCT
ejpam-841	693	2	12	12	NUM
ejpam-841	693	3	)	)	PUNCT
ejpam-841	693	4	lemma	lemma	PROPN
ejpam-841	693	5	7	7	NUM
ejpam-841	693	6	.	.	PUNCT
ejpam-841	693	7	det(n(p	det(n(p	PROPN
ejpam-841	693	8	,	,	PUNCT
ejpam-841	693	9	q	q	NOUN
ejpam-841	693	10	,	,	PUNCT
ejpam-841	693	11	r	r	NOUN
ejpam-841	693	12	)	)	PUNCT
ejpam-841	693	13	)	)	PUNCT
ejpam-841	693	14	is	be	AUX
ejpam-841	693	15	not	not	PART
ejpam-841	693	16	identically	identically	ADV
ejpam-841	693	17	zero	zero	NUM
ejpam-841	693	18	.	.	PUNCT
ejpam-841	694	1	proof	proof	NOUN
ejpam-841	694	2	.	.	PUNCT
ejpam-841	695	1	if	if	SCONJ
ejpam-841	695	2	det(n(p	det(n(p	PROPN
ejpam-841	695	3	,	,	PUNCT
ejpam-841	695	4	q	q	NOUN
ejpam-841	695	5	,	,	PUNCT
ejpam-841	695	6	r	r	NOUN
ejpam-841	695	7	)	)	PUNCT
ejpam-841	695	8	)	)	PUNCT
ejpam-841	696	1	≡	≡	PROPN
ejpam-841	696	2	0	0	NUM
ejpam-841	696	3	,	,	PUNCT
ejpam-841	696	4	then	then	ADV
ejpam-841	696	5	the	the	DET
ejpam-841	696	6	rows	row	NOUN
ejpam-841	696	7	of	of	ADP
ejpam-841	696	8	the	the	DET
ejpam-841	696	9	matrix	matrix	NOUN
ejpam-841	696	10	n	n	PRON
ejpam-841	696	11	are	be	AUX
ejpam-841	696	12	linearly	linearly	ADV
ejpam-841	696	13	dependent	dependent	ADJ
ejpam-841	696	14	over	over	ADP
ejpam-841	696	15	the	the	DET
ejpam-841	696	16	ring	ring	NOUN
ejpam-841	696	17	k[x	k[x	NOUN
ejpam-841	696	18	,	,	PUNCT
ejpam-841	696	19	y	y	PROPN
ejpam-841	696	20	,	,	PUNCT
ejpam-841	696	21	z	z	PROPN
ejpam-841	696	22	,	,	PUNCT
ejpam-841	696	23	w	w	NOUN
ejpam-841	696	24	]	]	X
ejpam-841	696	25	.	.	PUNCT
ejpam-841	697	1	without	without	ADP
ejpam-841	697	2	loss	loss	NOUN
ejpam-841	697	3	of	of	ADP
ejpam-841	697	4	generality	generality	NOUN
ejpam-841	697	5	,	,	PUNCT
ejpam-841	697	6	by	by	ADP
ejpam-841	697	7	lemma	lemma	PROPN
ejpam-841	697	8	3	3	NUM
ejpam-841	697	9	we	we	PRON
ejpam-841	697	10	can	can	AUX
ejpam-841	697	11	assume	assume	VERB
ejpam-841	697	12	that	that	SCONJ
ejpam-841	697	13	p1	p1	PROPN
ejpam-841	697	14	=	=	SYM
ejpam-841	697	15	y	y	PROPN
ejpam-841	697	16	,	,	PUNCT
ejpam-841	697	17	p0	p0	NOUN
ejpam-841	697	18	=	=	SYM
ejpam-841	697	19	−x	−x	NOUN
ejpam-841	697	20	,	,	PUNCT
ejpam-841	697	21	q1	q1	PROPN
ejpam-841	697	22	=	=	SYM
ejpam-841	697	23	w	w	PROPN
ejpam-841	697	24	,	,	PUNCT
ejpam-841	697	25	q0	q0	NOUN
ejpam-841	697	26	=	=	PUNCT
ejpam-841	697	27	−z	−z	NOUN
ejpam-841	697	28	.	.	PUNCT
ejpam-841	698	1	a	a	DET
ejpam-841	698	2	dependence	dependence	NOUN
ejpam-841	698	3	relation	relation	NOUN
ejpam-841	698	4	among	among	ADP
ejpam-841	698	5	the	the	DET
ejpam-841	698	6	three	three	NUM
ejpam-841	698	7	rows	row	NOUN
ejpam-841	698	8	of	of	ADP
ejpam-841	698	9	n(p	n(p	PROPN
ejpam-841	698	10	,	,	PUNCT
ejpam-841	698	11	q	q	NOUN
ejpam-841	698	12	,	,	PUNCT
ejpam-841	698	13	r	r	NOUN
ejpam-841	698	14	)	)	PUNCT
ejpam-841	698	15	would	would	AUX
ejpam-841	698	16	generate	generate	VERB
ejpam-841	698	17	three	three	NUM
ejpam-841	698	18	linear	linear	ADJ
ejpam-841	698	19	equations	equation	NOUN
ejpam-841	698	20	a	a	DET
ejpam-841	698	21	y	y	NOUN
ejpam-841	698	22	−	−	NOUN
ejpam-841	698	23	bx	bx	NOUN
ejpam-841	699	1	=	=	NOUN
ejpam-841	699	2	0	0	PUNCT
ejpam-841	699	3	bw	bw	NOUN
ejpam-841	699	4	−	−	PROPN
ejpam-841	699	5	cz	cz	NOUN
ejpam-841	699	6	=	=	NOUN
ejpam-841	699	7	0	0	PUNCT
ejpam-841	700	1	ar2	ar2	PROPN
ejpam-841	700	2	+	+	PROPN
ejpam-841	700	3	br1	br1	PROPN
ejpam-841	700	4	+	+	PROPN
ejpam-841	700	5	cr0	cr0	NOUN
ejpam-841	700	6	=	=	NOUN
ejpam-841	700	7	0	0	NUM
ejpam-841	700	8	with	with	ADP
ejpam-841	700	9	homogeneous	homogeneous	ADJ
ejpam-841	700	10	polynomials	polynomial	NOUN
ejpam-841	700	11	a(x	a(x	NOUN
ejpam-841	700	12	,	,	PUNCT
ejpam-841	700	13	y	y	PROPN
ejpam-841	700	14	,	,	PUNCT
ejpam-841	700	15	z	z	PROPN
ejpam-841	700	16	,	,	PUNCT
ejpam-841	700	17	w	w	NOUN
ejpam-841	700	18	)	)	PUNCT
ejpam-841	700	19	,	,	PUNCT
ejpam-841	700	20	b(x	b(x	NOUN
ejpam-841	700	21	,	,	PUNCT
ejpam-841	700	22	y	y	PROPN
ejpam-841	700	23	,	,	PUNCT
ejpam-841	700	24	z	z	PROPN
ejpam-841	700	25	,	,	PUNCT
ejpam-841	700	26	w	w	NOUN
ejpam-841	700	27	)	)	PUNCT
ejpam-841	700	28	,	,	PUNCT
ejpam-841	700	29	c(x	c(x	NOUN
ejpam-841	700	30	,	,	PUNCT
ejpam-841	700	31	y	y	PROPN
ejpam-841	700	32	,	,	PUNCT
ejpam-841	700	33	z	z	PROPN
ejpam-841	700	34	,	,	PUNCT
ejpam-841	700	35	w	w	NOUN
ejpam-841	700	36	)	)	PUNCT
ejpam-841	700	37	of	of	ADP
ejpam-841	700	38	the	the	DET
ejpam-841	700	39	same	same	ADJ
ejpam-841	700	40	degree	degree	NOUN
ejpam-841	700	41	in	in	ADP
ejpam-841	700	42	x	x	SYM
ejpam-841	700	43	,	,	PUNCT
ejpam-841	700	44	y	y	PROPN
ejpam-841	700	45	,	,	PUNCT
ejpam-841	700	46	z	z	PROPN
ejpam-841	700	47	,	,	PUNCT
ejpam-841	700	48	w	w	VERB
ejpam-841	700	49	at	at	ADV
ejpam-841	700	50	least	least	ADJ
ejpam-841	700	51	one	one	NUM
ejpam-841	700	52	of	of	ADP
ejpam-841	700	53	which	which	PRON
ejpam-841	700	54	is	be	AUX
ejpam-841	700	55	not	not	PART
ejpam-841	700	56	zero	zero	NUM
ejpam-841	700	57	.	.	PUNCT
ejpam-841	701	1	from	from	ADP
ejpam-841	701	2	the	the	DET
ejpam-841	701	3	first	first	ADJ
ejpam-841	701	4	equation	equation	NOUN
ejpam-841	701	5	,	,	PUNCT
ejpam-841	701	6	we	we	PRON
ejpam-841	701	7	find	find	VERB
ejpam-841	701	8	that	that	SCONJ
ejpam-841	701	9	a	a	DET
ejpam-841	701	10	=	=	X
ejpam-841	701	11	αx	αx	PROPN
ejpam-841	701	12	,	,	PUNCT
ejpam-841	701	13	b	b	X
ejpam-841	701	14	=	=	PROPN
ejpam-841	701	15	j.	j.	PROPN
ejpam-841	701	16	hoffman	hoffman	PROPN
ejpam-841	701	17	,	,	PUNCT
ejpam-841	701	18	h.	h.	PROPN
ejpam-841	701	19	wang	wang	PROPN
ejpam-841	701	20	,	,	PUNCT
ejpam-841	701	21	x.	x.	PROPN
ejpam-841	701	22	jia	jia	PROPN
ejpam-841	701	23	,	,	PUNCT
ejpam-841	701	24	r.	r.	PROPN
ejpam-841	701	25	goldman	goldman	PROPN
ejpam-841	701	26	/	/	SYM
ejpam-841	701	27	eur	eur	PROPN
ejpam-841	701	28	.	.	PUNCT
ejpam-841	702	1	j.	j.	PROPN
ejpam-841	702	2	pure	pure	PROPN
ejpam-841	702	3	appl	appl	PROPN
ejpam-841	702	4	.	.	PROPN
ejpam-841	702	5	math	math	PROPN
ejpam-841	702	6	,	,	PUNCT
ejpam-841	702	7	3	3	NUM
ejpam-841	702	8	(	(	PUNCT
ejpam-841	702	9	2010	2010	NUM
ejpam-841	702	10	)	)	PUNCT
ejpam-841	702	11	,	,	PUNCT
ejpam-841	702	12	602	602	NUM
ejpam-841	702	13	-	-	SYM
ejpam-841	702	14	632	632	NUM
ejpam-841	702	15	626	626	NUM
ejpam-841	702	16	αy	αy	NOUN
ejpam-841	702	17	.	.	PUNCT
ejpam-841	703	1	the	the	DET
ejpam-841	703	2	second	second	ADJ
ejpam-841	703	3	equation	equation	NOUN
ejpam-841	703	4	then	then	ADV
ejpam-841	703	5	shows	show	VERB
ejpam-841	703	6	that	that	SCONJ
ejpam-841	703	7	α	α	PRON
ejpam-841	703	8	=	=	SYM
ejpam-841	703	9	βz	βz	NOUN
ejpam-841	703	10	,	,	PUNCT
ejpam-841	703	11	c	c	NOUN
ejpam-841	703	12	=	=	SYM
ejpam-841	703	13	β	β	X
ejpam-841	703	14	yw	yw	PROPN
ejpam-841	703	15	.	.	PUNCT
ejpam-841	704	1	substituting	substitute	VERB
ejpam-841	704	2	these	these	DET
ejpam-841	704	3	results	result	NOUN
ejpam-841	704	4	into	into	ADP
ejpam-841	704	5	the	the	DET
ejpam-841	704	6	third	third	ADJ
ejpam-841	704	7	equation	equation	NOUN
ejpam-841	704	8	gives	give	VERB
ejpam-841	704	9	xzr2	xzr2	PROPN
ejpam-841	704	10	+	+	CCONJ
ejpam-841	704	11	yzr1	yzr1	VERB
ejpam-841	705	1	+	+	NUM
ejpam-841	705	2	ywr0	ywr0	NOUN
ejpam-841	705	3	=	=	SYM
ejpam-841	705	4	0	0	X
ejpam-841	705	5	.	.	PUNCT
ejpam-841	706	1	in	in	ADP
ejpam-841	706	2	other	other	ADJ
ejpam-841	706	3	words	word	NOUN
ejpam-841	706	4	,	,	PUNCT
ejpam-841	706	5	(	(	PUNCT
ejpam-841	706	6	r2	r2	PROPN
ejpam-841	706	7	,	,	PUNCT
ejpam-841	706	8	r1	r1	NOUN
ejpam-841	706	9	,	,	PUNCT
ejpam-841	706	10	r0	r0	NOUN
ejpam-841	706	11	)	)	PUNCT
ejpam-841	706	12	is	be	AUX
ejpam-841	706	13	a	a	DET
ejpam-841	706	14	syzygy	syzygy	NOUN
ejpam-841	706	15	of	of	ADP
ejpam-841	706	16	(	(	PUNCT
ejpam-841	706	17	xz	xz	PROPN
ejpam-841	706	18	,	,	PUNCT
ejpam-841	706	19	yz	yz	PROPN
ejpam-841	706	20	,	,	PUNCT
ejpam-841	706	21	yw	yw	PROPN
ejpam-841	706	22	)	)	PUNCT
ejpam-841	706	23	.	.	PUNCT
ejpam-841	707	1	but	but	CCONJ
ejpam-841	707	2	the	the	DET
ejpam-841	707	3	syzygy	syzygy	NOUN
ejpam-841	707	4	module	module	NOUN
ejpam-841	707	5	for	for	ADP
ejpam-841	707	6	(	(	PUNCT
ejpam-841	707	7	xz	xz	PROPN
ejpam-841	707	8	,	,	PUNCT
ejpam-841	707	9	yz	yz	PROPN
ejpam-841	707	10	,	,	PUNCT
ejpam-841	707	11	yw	yw	PROPN
ejpam-841	707	12	)	)	PUNCT
ejpam-841	707	13	is	be	AUX
ejpam-841	707	14	generated	generate	VERB
ejpam-841	707	15	by	by	ADP
ejpam-841	707	16	(	(	PUNCT
ejpam-841	707	17	y,−x	y,−x	NOUN
ejpam-841	707	18	,	,	PUNCT
ejpam-841	707	19	0	0	NUM
ejpam-841	707	20	)	)	PUNCT
ejpam-841	707	21	and	and	CCONJ
ejpam-841	707	22	(	(	PUNCT
ejpam-841	707	23	0	0	NUM
ejpam-841	707	24	,	,	PUNCT
ejpam-841	707	25	w,−z	w,−z	NOUN
ejpam-841	707	26	)	)	PUNCT
ejpam-841	707	27	.	.	PUNCT
ejpam-841	708	1	therefore	therefore	ADV
ejpam-841	708	2	,	,	PUNCT
ejpam-841	708	3	r2	r2	PROPN
ejpam-841	708	4	=	=	PUNCT
ejpam-841	708	5	my	my	PRON
ejpam-841	708	6	,	,	PUNCT
ejpam-841	708	7	r1	r1	NOUN
ejpam-841	708	8	=	=	SYM
ejpam-841	708	9	−mx	−mx	NOUN
ejpam-841	708	10	+	+	CCONJ
ejpam-841	708	11	nw	nw	NOUN
ejpam-841	708	12	,	,	PUNCT
ejpam-841	708	13	r0	r0	NOUN
ejpam-841	708	14	=	=	PUNCT
ejpam-841	708	15	−nz	−nz	PROPN
ejpam-841	708	16	where	where	SCONJ
ejpam-841	708	17	m	m	VERB
ejpam-841	708	18	,	,	PUNCT
ejpam-841	708	19	n	n	PRON
ejpam-841	708	20	are	be	AUX
ejpam-841	708	21	necessarily	necessarily	ADV
ejpam-841	708	22	constants	constant	NOUN
ejpam-841	708	23	because	because	SCONJ
ejpam-841	708	24	the	the	DET
ejpam-841	708	25	ri	ri	PROPN
ejpam-841	708	26	are	be	AUX
ejpam-841	708	27	linear	linear	ADJ
ejpam-841	708	28	in	in	ADP
ejpam-841	708	29	x	x	SYM
ejpam-841	708	30	,	,	PUNCT
ejpam-841	708	31	y	y	PROPN
ejpam-841	708	32	,	,	PUNCT
ejpam-841	708	33	z	z	PROPN
ejpam-841	708	34	,	,	PUNCT
ejpam-841	708	35	w.	w.	PROPN
ejpam-841	708	36	hence	hence	ADV
ejpam-841	708	37	,	,	PUNCT
ejpam-841	708	38	column	column	NOUN
ejpam-841	708	39	three	three	NUM
ejpam-841	708	40	of	of	ADP
ejpam-841	708	41	the	the	DET
ejpam-841	708	42	matrix	matrix	NOUN
ejpam-841	708	43	n	n	PRON
ejpam-841	708	44	is	be	AUX
ejpam-841	708	45	a	a	DET
ejpam-841	708	46	k	k	ADJ
ejpam-841	708	47	-	-	PUNCT
ejpam-841	708	48	linear	linear	ADJ
ejpam-841	708	49	combination	combination	NOUN
ejpam-841	708	50	of	of	ADP
ejpam-841	708	51	the	the	DET
ejpam-841	708	52	first	first	ADJ
ejpam-841	708	53	two	two	NUM
ejpam-841	708	54	columns	column	NOUN
ejpam-841	708	55	.	.	PUNCT
ejpam-841	709	1	thus	thus	ADV
ejpam-841	709	2	,	,	PUNCT
ejpam-841	709	3	the	the	DET
ejpam-841	709	4	moving	move	VERB
ejpam-841	709	5	plane	plane	NOUN
ejpam-841	709	6	r	r	NOUN
ejpam-841	709	7	is	be	AUX
ejpam-841	709	8	a	a	DET
ejpam-841	709	9	k	k	ADJ
ejpam-841	709	10	-	-	PUNCT
ejpam-841	709	11	linear	linear	ADJ
ejpam-841	709	12	combination	combination	NOUN
ejpam-841	709	13	of	of	ADP
ejpam-841	709	14	the	the	DET
ejpam-841	709	15	moving	move	VERB
ejpam-841	709	16	planes	plane	NOUN
ejpam-841	709	17	sp	sp	ADP
ejpam-841	709	18	and	and	CCONJ
ejpam-841	709	19	tq	tq	ADP
ejpam-841	709	20	.	.	PUNCT
ejpam-841	710	1	but	but	CCONJ
ejpam-841	710	2	this	this	PRON
ejpam-841	710	3	is	be	AUX
ejpam-841	710	4	impossible	impossible	ADJ
ejpam-841	710	5	because	because	SCONJ
ejpam-841	710	6	p	p	X
ejpam-841	710	7	,	,	PUNCT
ejpam-841	710	8	q	q	INTJ
ejpam-841	710	9	,	,	PUNCT
ejpam-841	710	10	r	r	NOUN
ejpam-841	710	11	are	be	AUX
ejpam-841	710	12	linearly	linearly	ADV
ejpam-841	710	13	independent	independent	ADJ
ejpam-841	710	14	over	over	ADP
ejpam-841	710	15	the	the	DET
ejpam-841	710	16	ring	ring	NOUN
ejpam-841	710	17	k[s	k[	NOUN
ejpam-841	710	18	,	,	PUNCT
ejpam-841	710	19	t	t	PROPN
ejpam-841	710	20	]	]	PUNCT
ejpam-841	710	21	.	.	PUNCT
ejpam-841	711	1	the	the	DET
ejpam-841	711	2	surface	surface	NOUN
ejpam-841	711	3	det(n(p	det(n(p	PROPN
ejpam-841	711	4	,	,	PUNCT
ejpam-841	711	5	q	q	NOUN
ejpam-841	711	6	,	,	PUNCT
ejpam-841	711	7	r	r	NOUN
ejpam-841	711	8	)	)	PUNCT
ejpam-841	711	9	)	)	PUNCT
ejpam-841	712	1	=	=	SYM
ejpam-841	712	2	0	0	NUM
ejpam-841	712	3	is	be	AUX
ejpam-841	712	4	a	a	DET
ejpam-841	712	5	cubic	cubic	ADJ
ejpam-841	712	6	surface	surface	NOUN
ejpam-841	712	7	containing	contain	VERB
ejpam-841	712	8	the	the	DET
ejpam-841	712	9	non	non	ADJ
ejpam-841	712	10	-	-	ADJ
ejpam-841	712	11	singular	singular	ADJ
ejpam-841	712	12	rational	rational	ADJ
ejpam-841	712	13	quartic	quartic	ADJ
ejpam-841	712	14	space	space	NOUN
ejpam-841	712	15	curve	curve	NOUN
ejpam-841	712	16	c	c	PROPN
ejpam-841	713	1	because	because	SCONJ
ejpam-841	713	2	the	the	DET
ejpam-841	713	3	moving	move	VERB
ejpam-841	713	4	planes	plane	NOUN
ejpam-841	713	5	p	p	X
ejpam-841	713	6	,	,	PUNCT
ejpam-841	713	7	q	q	INTJ
ejpam-841	713	8	,	,	PUNCT
ejpam-841	713	9	r	r	PRON
ejpam-841	713	10	follow	follow	VERB
ejpam-841	713	11	the	the	DET
ejpam-841	713	12	curve	curve	NOUN
ejpam-841	713	13	c	c	PROPN
ejpam-841	713	14	.	.	PUNCT
ejpam-841	714	1	moreover	moreover	ADV
ejpam-841	714	2	,	,	PUNCT
ejpam-841	714	3	we	we	PRON
ejpam-841	714	4	have	have	VERB
ejpam-841	714	5	the	the	DET
ejpam-841	714	6	following	follow	VERB
ejpam-841	714	7	result	result	NOUN
ejpam-841	714	8	.	.	PUNCT
ejpam-841	715	1	theorem	theorem	VERB
ejpam-841	715	2	7	7	NUM
ejpam-841	715	3	.	.	PUNCT
ejpam-841	716	1	res(p	res(p	PROPN
ejpam-841	716	2	,	,	PUNCT
ejpam-841	716	3	q	q	NOUN
ejpam-841	716	4	)	)	PUNCT
ejpam-841	716	5	=	=	SYM
ejpam-841	716	6	0	0	NUM
ejpam-841	716	7	,	,	PUNCT
ejpam-841	716	8	res(p	res(p	PROPN
ejpam-841	716	9	,	,	PUNCT
ejpam-841	716	10	r	r	NOUN
ejpam-841	716	11	)	)	PUNCT
ejpam-841	716	12	=	=	SYM
ejpam-841	716	13	0	0	NUM
ejpam-841	716	14	,	,	PUNCT
ejpam-841	716	15	res(q	res(q	NOUN
ejpam-841	716	16	,	,	PUNCT
ejpam-841	716	17	r	r	NOUN
ejpam-841	716	18	)	)	PUNCT
ejpam-841	716	19	=	=	SYM
ejpam-841	716	20	0	0	NUM
ejpam-841	716	21	,	,	PUNCT
ejpam-841	716	22	det(n(p	det(n(p	PROPN
ejpam-841	716	23	,	,	PUNCT
ejpam-841	716	24	q	q	NOUN
ejpam-841	716	25	,	,	PUNCT
ejpam-841	716	26	r	r	NOUN
ejpam-841	716	27	)	)	PUNCT
ejpam-841	716	28	)	)	PUNCT
ejpam-841	717	1	=	=	SYM
ejpam-841	717	2	0	0	NUM
ejpam-841	717	3	are	be	AUX
ejpam-841	717	4	the	the	DET
ejpam-841	717	5	implicit	implicit	ADJ
ejpam-841	717	6	equations	equation	NOUN
ejpam-841	717	7	of	of	ADP
ejpam-841	717	8	the	the	DET
ejpam-841	717	9	non	non	ADJ
ejpam-841	717	10	-	-	ADJ
ejpam-841	717	11	singular	singular	ADJ
ejpam-841	717	12	rational	rational	ADJ
ejpam-841	717	13	quartic	quartic	ADJ
ejpam-841	717	14	space	space	NOUN
ejpam-841	717	15	curve	curve	NOUN
ejpam-841	717	16	with	with	ADP
ejpam-841	717	17	µ-basis	µ-basis	NOUN
ejpam-841	717	18	p	p	PRON
ejpam-841	717	19	,	,	PUNCT
ejpam-841	717	20	q	q	NOUN
ejpam-841	717	21	,	,	PUNCT
ejpam-841	717	22	r.	r.	NOUN
ejpam-841	717	23	proof	proof	NOUN
ejpam-841	717	24	.	.	PUNCT
ejpam-841	718	1	since	since	SCONJ
ejpam-841	718	2	p	p	PRON
ejpam-841	718	3	,	,	PUNCT
ejpam-841	718	4	q	q	ADJ
ejpam-841	718	5	,	,	PUNCT
ejpam-841	718	6	r	r	NOUN
ejpam-841	718	7	are	be	AUX
ejpam-841	718	8	a	a	DET
ejpam-841	718	9	µ-basis	µ-basis	NOUN
ejpam-841	718	10	,	,	PUNCT
ejpam-841	718	11	the	the	DET
ejpam-841	718	12	space	space	NOUN
ejpam-841	718	13	curve	curve	NOUN
ejpam-841	718	14	c	c	PROPN
ejpam-841	718	15	is	be	AUX
ejpam-841	718	16	contained	contain	VERB
ejpam-841	718	17	in	in	ADP
ejpam-841	718	18	the	the	DET
ejpam-841	718	19	surface	surface	NOUN
ejpam-841	718	20	det(n(p	det(n(p	PROPN
ejpam-841	718	21	,	,	PUNCT
ejpam-841	718	22	q	q	NOUN
ejpam-841	718	23	,	,	PUNCT
ejpam-841	718	24	r	r	NOUN
ejpam-841	718	25	)	)	PUNCT
ejpam-841	718	26	)	)	PUNCT
ejpam-841	719	1	=	=	SYM
ejpam-841	719	2	0	0	X
ejpam-841	719	3	.	.	PUNCT
ejpam-841	720	1	by	by	ADP
ejpam-841	720	2	remark	remark	NOUN
ejpam-841	720	3	4	4	NUM
ejpam-841	720	4	,	,	PUNCT
ejpam-841	720	5	the	the	DET
ejpam-841	720	6	space	space	NOUN
ejpam-841	720	7	curve	curve	NOUN
ejpam-841	720	8	c	c	PROPN
ejpam-841	720	9	is	be	AUX
ejpam-841	720	10	also	also	ADV
ejpam-841	720	11	contained	contain	VERB
ejpam-841	720	12	in	in	ADP
ejpam-841	720	13	the	the	DET
ejpam-841	720	14	surfaces	surface	NOUN
ejpam-841	720	15	res(p	res(p	PROPN
ejpam-841	720	16	,	,	PUNCT
ejpam-841	720	17	q	q	NOUN
ejpam-841	720	18	)	)	PUNCT
ejpam-841	720	19	=	=	SYM
ejpam-841	720	20	0	0	NUM
ejpam-841	720	21	,	,	PUNCT
ejpam-841	720	22	res(p	res(p	PROPN
ejpam-841	720	23	,	,	PUNCT
ejpam-841	720	24	r	r	NOUN
ejpam-841	720	25	)	)	PUNCT
ejpam-841	720	26	=	=	SYM
ejpam-841	720	27	0	0	NUM
ejpam-841	720	28	,	,	PUNCT
ejpam-841	720	29	and	and	CCONJ
ejpam-841	720	30	res(q	res(q	NOUN
ejpam-841	720	31	,	,	PUNCT
ejpam-841	720	32	r	r	NOUN
ejpam-841	720	33	)	)	PUNCT
ejpam-841	720	34	=	=	SYM
ejpam-841	721	1	0	0	X
ejpam-841	721	2	.	.	PUNCT
ejpam-841	722	1	hence	hence	ADV
ejpam-841	722	2	the	the	DET
ejpam-841	722	3	polynomials	polynomial	NOUN
ejpam-841	722	4	res(p	res(p	PROPN
ejpam-841	722	5	,	,	PUNCT
ejpam-841	722	6	q	q	NOUN
ejpam-841	722	7	)	)	PUNCT
ejpam-841	722	8	,	,	PUNCT
ejpam-841	722	9	res(p	res(p	PROPN
ejpam-841	722	10	,	,	PUNCT
ejpam-841	722	11	r	r	NOUN
ejpam-841	722	12	)	)	PUNCT
ejpam-841	722	13	,	,	PUNCT
ejpam-841	722	14	res(q	res(q	PROPN
ejpam-841	722	15	,	,	PUNCT
ejpam-841	722	16	r	r	NOUN
ejpam-841	722	17	)	)	PUNCT
ejpam-841	722	18	,	,	PUNCT
ejpam-841	722	19	and	and	CCONJ
ejpam-841	722	20	det(n(p	det(n(p	ADJ
ejpam-841	722	21	,	,	PUNCT
ejpam-841	722	22	q	q	NOUN
ejpam-841	722	23	,	,	PUNCT
ejpam-841	722	24	r	r	NOUN
ejpam-841	722	25	)	)	PUNCT
ejpam-841	722	26	)	)	PUNCT
ejpam-841	722	27	are	be	AUX
ejpam-841	722	28	contained	contain	VERB
ejpam-841	722	29	in	in	ADP
ejpam-841	722	30	the	the	DET
ejpam-841	722	31	ideal	ideal	NOUN
ejpam-841	722	32	of	of	ADP
ejpam-841	722	33	the	the	DET
ejpam-841	722	34	quartic	quartic	ADJ
ejpam-841	722	35	space	space	NOUN
ejpam-841	722	36	curve	curve	NOUN
ejpam-841	722	37	c	c	PROPN
ejpam-841	722	38	,	,	PUNCT
ejpam-841	722	39	which	which	PRON
ejpam-841	722	40	is	be	AUX
ejpam-841	722	41	generated	generate	VERB
ejpam-841	722	42	by	by	ADP
ejpam-841	722	43	the	the	DET
ejpam-841	722	44	four	four	NUM
ejpam-841	722	45	polynomials	polynomial	NOUN
ejpam-841	722	46	f1	f1	NOUN
ejpam-841	722	47	,	,	PUNCT
ejpam-841	722	48	f2	f2	PROPN
ejpam-841	722	49	,	,	PUNCT
ejpam-841	722	50	f3	f3	PROPN
ejpam-841	722	51	,	,	PUNCT
ejpam-841	722	52	f4	f4	NUM
ejpam-841	722	53	in	in	ADP
ejpam-841	722	54	lemma	lemma	PROPN
ejpam-841	722	55	5	5	NUM
ejpam-841	722	56	.	.	PUNCT
ejpam-841	722	57	to	to	PART
ejpam-841	722	58	simplify	simplify	VERB
ejpam-841	722	59	our	our	PRON
ejpam-841	722	60	notation	notation	NOUN
ejpam-841	722	61	,	,	PUNCT
ejpam-841	722	62	let	let	VERB
ejpam-841	722	63	n	n	PRON
ejpam-841	722	64	=	=	SYM
ejpam-841	722	65	n(p	n(p	PROPN
ejpam-841	722	66	,	,	PUNCT
ejpam-841	722	67	q	q	NOUN
ejpam-841	722	68	,	,	PUNCT
ejpam-841	722	69	r	r	NOUN
ejpam-841	722	70	)	)	PUNCT
ejpam-841	722	71	.	.	PUNCT
ejpam-841	723	1	now	now	ADV
ejpam-841	723	2	we	we	PRON
ejpam-841	723	3	claim	claim	VERB
ejpam-841	723	4	that	that	SCONJ
ejpam-841	723	5	the	the	DET
ejpam-841	723	6	cubic	cubic	ADJ
ejpam-841	723	7	surface	surface	PROPN
ejpam-841	723	8	det(n	det(n	PROPN
ejpam-841	723	9	)	)	PUNCT
ejpam-841	723	10	=	=	SYM
ejpam-841	723	11	0	0	NUM
ejpam-841	723	12	is	be	AUX
ejpam-841	723	13	irreducible	irreducible	ADJ
ejpam-841	723	14	.	.	PUNCT
ejpam-841	724	1	otherwise	otherwise	ADV
ejpam-841	724	2	,	,	PUNCT
ejpam-841	724	3	the	the	DET
ejpam-841	724	4	cubic	cubic	ADJ
ejpam-841	724	5	det(n	det(n	PROPN
ejpam-841	724	6	)	)	PUNCT
ejpam-841	724	7	would	would	AUX
ejpam-841	724	8	have	have	VERB
ejpam-841	724	9	res(p	res(p	PROPN
ejpam-841	724	10	,	,	PUNCT
ejpam-841	724	11	q	q	NOUN
ejpam-841	724	12	)	)	PUNCT
ejpam-841	724	13	as	as	ADP
ejpam-841	724	14	a	a	DET
ejpam-841	724	15	factor	factor	NOUN
ejpam-841	724	16	,	,	PUNCT
ejpam-841	724	17	since	since	SCONJ
ejpam-841	724	18	a	a	DET
ejpam-841	724	19	nonsingular	nonsingular	ADJ
ejpam-841	724	20	rational	rational	ADJ
ejpam-841	724	21	quartic	quartic	ADJ
ejpam-841	724	22	space	space	NOUN
ejpam-841	724	23	curve	curve	NOUN
ejpam-841	724	24	c	c	PROPN
ejpam-841	724	25	is	be	AUX
ejpam-841	724	26	contained	contain	VERB
ejpam-841	724	27	in	in	ADP
ejpam-841	724	28	exactly	exactly	ADV
ejpam-841	724	29	one	one	NUM
ejpam-841	724	30	quadric	quadric	ADJ
ejpam-841	724	31	surface	surface	NOUN
ejpam-841	724	32	.	.	PUNCT
ejpam-841	725	1	but	but	CCONJ
ejpam-841	725	2	this	this	PRON
ejpam-841	725	3	is	be	AUX
ejpam-841	725	4	impossible	impossible	ADJ
ejpam-841	725	5	,	,	PUNCT
ejpam-841	725	6	since	since	SCONJ
ejpam-841	725	7	there	there	PRON
ejpam-841	725	8	exists	exist	VERB
ejpam-841	725	9	at	at	ADV
ejpam-841	725	10	least	least	ADV
ejpam-841	725	11	one	one	NUM
ejpam-841	725	12	point	point	NOUN
ejpam-841	725	13	which	which	PRON
ejpam-841	725	14	lies	lie	VERB
ejpam-841	725	15	on	on	ADP
ejpam-841	725	16	the	the	DET
ejpam-841	725	17	surface	surface	NOUN
ejpam-841	725	18	res(p	res(p	PROPN
ejpam-841	725	19	,	,	PUNCT
ejpam-841	725	20	q	q	NOUN
ejpam-841	725	21	)	)	PUNCT
ejpam-841	725	22	=	=	SYM
ejpam-841	725	23	0	0	NUM
ejpam-841	725	24	,	,	PUNCT
ejpam-841	725	25	but	but	CCONJ
ejpam-841	725	26	not	not	PART
ejpam-841	725	27	on	on	ADP
ejpam-841	725	28	the	the	DET
ejpam-841	725	29	surface	surface	NOUN
ejpam-841	725	30	det(n	det(n	PROPN
ejpam-841	725	31	)	)	PUNCT
ejpam-841	725	32	=	=	NOUN
ejpam-841	726	1	0	0	X
ejpam-841	726	2	.	.	PUNCT
ejpam-841	726	3	to	to	PART
ejpam-841	726	4	see	see	VERB
ejpam-841	726	5	that	that	SCONJ
ejpam-841	726	6	such	such	DET
ejpam-841	726	7	a	a	DET
ejpam-841	726	8	point	point	NOUN
ejpam-841	726	9	exists	exist	VERB
ejpam-841	726	10	,	,	PUNCT
ejpam-841	726	11	recall	recall	VERB
ejpam-841	726	12	that	that	SCONJ
ejpam-841	726	13	by	by	ADP
ejpam-841	726	14	lemma	lemma	PROPN
ejpam-841	726	15	3	3	NUM
ejpam-841	726	16	there	there	PRON
ejpam-841	726	17	is	be	VERB
ejpam-841	726	18	a	a	DET
ejpam-841	726	19	linear	linear	ADJ
ejpam-841	726	20	transformation	transformation	NOUN
ejpam-841	726	21	on	on	ADP
ejpam-841	726	22	the	the	DET
ejpam-841	726	23	µ-basis	µ-basis	NOUN
ejpam-841	726	24	elements	element	NOUN
ejpam-841	726	25	p	p	NOUN
ejpam-841	726	26	,	,	PUNCT
ejpam-841	726	27	q	q	X
ejpam-841	726	28	and	and	CCONJ
ejpam-841	726	29	a	a	DET
ejpam-841	726	30	projective	projective	ADJ
ejpam-841	726	31	change	change	NOUN
ejpam-841	726	32	of	of	ADP
ejpam-841	726	33	coordinates	coordinate	NOUN
ejpam-841	726	34	so	so	SCONJ
ejpam-841	726	35	that	that	SCONJ
ejpam-841	726	36	p	p	X
ejpam-841	726	37	=	=	X
ejpam-841	726	38	ys	ys	NOUN
ejpam-841	726	39	−	−	PROPN
ejpam-841	726	40	x	x	SYM
ejpam-841	726	41	t	t	PROPN
ejpam-841	726	42	,	,	PUNCT
ejpam-841	726	43	q	q	X
ejpam-841	726	44	=	=	SYM
ejpam-841	726	45	ws−	ws−	PUNCT
ejpam-841	726	46	zt	zt	PROPN
ejpam-841	726	47	,	,	PUNCT
ejpam-841	726	48	r	r	NOUN
ejpam-841	726	49	=	=	PUNCT
ejpam-841	726	50	r2s2	r2s2	PROPN
ejpam-841	726	51	+	+	NOUN
ejpam-841	726	52	r1st	r1st	PUNCT
ejpam-841	727	1	+	+	CCONJ
ejpam-841	727	2	r0	r0	NOUN
ejpam-841	727	3	t2	t2	NOUN
ejpam-841	727	4	,	,	PUNCT
ejpam-841	727	5	and	and	CCONJ
ejpam-841	727	6	det(n	det(n	PROPN
ejpam-841	727	7	)	)	PUNCT
ejpam-841	727	8	=	=	SYM
ejpam-841	727	9	r2	r2	PROPN
ejpam-841	727	10	xz	xz	PROPN
ejpam-841	727	11	+	+	CCONJ
ejpam-841	727	12	r1	r1	PROPN
ejpam-841	727	13	yz	yz	PROPN
ejpam-841	727	14	+	+	CCONJ
ejpam-841	727	15	r0	r0	PROPN
ejpam-841	727	16	yw	yw	PROPN
ejpam-841	727	17	.	.	PUNCT
ejpam-841	728	1	now	now	ADV
ejpam-841	728	2	observe	observe	VERB
ejpam-841	728	3	that	that	SCONJ
ejpam-841	728	4	the	the	DET
ejpam-841	728	5	surface	surface	NOUN
ejpam-841	728	6	res(p	res(p	PROPN
ejpam-841	728	7	,	,	PUNCT
ejpam-841	728	8	q	q	NOUN
ejpam-841	728	9	)	)	PUNCT
ejpam-841	728	10	=	=	SYM
ejpam-841	728	11	xw	xw	PROPN
ejpam-841	729	1	−	−	PROPN
ejpam-841	729	2	yz	yz	PROPN
ejpam-841	729	3	=	=	SYM
ejpam-841	729	4	0	0	PROPN
ejpam-841	729	5	contains	contain	VERB
ejpam-841	729	6	the	the	DET
ejpam-841	729	7	line	line	NOUN
ejpam-841	729	8	x	x	PUNCT
ejpam-841	730	1	=	=	PUNCT
ejpam-841	730	2	z	z	NOUN
ejpam-841	730	3	=	=	SYM
ejpam-841	730	4	0	0	PROPN
ejpam-841	730	5	.	.	PUNCT
ejpam-841	731	1	but	but	CCONJ
ejpam-841	731	2	there	there	PRON
ejpam-841	731	3	must	must	AUX
ejpam-841	731	4	be	be	AUX
ejpam-841	731	5	a	a	DET
ejpam-841	731	6	point	point	NOUN
ejpam-841	731	7	p	p	X
ejpam-841	731	8	=	=	X
ejpam-841	731	9	(	(	PUNCT
ejpam-841	731	10	0	0	NUM
ejpam-841	731	11	,	,	PUNCT
ejpam-841	731	12	a	a	DET
ejpam-841	731	13	,	,	PUNCT
ejpam-841	731	14	0	0	NUM
ejpam-841	731	15	,	,	PUNCT
ejpam-841	731	16	b	b	NOUN
ejpam-841	731	17	)	)	PUNCT
ejpam-841	731	18	on	on	ADP
ejpam-841	731	19	the	the	DET
ejpam-841	731	20	line	line	NOUN
ejpam-841	731	21	x	x	PUNCT
ejpam-841	732	1	=	=	PUNCT
ejpam-841	732	2	z	z	NOUN
ejpam-841	732	3	=	=	SYM
ejpam-841	732	4	0	0	NUM
ejpam-841	732	5	such	such	ADJ
ejpam-841	732	6	that	that	SCONJ
ejpam-841	732	7	det(n)(p	det(n)(p	VERB
ejpam-841	732	8	)	)	PUNCT
ejpam-841	732	9	=	=	PRON
ejpam-841	732	10	r0(p)ab	r0(p)ab	ADJ
ejpam-841	732	11	6=	6=	NUM
ejpam-841	732	12	0	0	X
ejpam-841	732	13	.	.	PUNCT
ejpam-841	733	1	if	if	SCONJ
ejpam-841	733	2	not	not	PART
ejpam-841	733	3	,	,	PUNCT
ejpam-841	733	4	then	then	ADV
ejpam-841	733	5	r0(p	r0(p	ADJ
ejpam-841	733	6	)	)	PUNCT
ejpam-841	733	7	=	=	SYM
ejpam-841	733	8	0	0	NUM
ejpam-841	733	9	,	,	PUNCT
ejpam-841	733	10	so	so	ADV
ejpam-841	733	11	deg(gcd(p	deg(gcd(p	PROPN
ejpam-841	733	12	·	·	PUNCT
ejpam-841	734	1	p	p	X
ejpam-841	734	2	,	,	PUNCT
ejpam-841	734	3	q	q	X
ejpam-841	734	4	·	·	PUNCT
ejpam-841	734	5	p	p	X
ejpam-841	734	6	,	,	PUNCT
ejpam-841	734	7	r	r	NOUN
ejpam-841	734	8	·	·	PUNCT
ejpam-841	734	9	p	p	X
ejpam-841	734	10	)	)	PUNCT
ejpam-841	734	11	)	)	PUNCT
ejpam-841	734	12	=	=	SYM
ejpam-841	735	1	deg(gcd(as	deg(gcd(as	PROPN
ejpam-841	735	2	,	,	PUNCT
ejpam-841	735	3	bs	bs	NOUN
ejpam-841	735	4	,	,	PUNCT
ejpam-841	735	5	r2s2	r2s2	NOUN
ejpam-841	735	6	+	+	NOUN
ejpam-841	735	7	r1st	r1st	NUM
ejpam-841	735	8	)	)	PUNCT
ejpam-841	735	9	)	)	PUNCT
ejpam-841	736	1	=	=	PUNCT
ejpam-841	736	2	1	1	X
ejpam-841	736	3	.	.	PUNCT
ejpam-841	736	4	hence	hence	ADV
ejpam-841	736	5	by	by	ADP
ejpam-841	736	6	proposition	proposition	NOUN
ejpam-841	736	7	1	1	NUM
ejpam-841	736	8	,	,	PUNCT
ejpam-841	736	9	c	c	PROPN
ejpam-841	736	10	must	must	AUX
ejpam-841	736	11	contain	contain	VERB
ejpam-841	736	12	a	a	DET
ejpam-841	736	13	line	line	NOUN
ejpam-841	736	14	,	,	PUNCT
ejpam-841	736	15	contradicting	contradict	VERB
ejpam-841	736	16	the	the	DET
ejpam-841	736	17	assumption	assumption	NOUN
ejpam-841	736	18	that	that	SCONJ
ejpam-841	736	19	c	c	PROPN
ejpam-841	736	20	is	be	AUX
ejpam-841	736	21	a	a	DET
ejpam-841	736	22	space	space	NOUN
ejpam-841	736	23	curve	curve	NOUN
ejpam-841	736	24	.	.	PUNCT
ejpam-841	737	1	moreover	moreover	ADV
ejpam-841	737	2	,	,	PUNCT
ejpam-841	737	3	the	the	DET
ejpam-841	737	4	three	three	NUM
ejpam-841	737	5	cubic	cubic	ADJ
ejpam-841	737	6	surfaces	surface	NOUN
ejpam-841	737	7	det(n	det(n	PROPN
ejpam-841	737	8	)	)	PUNCT
ejpam-841	737	9	=	=	SYM
ejpam-841	737	10	0	0	NUM
ejpam-841	737	11	,	,	PUNCT
ejpam-841	737	12	res(p	res(p	PROPN
ejpam-841	737	13	,	,	PUNCT
ejpam-841	737	14	r	r	NOUN
ejpam-841	737	15	)	)	PUNCT
ejpam-841	737	16	=	=	SYM
ejpam-841	737	17	0	0	NUM
ejpam-841	737	18	and	and	CCONJ
ejpam-841	737	19	res(q	res(q	NOUN
ejpam-841	737	20	,	,	PUNCT
ejpam-841	737	21	r	r	NOUN
ejpam-841	737	22	)	)	PUNCT
ejpam-841	737	23	=	=	SYM
ejpam-841	737	24	0	0	NUM
ejpam-841	737	25	are	be	AUX
ejpam-841	737	26	distinct	distinct	ADJ
ejpam-841	737	27	.	.	PUNCT
ejpam-841	738	1	indeed	indeed	ADV
ejpam-841	738	2	,	,	PUNCT
ejpam-841	738	3	since	since	SCONJ
ejpam-841	738	4	both	both	CCONJ
ejpam-841	738	5	p	p	NOUN
ejpam-841	738	6	and	and	CCONJ
ejpam-841	738	7	q	q	NOUN
ejpam-841	738	8	are	be	AUX
ejpam-841	738	9	axial	axial	ADJ
ejpam-841	738	10	moving	move	VERB
ejpam-841	738	11	planes	plane	NOUN
ejpam-841	738	12	,	,	PUNCT
ejpam-841	738	13	the	the	DET
ejpam-841	738	14	axis	axis	NOUN
ejpam-841	738	15	of	of	ADP
ejpam-841	738	16	p	p	NOUN
ejpam-841	738	17	is	be	AUX
ejpam-841	738	18	contained	contain	VERB
ejpam-841	738	19	in	in	ADP
ejpam-841	738	20	the	the	DET
ejpam-841	738	21	surfaces	surface	NOUN
ejpam-841	738	22	res(p	res(p	PROPN
ejpam-841	738	23	,	,	PUNCT
ejpam-841	738	24	r	r	NOUN
ejpam-841	738	25	)	)	PUNCT
ejpam-841	738	26	=	=	SYM
ejpam-841	738	27	0	0	NUM
ejpam-841	738	28	and	and	CCONJ
ejpam-841	738	29	det(n	det(n	PROPN
ejpam-841	738	30	)	)	PUNCT
ejpam-841	738	31	=	=	SYM
ejpam-841	738	32	0	0	NUM
ejpam-841	738	33	,	,	PUNCT
ejpam-841	738	34	but	but	CCONJ
ejpam-841	738	35	is	be	AUX
ejpam-841	738	36	not	not	PART
ejpam-841	738	37	contained	contain	VERB
ejpam-841	738	38	in	in	ADP
ejpam-841	738	39	the	the	DET
ejpam-841	738	40	surface	surface	NOUN
ejpam-841	738	41	res(q	res(q	NOUN
ejpam-841	738	42	,	,	PUNCT
ejpam-841	738	43	r	r	NOUN
ejpam-841	738	44	)	)	PUNCT
ejpam-841	738	45	=	=	SYM
ejpam-841	738	46	0	0	NUM
ejpam-841	738	47	;	;	PUNCT
ejpam-841	738	48	and	and	CCONJ
ejpam-841	738	49	similarly	similarly	ADV
ejpam-841	738	50	,	,	PUNCT
ejpam-841	738	51	the	the	DET
ejpam-841	738	52	axis	axis	NOUN
ejpam-841	738	53	of	of	ADP
ejpam-841	738	54	q	q	NOUN
ejpam-841	738	55	is	be	AUX
ejpam-841	738	56	contained	contain	VERB
ejpam-841	738	57	in	in	ADP
ejpam-841	738	58	the	the	DET
ejpam-841	738	59	surfaces	surface	NOUN
ejpam-841	738	60	res(q	res(q	NOUN
ejpam-841	738	61	,	,	PUNCT
ejpam-841	738	62	r	r	NOUN
ejpam-841	738	63	)	)	PUNCT
ejpam-841	739	1	=	=	SYM
ejpam-841	739	2	0	0	NUM
ejpam-841	739	3	and	and	CCONJ
ejpam-841	739	4	det(n	det(n	PROPN
ejpam-841	739	5	)	)	PUNCT
ejpam-841	739	6	=	=	SYM
ejpam-841	739	7	0	0	NUM
ejpam-841	739	8	,	,	PUNCT
ejpam-841	739	9	but	but	CCONJ
ejpam-841	739	10	is	be	AUX
ejpam-841	739	11	not	not	PART
ejpam-841	739	12	contained	contain	VERB
ejpam-841	739	13	in	in	ADP
ejpam-841	739	14	the	the	DET
ejpam-841	739	15	surface	surface	NOUN
ejpam-841	739	16	res(p	res(p	PROPN
ejpam-841	739	17	,	,	PUNCT
ejpam-841	739	18	r	r	NOUN
ejpam-841	739	19	)	)	PUNCT
ejpam-841	739	20	=	=	SYM
ejpam-841	740	1	0	0	X
ejpam-841	740	2	.	.	PUNCT
ejpam-841	741	1	indeed	indeed	ADV
ejpam-841	741	2	a	a	DET
ejpam-841	741	3	point	point	NOUN
ejpam-841	741	4	x	x	PUNCT
ejpam-841	741	5	is	be	AUX
ejpam-841	741	6	on	on	ADP
ejpam-841	741	7	res(q	res(q	NOUN
ejpam-841	741	8	,	,	PUNCT
ejpam-841	741	9	r	r	NOUN
ejpam-841	741	10	)	)	PUNCT
ejpam-841	741	11	=	=	SYM
ejpam-841	741	12	0	0	PUNCT
ejpam-841	742	1	if	if	SCONJ
ejpam-841	742	2	and	and	CCONJ
ejpam-841	742	3	only	only	ADV
ejpam-841	742	4	if	if	SCONJ
ejpam-841	742	5	there	there	PRON
ejpam-841	742	6	is	be	VERB
ejpam-841	742	7	an	an	DET
ejpam-841	742	8	(	(	PUNCT
ejpam-841	742	9	s	s	PROPN
ejpam-841	742	10	,	,	PUNCT
ejpam-841	742	11	t	t	PROPN
ejpam-841	742	12	)	)	PUNCT
ejpam-841	742	13	such	such	ADJ
ejpam-841	742	14	that	that	SCONJ
ejpam-841	742	15	x	x	PRON
ejpam-841	742	16	is	be	AUX
ejpam-841	742	17	on	on	ADP
ejpam-841	742	18	the	the	DET
ejpam-841	742	19	planes	plane	NOUN
ejpam-841	742	20	q(s	q(s	PROPN
ejpam-841	742	21	,	,	PUNCT
ejpam-841	742	22	t	t	PROPN
ejpam-841	742	23	)	)	PUNCT
ejpam-841	742	24	=	=	SYM
ejpam-841	742	25	0	0	NUM
ejpam-841	742	26	,	,	PUNCT
ejpam-841	742	27	r(s	r(s	PROPN
ejpam-841	742	28	,	,	PUNCT
ejpam-841	742	29	t	t	PROPN
ejpam-841	742	30	)	)	PUNCT
ejpam-841	742	31	=	=	NOUN
ejpam-841	743	1	0	0	X
ejpam-841	743	2	.	.	PUNCT
ejpam-841	744	1	if	if	SCONJ
ejpam-841	744	2	x	x	PRON
ejpam-841	744	3	is	be	AUX
ejpam-841	744	4	also	also	ADV
ejpam-841	744	5	on	on	ADP
ejpam-841	744	6	the	the	DET
ejpam-841	744	7	axis	axis	NOUN
ejpam-841	744	8	of	of	ADP
ejpam-841	744	9	p	p	NOUN
ejpam-841	744	10	,	,	PUNCT
ejpam-841	744	11	then	then	ADV
ejpam-841	744	12	the	the	DET
ejpam-841	744	13	point	point	NOUN
ejpam-841	744	14	x	x	PUNCT
ejpam-841	744	15	would	would	AUX
ejpam-841	744	16	also	also	ADV
ejpam-841	744	17	be	be	AUX
ejpam-841	744	18	on	on	ADP
ejpam-841	744	19	p(s	p(s	PROPN
ejpam-841	744	20	,	,	PUNCT
ejpam-841	744	21	t	t	PROPN
ejpam-841	744	22	)	)	PUNCT
ejpam-841	744	23	=	=	SYM
ejpam-841	744	24	0	0	NUM
ejpam-841	744	25	,	,	PUNCT
ejpam-841	744	26	and	and	CCONJ
ejpam-841	744	27	thus	thus	ADV
ejpam-841	744	28	by	by	ADP
ejpam-841	744	29	proposition	proposition	NOUN
ejpam-841	744	30	1	1	NUM
ejpam-841	744	31	on	on	ADP
ejpam-841	744	32	the	the	DET
ejpam-841	744	33	curve	curve	NOUN
ejpam-841	744	34	c	c	PROPN
ejpam-841	744	35	.	.	PUNCT
ejpam-841	745	1	therefore	therefore	ADV
ejpam-841	745	2	the	the	DET
ejpam-841	745	3	entire	entire	ADJ
ejpam-841	745	4	axis	axis	NOUN
ejpam-841	745	5	of	of	ADP
ejpam-841	745	6	p	p	NOUN
ejpam-841	745	7	would	would	AUX
ejpam-841	745	8	be	be	AUX
ejpam-841	745	9	on	on	ADP
ejpam-841	745	10	c	c	PROPN
ejpam-841	745	11	,	,	PUNCT
ejpam-841	745	12	which	which	PRON
ejpam-841	745	13	is	be	AUX
ejpam-841	745	14	absurd	absurd	ADJ
ejpam-841	745	15	.	.	PUNCT
ejpam-841	746	1	hence	hence	ADV
ejpam-841	746	2	the	the	DET
ejpam-841	746	3	three	three	NUM
ejpam-841	746	4	surfaces	surface	NOUN
ejpam-841	746	5	det(n	det(n	PROPN
ejpam-841	746	6	)	)	PUNCT
ejpam-841	746	7	=	=	SYM
ejpam-841	746	8	0	0	NUM
ejpam-841	746	9	,	,	PUNCT
ejpam-841	746	10	res(p	res(p	PROPN
ejpam-841	746	11	,	,	PUNCT
ejpam-841	746	12	r	r	NOUN
ejpam-841	746	13	)	)	PUNCT
ejpam-841	746	14	=	=	SYM
ejpam-841	746	15	0	0	NUM
ejpam-841	746	16	and	and	CCONJ
ejpam-841	746	17	res(q	res(q	NOUN
ejpam-841	746	18	,	,	PUNCT
ejpam-841	746	19	r	r	NOUN
ejpam-841	746	20	)	)	PUNCT
ejpam-841	746	21	=	=	SYM
ejpam-841	746	22	0	0	NUM
ejpam-841	746	23	are	be	AUX
ejpam-841	746	24	different	different	ADJ
ejpam-841	746	25	from	from	ADP
ejpam-841	746	26	each	each	DET
ejpam-841	746	27	other	other	ADJ
ejpam-841	746	28	.	.	PUNCT
ejpam-841	747	1	furthermore	furthermore	ADV
ejpam-841	747	2	,	,	PUNCT
ejpam-841	747	3	the	the	DET
ejpam-841	747	4	three	three	NUM
ejpam-841	747	5	cubic	cubic	ADJ
ejpam-841	747	6	surfaces	surface	NOUN
ejpam-841	747	7	det(n	det(n	PROPN
ejpam-841	747	8	)	)	PUNCT
ejpam-841	747	9	=	=	SYM
ejpam-841	748	1	0	0	NUM
ejpam-841	748	2	,	,	PUNCT
ejpam-841	748	3	res(p	res(p	PROPN
ejpam-841	748	4	,	,	PUNCT
ejpam-841	748	5	r	r	NOUN
ejpam-841	748	6	)	)	PUNCT
ejpam-841	748	7	=	=	SYM
ejpam-841	748	8	0	0	NUM
ejpam-841	748	9	and	and	CCONJ
ejpam-841	748	10	res(q	res(q	NOUN
ejpam-841	748	11	,	,	PUNCT
ejpam-841	748	12	r	r	NOUN
ejpam-841	748	13	)	)	PUNCT
ejpam-841	748	14	=	=	SYM
ejpam-841	748	15	0	0	NUM
ejpam-841	748	16	are	be	AUX
ejpam-841	748	17	linearly	linearly	ADV
ejpam-841	748	18	independent	independent	ADJ
ejpam-841	748	19	.	.	PUNCT
ejpam-841	749	1	for	for	AUX
ejpam-841	749	2	suppose	suppose	VERB
ejpam-841	749	3	that	that	SCONJ
ejpam-841	749	4	there	there	PRON
ejpam-841	749	5	exist	exist	VERB
ejpam-841	749	6	some	some	DET
ejpam-841	749	7	nonzero	nonzero	PROPN
ejpam-841	749	8	constants	constant	NOUN
ejpam-841	749	9	a	a	DET
ejpam-841	749	10	,	,	PUNCT
ejpam-841	749	11	b	b	NOUN
ejpam-841	749	12	,	,	PUNCT
ejpam-841	749	13	c	c	X
ejpam-841	749	14	such	such	ADJ
ejpam-841	749	15	that	that	PRON
ejpam-841	749	16	ares(p	ares(p	PROPN
ejpam-841	749	17	,	,	PUNCT
ejpam-841	749	18	r	r	NOUN
ejpam-841	749	19	)	)	PUNCT
ejpam-841	749	20	+	+	CCONJ
ejpam-841	749	21	bres(q	bres(q	PROPN
ejpam-841	749	22	,	,	PUNCT
ejpam-841	749	23	r)+	r)+	VERB
ejpam-841	749	24	c	c	NOUN
ejpam-841	749	25	det(n)≡	det(n)≡	PROPN
ejpam-841	749	26	0	0	PROPN
ejpam-841	749	27	.	.	PUNCT
ejpam-841	749	28	j.	j.	PROPN
ejpam-841	749	29	hoffman	hoffman	PROPN
ejpam-841	749	30	,	,	PUNCT
ejpam-841	749	31	h.	h.	PROPN
ejpam-841	749	32	wang	wang	PROPN
ejpam-841	749	33	,	,	PUNCT
ejpam-841	749	34	x.	x.	PROPN
ejpam-841	749	35	jia	jia	PROPN
ejpam-841	749	36	,	,	PUNCT
ejpam-841	749	37	r.	r.	PROPN
ejpam-841	749	38	goldman	goldman	PROPN
ejpam-841	749	39	/	/	SYM
ejpam-841	749	40	eur	eur	PROPN
ejpam-841	749	41	.	.	PUNCT
ejpam-841	750	1	j.	j.	PROPN
ejpam-841	750	2	pure	pure	PROPN
ejpam-841	750	3	appl	appl	PROPN
ejpam-841	750	4	.	.	PROPN
ejpam-841	750	5	math	math	PROPN
ejpam-841	750	6	,	,	PUNCT
ejpam-841	750	7	3	3	NUM
ejpam-841	750	8	(	(	PUNCT
ejpam-841	750	9	2010	2010	NUM
ejpam-841	750	10	)	)	PUNCT
ejpam-841	750	11	,	,	PUNCT
ejpam-841	750	12	602	602	NUM
ejpam-841	750	13	-	-	SYM
ejpam-841	750	14	632	632	NUM
ejpam-841	750	15	627	627	NUM
ejpam-841	750	16	then	then	ADV
ejpam-841	750	17	〈	〈	PROPN
ejpam-841	750	18	res(p	res(p	PROPN
ejpam-841	750	19	,	,	PUNCT
ejpam-841	750	20	r	r	NOUN
ejpam-841	750	21	)	)	PUNCT
ejpam-841	750	22	,	,	PUNCT
ejpam-841	750	23	det(n)〉=	det(n)〉=	X
ejpam-841	750	24	〈	〈	PROPN
ejpam-841	750	25	res(p	res(p	PROPN
ejpam-841	750	26	,	,	PUNCT
ejpam-841	750	27	r),res(q	r),res(q	NOUN
ejpam-841	750	28	,	,	PUNCT
ejpam-841	750	29	r	r	NOUN
ejpam-841	750	30	)	)	PUNCT
ejpam-841	750	31	〉	〉	NOUN
ejpam-841	750	32	,	,	PUNCT
ejpam-841	750	33	so	so	ADV
ejpam-841	750	34	v(〈res(p	v(〈res(p	PROPN
ejpam-841	750	35	,	,	PUNCT
ejpam-841	750	36	r	r	NOUN
ejpam-841	750	37	)	)	PUNCT
ejpam-841	750	38	,	,	PUNCT
ejpam-841	750	39	det(n	det(n	NOUN
ejpam-841	750	40	)	)	PUNCT
ejpam-841	750	41	〉	〉	NOUN
ejpam-841	750	42	)	)	PUNCT
ejpam-841	750	43	=	=	SYM
ejpam-841	750	44	v(res(p	v(res(p	NOUN
ejpam-841	750	45	,	,	PUNCT
ejpam-841	750	46	r),res(q	r),res(q	NOUN
ejpam-841	750	47	,	,	PUNCT
ejpam-841	750	48	r	r	NOUN
ejpam-841	750	49	)	)	PUNCT
ejpam-841	750	50	〉	〉	NOUN
ejpam-841	750	51	)	)	PUNCT
ejpam-841	750	52	.	.	PUNCT
ejpam-841	751	1	hence	hence	ADV
ejpam-841	751	2	the	the	DET
ejpam-841	751	3	axis	axis	NOUN
ejpam-841	751	4	of	of	ADP
ejpam-841	751	5	p	p	NOUN
ejpam-841	751	6	is	be	AUX
ejpam-841	751	7	contained	contain	VERB
ejpam-841	751	8	in	in	ADP
ejpam-841	751	9	the	the	DET
ejpam-841	751	10	surface	surface	NOUN
ejpam-841	751	11	res(q	res(q	NOUN
ejpam-841	751	12	,	,	PUNCT
ejpam-841	751	13	r	r	NOUN
ejpam-841	751	14	)	)	PUNCT
ejpam-841	751	15	=	=	SYM
ejpam-841	752	1	0	0	X
ejpam-841	752	2	.	.	PUNCT
ejpam-841	753	1	contradiction	contradiction	NOUN
ejpam-841	753	2	.	.	PUNCT
ejpam-841	754	1	therefore	therefore	ADV
ejpam-841	754	2	,	,	PUNCT
ejpam-841	754	3	the	the	DET
ejpam-841	754	4	three	three	NUM
ejpam-841	754	5	cubic	cubic	ADJ
ejpam-841	754	6	surfaces	surface	NOUN
ejpam-841	754	7	det(n	det(n	PROPN
ejpam-841	754	8	)	)	PUNCT
ejpam-841	754	9	=	=	SYM
ejpam-841	755	1	0	0	NUM
ejpam-841	755	2	,	,	PUNCT
ejpam-841	755	3	res(p	res(p	PROPN
ejpam-841	755	4	,	,	PUNCT
ejpam-841	755	5	r	r	NOUN
ejpam-841	755	6	)	)	PUNCT
ejpam-841	755	7	=	=	SYM
ejpam-841	755	8	0	0	NUM
ejpam-841	755	9	and	and	CCONJ
ejpam-841	755	10	res(q	res(q	NOUN
ejpam-841	755	11	,	,	PUNCT
ejpam-841	755	12	r	r	NOUN
ejpam-841	755	13	)	)	PUNCT
ejpam-841	755	14	=	=	SYM
ejpam-841	755	15	0	0	NUM
ejpam-841	755	16	are	be	AUX
ejpam-841	755	17	linearly	linearly	ADV
ejpam-841	755	18	independent	independent	ADJ
ejpam-841	755	19	.	.	PUNCT
ejpam-841	756	1	thus	thus	ADV
ejpam-841	756	2	,	,	PUNCT
ejpam-841	756	3	by	by	ADP
ejpam-841	756	4	lemma	lemma	PROPN
ejpam-841	756	5	5	5	NUM
ejpam-841	756	6	{	{	PUNCT
ejpam-841	756	7	res(p	res(p	PROPN
ejpam-841	756	8	,	,	PUNCT
ejpam-841	756	9	q	q	NOUN
ejpam-841	756	10	)	)	PUNCT
ejpam-841	756	11	,	,	PUNCT
ejpam-841	756	12	res(p	res(p	PROPN
ejpam-841	756	13	,	,	PUNCT
ejpam-841	756	14	r	r	NOUN
ejpam-841	756	15	)	)	PUNCT
ejpam-841	756	16	,	,	PUNCT
ejpam-841	756	17	res(q	res(q	PROPN
ejpam-841	756	18	,	,	PUNCT
ejpam-841	756	19	r	r	NOUN
ejpam-841	756	20	)	)	PUNCT
ejpam-841	756	21	,	,	PUNCT
ejpam-841	756	22	det(n	det(n	PROPN
ejpam-841	756	23	)	)	PUNCT
ejpam-841	756	24	}	}	PUNCT
ejpam-841	756	25	must	must	AUX
ejpam-841	756	26	be	be	AUX
ejpam-841	756	27	a	a	DET
ejpam-841	756	28	set	set	NOUN
ejpam-841	756	29	of	of	ADP
ejpam-841	756	30	generators	generator	NOUN
ejpam-841	756	31	for	for	ADP
ejpam-841	756	32	the	the	DET
ejpam-841	756	33	ideal	ideal	NOUN
ejpam-841	756	34	of	of	ADP
ejpam-841	756	35	the	the	DET
ejpam-841	756	36	non	non	ADJ
ejpam-841	756	37	-	-	ADJ
ejpam-841	756	38	singular	singular	ADJ
ejpam-841	756	39	rational	rational	ADJ
ejpam-841	756	40	quartic	quartic	ADJ
ejpam-841	756	41	space	space	NOUN
ejpam-841	756	42	curve	curve	NOUN
ejpam-841	756	43	c	c	PROPN
ejpam-841	756	44	.	.	PUNCT
ejpam-841	757	1	therefore	therefore	ADV
ejpam-841	757	2	,	,	PUNCT
ejpam-841	757	3	res(p	res(p	PROPN
ejpam-841	757	4	,	,	PUNCT
ejpam-841	757	5	q	q	NOUN
ejpam-841	757	6	)	)	PUNCT
ejpam-841	757	7	=	=	SYM
ejpam-841	757	8	0	0	NUM
ejpam-841	757	9	,	,	PUNCT
ejpam-841	757	10	res(p	res(p	PROPN
ejpam-841	757	11	,	,	PUNCT
ejpam-841	757	12	r	r	NOUN
ejpam-841	757	13	)	)	PUNCT
ejpam-841	757	14	=	=	SYM
ejpam-841	757	15	0	0	NUM
ejpam-841	757	16	,	,	PUNCT
ejpam-841	757	17	res(q	res(q	NOUN
ejpam-841	757	18	,	,	PUNCT
ejpam-841	757	19	r	r	NOUN
ejpam-841	757	20	)	)	PUNCT
ejpam-841	757	21	=	=	SYM
ejpam-841	757	22	0	0	NUM
ejpam-841	757	23	,	,	PUNCT
ejpam-841	757	24	det(n	det(n	PROPN
ejpam-841	757	25	)	)	PUNCT
ejpam-841	757	26	=	=	SYM
ejpam-841	757	27	0	0	NUM
ejpam-841	757	28	are	be	AUX
ejpam-841	757	29	the	the	DET
ejpam-841	757	30	implicit	implicit	ADJ
ejpam-841	757	31	equations	equation	NOUN
ejpam-841	757	32	of	of	ADP
ejpam-841	757	33	the	the	DET
ejpam-841	757	34	non	non	ADJ
ejpam-841	757	35	-	-	ADJ
ejpam-841	757	36	singular	singular	ADJ
ejpam-841	757	37	rational	rational	ADJ
ejpam-841	757	38	quartic	quartic	ADJ
ejpam-841	757	39	space	space	NOUN
ejpam-841	757	40	curve	curve	NOUN
ejpam-841	757	41	.	.	PUNCT
ejpam-841	758	1	we	we	PRON
ejpam-841	758	2	illustrate	illustrate	VERB
ejpam-841	758	3	our	our	PRON
ejpam-841	758	4	method	method	NOUN
ejpam-841	758	5	for	for	ADP
ejpam-841	758	6	finding	find	VERB
ejpam-841	758	7	the	the	DET
ejpam-841	758	8	implicit	implicit	ADJ
ejpam-841	758	9	equations	equation	NOUN
ejpam-841	758	10	of	of	ADP
ejpam-841	758	11	a	a	DET
ejpam-841	758	12	non	non	ADJ
ejpam-841	758	13	-	-	ADJ
ejpam-841	758	14	singular	singular	ADJ
ejpam-841	758	15	rational	rational	ADJ
ejpam-841	758	16	quartic	quartic	ADJ
ejpam-841	758	17	space	space	NOUN
ejpam-841	758	18	curve	curve	NOUN
ejpam-841	758	19	with	with	ADP
ejpam-841	758	20	the	the	DET
ejpam-841	758	21	following	follow	VERB
ejpam-841	758	22	simple	simple	ADJ
ejpam-841	758	23	example	example	NOUN
ejpam-841	758	24	.	.	PUNCT
ejpam-841	759	1	example	example	NOUN
ejpam-841	760	1	4	4	NUM
ejpam-841	760	2	.	.	PUNCT
ejpam-841	760	3	let	let	VERB
ejpam-841	760	4	the	the	DET
ejpam-841	760	5	non	non	ADJ
ejpam-841	760	6	-	-	ADJ
ejpam-841	760	7	singular	singular	ADJ
ejpam-841	760	8	rational	rational	ADJ
ejpam-841	760	9	quartic	quartic	ADJ
ejpam-841	760	10	space	space	NOUN
ejpam-841	760	11	curve	curve	NOUN
ejpam-841	760	12	c	c	AUX
ejpam-841	760	13	be	be	AUX
ejpam-841	760	14	given	give	VERB
ejpam-841	760	15	as	as	ADP
ejpam-841	760	16	the	the	DET
ejpam-841	760	17	image	image	NOUN
ejpam-841	760	18	of	of	ADP
ejpam-841	760	19	the	the	DET
ejpam-841	760	20	parameterization	parameterization	NOUN
ejpam-841	760	21	:	:	PUNCT
ejpam-841	760	22	(	(	PUNCT
ejpam-841	760	23	x	x	X
ejpam-841	760	24	,	,	PUNCT
ejpam-841	760	25	y	y	PROPN
ejpam-841	760	26	,	,	PUNCT
ejpam-841	760	27	z	z	PROPN
ejpam-841	760	28	,	,	PUNCT
ejpam-841	760	29	w	w	NOUN
ejpam-841	760	30	)	)	PUNCT
ejpam-841	760	31	=	=	SYM
ejpam-841	760	32	(	(	PUNCT
ejpam-841	760	33	s4	s4	PROPN
ejpam-841	760	34	,	,	PUNCT
ejpam-841	760	35	s2	s2	NOUN
ejpam-841	760	36	t(s+	t(s+	NOUN
ejpam-841	760	37	t	t	PROPN
ejpam-841	760	38	)	)	PUNCT
ejpam-841	760	39	,	,	PUNCT
ejpam-841	760	40	st2(s−	st2(s−	PROPN
ejpam-841	760	41	t	t	PROPN
ejpam-841	760	42	)	)	PUNCT
ejpam-841	760	43	,	,	PUNCT
ejpam-841	760	44	t4	t4	PROPN
ejpam-841	760	45	)	)	PUNCT
ejpam-841	760	46	.	.	PUNCT
ejpam-841	761	1	compute	compute	VERB
ejpam-841	761	2	a	a	DET
ejpam-841	761	3	µ-basis	µ-basis	NOUN
ejpam-841	761	4	using	use	VERB
ejpam-841	761	5	the	the	DET
ejpam-841	761	6	algorithm	algorithm	NOUN
ejpam-841	761	7	in	in	ADP
ejpam-841	761	8	[	[	X
ejpam-841	761	9	24	24	NUM
ejpam-841	761	10	]	]	X
ejpam-841	761	11	p	p	X
ejpam-841	761	12	=	=	PUNCT
ejpam-841	761	13	2	2	NUM
ejpam-841	761	14	t	t	NOUN
ejpam-841	761	15	x	x	SYM
ejpam-841	762	1	+	+	CCONJ
ejpam-841	762	2	(	(	PUNCT
ejpam-841	762	3	−2s+	−2s+	NOUN
ejpam-841	762	4	t)y	t)y	X
ejpam-841	763	1	+	+	CCONJ
ejpam-841	763	2	sz	sz	NOUN
ejpam-841	763	3	,	,	PUNCT
ejpam-841	763	4	q	q	PROPN
ejpam-841	763	5	=	=	PUNCT
ejpam-841	763	6	t	t	X
ejpam-841	763	7	x	x	PUNCT
ejpam-841	763	8	−	−	PROPN
ejpam-841	763	9	s	s	PART
ejpam-841	763	10	y	y	PROPN
ejpam-841	763	11	+	+	CCONJ
ejpam-841	763	12	(	(	PUNCT
ejpam-841	763	13	s+	s+	X
ejpam-841	763	14	t)z	t)z	PUNCT
ejpam-841	763	15	+	+	CCONJ
ejpam-841	763	16	sw	sw	PROPN
ejpam-841	763	17	,	,	PUNCT
ejpam-841	763	18	r	r	NOUN
ejpam-841	763	19	=	=	SYM
ejpam-841	763	20	(	(	PUNCT
ejpam-841	763	21	−t2	−t2	PROPN
ejpam-841	763	22	−	−	PROPN
ejpam-841	763	23	st)x	st)x	PROPN
ejpam-841	763	24	+	+	PROPN
ejpam-841	763	25	s2	s2	PROPN
ejpam-841	763	26	y.	y.	NOUN
ejpam-841	763	27	then	then	ADV
ejpam-841	763	28	res(p	res(p	PROPN
ejpam-841	763	29	,	,	PUNCT
ejpam-841	763	30	q	q	NOUN
ejpam-841	763	31	)	)	PUNCT
ejpam-841	763	32	=	=	PUNCT
ejpam-841	764	1	y2	y2	PROPN
ejpam-841	764	2	−	−	PROPN
ejpam-841	764	3	xz−	xz−	PUNCT
ejpam-841	765	1	3yz	3yz	NOUN
ejpam-841	765	2	+	+	CCONJ
ejpam-841	765	3	z2	z2	PROPN
ejpam-841	765	4	−	−	PROPN
ejpam-841	765	5	2xw−	2xw−	NUM
ejpam-841	765	6	yw	yw	PROPN
ejpam-841	765	7	,	,	PUNCT
ejpam-841	765	8	res(p	res(p	PROPN
ejpam-841	765	9	,	,	PUNCT
ejpam-841	765	10	r	r	NOUN
ejpam-841	765	11	)	)	PUNCT
ejpam-841	765	12	=	=	SYM
ejpam-841	765	13	z3	z3	NOUN
ejpam-841	765	14	−	−	PROPN
ejpam-841	765	15	xzw	xzw	PROPN
ejpam-841	766	1	+	+	CCONJ
ejpam-841	766	2	2yzw	2yzw	NUM
ejpam-841	766	3	−	−	NOUN
ejpam-841	767	1	z2w	z2w	PROPN
ejpam-841	767	2	−	−	PROPN
ejpam-841	767	3	2xw2	2xw2	NUM
ejpam-841	767	4	+	+	NUM
ejpam-841	767	5	yw2	yw2	NOUN
ejpam-841	767	6	,	,	PUNCT
ejpam-841	767	7	res(q	res(q	NOUN
ejpam-841	767	8	,	,	PUNCT
ejpam-841	767	9	r	r	NOUN
ejpam-841	767	10	)	)	PUNCT
ejpam-841	767	11	=	=	PUNCT
ejpam-841	768	1	yz2	yz2	ADJ
ejpam-841	769	1	−	−	NOUN
ejpam-841	769	2	x	x	SYM
ejpam-841	769	3	yw+	yw+	PROPN
ejpam-841	769	4	2xzw	2xzw	PROPN
ejpam-841	769	5	+	+	PROPN
ejpam-841	769	6	3yzw	3yzw	NUM
ejpam-841	769	7	−	−	NOUN
ejpam-841	769	8	z2w	z2w	PROPN
ejpam-841	770	1	+	+	CCONJ
ejpam-841	770	2	yw2	yw2	PROPN
ejpam-841	770	3	,	,	PUNCT
ejpam-841	770	4	det(n(p	det(n(p	PROPN
ejpam-841	770	5	,	,	PUNCT
ejpam-841	770	6	q	q	NOUN
ejpam-841	770	7	,	,	PUNCT
ejpam-841	770	8	r	r	NOUN
ejpam-841	770	9	)	)	PUNCT
ejpam-841	770	10	)	)	PUNCT
ejpam-841	771	1	=	=	PUNCT
ejpam-841	771	2	xz2	xz2	X
ejpam-841	772	1	−	−	PROPN
ejpam-841	772	2	x2w	x2w	PUNCT
ejpam-841	773	1	+	+	PUNCT
ejpam-841	773	2	2x	2x	NUM
ejpam-841	773	3	yw−	yw−	NUM
ejpam-841	773	4	3xzw	3xzw	NUM
ejpam-841	773	5	−	−	PROPN
ejpam-841	773	6	3yzw	3yzw	PROPN
ejpam-841	773	7	+	+	PROPN
ejpam-841	773	8	z2w	z2w	PROPN
ejpam-841	773	9	−	−	PROPN
ejpam-841	773	10	xw2	xw2	NOUN
ejpam-841	774	1	−	−	PROPN
ejpam-841	774	2	yw2	yw2	PROPN
ejpam-841	774	3	are	be	AUX
ejpam-841	774	4	the	the	DET
ejpam-841	774	5	implicit	implicit	ADJ
ejpam-841	774	6	equations	equation	NOUN
ejpam-841	774	7	of	of	ADP
ejpam-841	774	8	the	the	DET
ejpam-841	774	9	curve	curve	NOUN
ejpam-841	774	10	c	c	PROPN
ejpam-841	774	11	(	(	PUNCT
ejpam-841	774	12	see	see	VERB
ejpam-841	774	13	figure	figure	NOUN
ejpam-841	774	14	2	2	NUM
ejpam-841	774	15	)	)	PUNCT
ejpam-841	774	16	.	.	PUNCT
ejpam-841	775	1	figure	figure	NOUN
ejpam-841	775	2	2	2	NUM
ejpam-841	775	3	:	:	PUNCT
ejpam-841	775	4	set	set	VERB
ejpam-841	775	5	-	-	PUNCT
ejpam-841	775	6	theoreti	theoreti	NOUN
ejpam-841	775	7	generators	generator	NOUN
ejpam-841	775	8	of	of	ADP
ejpam-841	775	9	a	a	DET
ejpam-841	775	10	non	non	ADJ
ejpam-841	775	11	-	-	ADJ
ejpam-841	775	12	singular	singular	ADJ
ejpam-841	775	13	rational	rational	ADJ
ejpam-841	775	14	quarti	quarti	PROPN
ejpam-841	775	15	spa	spa	NOUN
ejpam-841	775	16	e	e	NOUN
ejpam-841	775	17	urve	urve	NOUN
ejpam-841	775	18	.	.	PUNCT
ejpam-841	776	1	j.	j.	PROPN
ejpam-841	776	2	hoffman	hoffman	PROPN
ejpam-841	776	3	,	,	PUNCT
ejpam-841	776	4	h.	h.	PROPN
ejpam-841	776	5	wang	wang	PROPN
ejpam-841	776	6	,	,	PUNCT
ejpam-841	776	7	x.	x.	PROPN
ejpam-841	776	8	jia	jia	PROPN
ejpam-841	776	9	,	,	PUNCT
ejpam-841	776	10	r.	r.	PROPN
ejpam-841	776	11	goldman	goldman	PROPN
ejpam-841	776	12	/	/	SYM
ejpam-841	776	13	eur	eur	PROPN
ejpam-841	776	14	.	.	PUNCT
ejpam-841	777	1	j.	j.	PROPN
ejpam-841	777	2	pure	pure	PROPN
ejpam-841	777	3	appl	appl	PROPN
ejpam-841	777	4	.	.	PROPN
ejpam-841	777	5	math	math	PROPN
ejpam-841	777	6	,	,	PUNCT
ejpam-841	777	7	3	3	NUM
ejpam-841	777	8	(	(	PUNCT
ejpam-841	777	9	2010	2010	NUM
ejpam-841	777	10	)	)	PUNCT
ejpam-841	777	11	,	,	PUNCT
ejpam-841	777	12	602	602	NUM
ejpam-841	777	13	-	-	SYM
ejpam-841	777	14	632	632	NUM
ejpam-841	777	15	628	628	NUM
ejpam-841	777	16	4.3	4.3	NUM
ejpam-841	777	17	.	.	PUNCT
ejpam-841	778	1	generators	generator	NOUN
ejpam-841	778	2	for	for	ADP
ejpam-841	778	3	the	the	DET
ejpam-841	778	4	rees	rees	PROPN
ejpam-841	778	5	algebra	algebra	NOUN
ejpam-841	778	6	associated	associate	VERB
ejpam-841	778	7	to	to	ADP
ejpam-841	778	8	rational	rational	ADJ
ejpam-841	778	9	quartic	quartic	ADJ
ejpam-841	778	10	space	space	NOUN
ejpam-841	778	11	curves	curve	NOUN
ejpam-841	778	12	now	now	ADV
ejpam-841	778	13	we	we	PRON
ejpam-841	778	14	can	can	AUX
ejpam-841	778	15	give	give	VERB
ejpam-841	778	16	a	a	DET
ejpam-841	778	17	minimal	minimal	ADJ
ejpam-841	778	18	set	set	NOUN
ejpam-841	778	19	of	of	ADP
ejpam-841	778	20	generators	generator	NOUN
ejpam-841	778	21	for	for	ADP
ejpam-841	778	22	the	the	DET
ejpam-841	778	23	rees	rees	PROPN
ejpam-841	778	24	algebra	algebra	NOUN
ejpam-841	778	25	associated	associate	VERB
ejpam-841	778	26	to	to	ADP
ejpam-841	778	27	a	a	DET
ejpam-841	778	28	rational	rational	ADJ
ejpam-841	778	29	quartic	quartic	ADJ
ejpam-841	778	30	space	space	NOUN
ejpam-841	778	31	curve	curve	NOUN
ejpam-841	778	32	using	use	VERB
ejpam-841	778	33	only	only	ADV
ejpam-841	778	34	the	the	DET
ejpam-841	778	35	µ-basis	µ-basis	NOUN
ejpam-841	778	36	of	of	ADP
ejpam-841	778	37	the	the	DET
ejpam-841	778	38	curve	curve	NOUN
ejpam-841	778	39	.	.	PUNCT
ejpam-841	779	1	theorem	theorem	ADJ
ejpam-841	779	2	8	8	NUM
ejpam-841	779	3	.	.	PUNCT
ejpam-841	780	1	let	let	VERB
ejpam-841	780	2	p	p	PRON
ejpam-841	780	3	,	,	PUNCT
ejpam-841	780	4	q	q	ADJ
ejpam-841	780	5	,	,	PUNCT
ejpam-841	780	6	r	r	NOUN
ejpam-841	780	7	be	be	AUX
ejpam-841	780	8	a	a	DET
ejpam-841	780	9	µ-basis	µ-basis	NOUN
ejpam-841	780	10	for	for	ADP
ejpam-841	780	11	a	a	DET
ejpam-841	780	12	rational	rational	ADJ
ejpam-841	780	13	quartic	quartic	ADJ
ejpam-841	780	14	space	space	NOUN
ejpam-841	780	15	curve	curve	NOUN
ejpam-841	780	16	.	.	PUNCT
ejpam-841	781	1	then	then	ADV
ejpam-841	781	2	:	:	PUNCT
ejpam-841	781	3	1	1	X
ejpam-841	781	4	.	.	X
ejpam-841	781	5	a	a	DET
ejpam-841	781	6	minimal	minimal	ADJ
ejpam-841	781	7	set	set	NOUN
ejpam-841	781	8	of	of	ADP
ejpam-841	781	9	generators	generator	NOUN
ejpam-841	781	10	for	for	ADP
ejpam-841	781	11	the	the	DET
ejpam-841	781	12	kernel	kernel	PROPN
ejpam-841	781	13	k	k	PROPN
ejpam-841	781	14	of	of	ADP
ejpam-841	781	15	the	the	DET
ejpam-841	781	16	rees	rees	PROPN
ejpam-841	781	17	algebra	algebra	NOUN
ejpam-841	781	18	for	for	ADP
ejpam-841	781	19	a	a	DET
ejpam-841	781	20	singular	singular	ADJ
ejpam-841	781	21	rational	rational	ADJ
ejpam-841	781	22	quartic	quartic	ADJ
ejpam-841	781	23	space	space	NOUN
ejpam-841	781	24	curve	curve	NOUN
ejpam-841	781	25	is	be	AUX
ejpam-841	781	26	given	give	VERB
ejpam-841	781	27	by	by	ADP
ejpam-841	781	28	p	p	X
ejpam-841	781	29	,	,	PUNCT
ejpam-841	781	30	q	q	ADJ
ejpam-841	781	31	,	,	PUNCT
ejpam-841	781	32	r	r	NOUN
ejpam-841	781	33	,	,	PUNCT
ejpam-841	781	34	sylvs	sylvs	NOUN
ejpam-841	781	35	,	,	PUNCT
ejpam-841	781	36	t(p	t(p	PROPN
ejpam-841	781	37	,	,	PUNCT
ejpam-841	781	38	q	q	NOUN
ejpam-841	781	39	)	)	PUNCT
ejpam-841	781	40	,	,	PUNCT
ejpam-841	781	41	det(m(p	det(m(p	NOUN
ejpam-841	781	42	,	,	PUNCT
ejpam-841	781	43	q	q	NOUN
ejpam-841	781	44	,	,	PUNCT
ejpam-841	781	45	r	r	NOUN
ejpam-841	781	46	)	)	PUNCT
ejpam-841	781	47	)	)	PUNCT
ejpam-841	782	1	a(x	a(x	PROPN
ejpam-841	782	2	,	,	PUNCT
ejpam-841	782	3	y	y	PROPN
ejpam-841	782	4	,	,	PUNCT
ejpam-841	782	5	z	z	PROPN
ejpam-841	782	6	,	,	PUNCT
ejpam-841	782	7	w	w	PROPN
ejpam-841	782	8	)	)	PUNCT
ejpam-841	782	9	,	,	PUNCT
ejpam-841	782	10	where	where	SCONJ
ejpam-841	782	11	a	a	PRON
ejpam-841	782	12	and	and	CCONJ
ejpam-841	782	13	m	m	NOUN
ejpam-841	782	14	are	be	AUX
ejpam-841	782	15	defined	define	VERB
ejpam-841	782	16	in	in	ADP
ejpam-841	782	17	equations	equation	NOUN
ejpam-841	782	18	(	(	PUNCT
ejpam-841	782	19	10	10	NUM
ejpam-841	782	20	)	)	PUNCT
ejpam-841	782	21	and	and	CCONJ
ejpam-841	782	22	(	(	PUNCT
ejpam-841	782	23	11	11	NUM
ejpam-841	782	24	)	)	PUNCT
ejpam-841	782	25	.	.	PUNCT
ejpam-841	783	1	2	2	X
ejpam-841	783	2	.	.	X
ejpam-841	783	3	a	a	DET
ejpam-841	783	4	minimal	minimal	ADJ
ejpam-841	783	5	set	set	NOUN
ejpam-841	783	6	of	of	ADP
ejpam-841	783	7	generators	generator	NOUN
ejpam-841	783	8	for	for	ADP
ejpam-841	783	9	the	the	DET
ejpam-841	783	10	kernel	kernel	PROPN
ejpam-841	783	11	k	k	PROPN
ejpam-841	783	12	of	of	ADP
ejpam-841	783	13	the	the	DET
ejpam-841	783	14	rees	rees	PROPN
ejpam-841	783	15	algebra	algebra	NOUN
ejpam-841	783	16	for	for	ADP
ejpam-841	783	17	a	a	DET
ejpam-841	783	18	non	non	ADJ
ejpam-841	783	19	-	-	ADJ
ejpam-841	783	20	singular	singular	ADJ
ejpam-841	783	21	rational	rational	ADJ
ejpam-841	783	22	quartic	quartic	ADJ
ejpam-841	783	23	space	space	NOUN
ejpam-841	783	24	curve	curve	NOUN
ejpam-841	783	25	is	be	AUX
ejpam-841	783	26	given	give	VERB
ejpam-841	783	27	by	by	ADP
ejpam-841	783	28	p	p	X
ejpam-841	783	29	,	,	PUNCT
ejpam-841	783	30	q	q	ADJ
ejpam-841	783	31	,	,	PUNCT
ejpam-841	783	32	r	r	NOUN
ejpam-841	783	33	,	,	PUNCT
ejpam-841	783	34	sylvs	sylvs	NOUN
ejpam-841	783	35	,	,	PUNCT
ejpam-841	783	36	t(p	t(p	PROPN
ejpam-841	783	37	,	,	PUNCT
ejpam-841	783	38	r	r	NOUN
ejpam-841	783	39	)	)	PUNCT
ejpam-841	783	40	,	,	PUNCT
ejpam-841	783	41	sylvs	sylvs	PROPN
ejpam-841	783	42	,	,	PUNCT
ejpam-841	783	43	t(q	t(q	PROPN
ejpam-841	783	44	,	,	PUNCT
ejpam-841	783	45	r	r	NOUN
ejpam-841	783	46	)	)	PUNCT
ejpam-841	783	47	,	,	PUNCT
ejpam-841	783	48	res(p	res(p	PROPN
ejpam-841	783	49	,	,	PUNCT
ejpam-841	783	50	q	q	NOUN
ejpam-841	783	51	)	)	PUNCT
ejpam-841	783	52	,	,	PUNCT
ejpam-841	783	53	res(p	res(p	PROPN
ejpam-841	783	54	,	,	PUNCT
ejpam-841	783	55	r	r	NOUN
ejpam-841	783	56	)	)	PUNCT
ejpam-841	783	57	,	,	PUNCT
ejpam-841	783	58	res(q	res(q	PROPN
ejpam-841	783	59	,	,	PUNCT
ejpam-841	783	60	r	r	NOUN
ejpam-841	783	61	)	)	PUNCT
ejpam-841	783	62	,	,	PUNCT
ejpam-841	783	63	det(n(p	det(n(p	PROPN
ejpam-841	783	64	,	,	PUNCT
ejpam-841	783	65	q	q	NOUN
ejpam-841	783	66	,	,	PUNCT
ejpam-841	783	67	r	r	NOUN
ejpam-841	783	68	)	)	PUNCT
ejpam-841	783	69	)	)	PUNCT
ejpam-841	783	70	,	,	PUNCT
ejpam-841	783	71	where	where	SCONJ
ejpam-841	783	72	n(p	n(p	PROPN
ejpam-841	783	73	,	,	PUNCT
ejpam-841	783	74	q	q	NOUN
ejpam-841	783	75	,	,	PUNCT
ejpam-841	783	76	r	r	NOUN
ejpam-841	783	77	)	)	PUNCT
ejpam-841	783	78	is	be	AUX
ejpam-841	783	79	defined	define	VERB
ejpam-841	783	80	in	in	ADP
ejpam-841	783	81	equation	equation	NOUN
ejpam-841	783	82	(	(	PUNCT
ejpam-841	783	83	12	12	NUM
ejpam-841	783	84	)	)	PUNCT
ejpam-841	783	85	.	.	PUNCT
ejpam-841	784	1	proof	proof	NOUN
ejpam-841	784	2	.	.	PUNCT
ejpam-841	785	1	we	we	PRON
ejpam-841	785	2	will	will	AUX
ejpam-841	785	3	prove	prove	VERB
ejpam-841	785	4	the	the	DET
ejpam-841	785	5	claim	claim	NOUN
ejpam-841	785	6	by	by	ADP
ejpam-841	785	7	comparing	compare	VERB
ejpam-841	785	8	the	the	DET
ejpam-841	785	9	generators	generator	NOUN
ejpam-841	785	10	listed	list	VERB
ejpam-841	785	11	above	above	ADV
ejpam-841	785	12	against	against	ADP
ejpam-841	785	13	the	the	DET
ejpam-841	785	14	generators	generator	NOUN
ejpam-841	785	15	described	describe	VERB
ejpam-841	785	16	in	in	ADP
ejpam-841	785	17	theorems	theorem	NOUN
ejpam-841	785	18	1	1	NUM
ejpam-841	785	19	and	and	CCONJ
ejpam-841	785	20	3	3	NUM
ejpam-841	785	21	.	.	X
ejpam-841	786	1	first	first	ADV
ejpam-841	786	2	,	,	PUNCT
ejpam-841	786	3	we	we	PRON
ejpam-841	786	4	note	note	VERB
ejpam-841	786	5	that	that	SCONJ
ejpam-841	786	6	the	the	DET
ejpam-841	786	7	µ-basis	µ-basis	NOUN
ejpam-841	786	8	elements	element	NOUN
ejpam-841	786	9	p	p	X
ejpam-841	786	10	,	,	PUNCT
ejpam-841	786	11	q	q	ADJ
ejpam-841	786	12	,	,	PUNCT
ejpam-841	786	13	r	r	NOUN
ejpam-841	786	14	and	and	CCONJ
ejpam-841	786	15	sylvs	sylvs	NOUN
ejpam-841	786	16	,	,	PUNCT
ejpam-841	786	17	t(p	t(p	PROPN
ejpam-841	786	18	,	,	PUNCT
ejpam-841	786	19	q	q	NOUN
ejpam-841	786	20	)	)	PUNCT
ejpam-841	786	21	=	=	SYM
ejpam-841	786	22	res(p	res(p	PROPN
ejpam-841	786	23	,	,	PUNCT
ejpam-841	786	24	q	q	NOUN
ejpam-841	786	25	)	)	PUNCT
ejpam-841	786	26	are	be	AUX
ejpam-841	786	27	among	among	ADP
ejpam-841	786	28	the	the	DET
ejpam-841	786	29	generators	generator	NOUN
ejpam-841	786	30	in	in	ADP
ejpam-841	786	31	both	both	DET
ejpam-841	786	32	theorems	theorem	NOUN
ejpam-841	786	33	.	.	PUNCT
ejpam-841	787	1	we	we	PRON
ejpam-841	787	2	will	will	AUX
ejpam-841	787	3	focus	focus	VERB
ejpam-841	787	4	therefore	therefore	ADV
ejpam-841	787	5	on	on	ADP
ejpam-841	787	6	the	the	DET
ejpam-841	787	7	other	other	ADJ
ejpam-841	787	8	generators	generator	NOUN
ejpam-841	787	9	.	.	PUNCT
ejpam-841	788	1	for	for	ADP
ejpam-841	788	2	singular	singular	ADJ
ejpam-841	788	3	quartic	quartic	ADJ
ejpam-841	788	4	space	space	NOUN
ejpam-841	788	5	curves	curve	NOUN
ejpam-841	788	6	,	,	PUNCT
ejpam-841	788	7	recall	recall	VERB
ejpam-841	788	8	that	that	SCONJ
ejpam-841	788	9	by	by	ADP
ejpam-841	788	10	lemma	lemma	PROPN
ejpam-841	788	11	2	2	NUM
ejpam-841	788	12	there	there	PRON
ejpam-841	788	13	is	be	VERB
ejpam-841	788	14	a	a	DET
ejpam-841	788	15	linear	linear	ADJ
ejpam-841	788	16	transformation	transformation	NOUN
ejpam-841	788	17	on	on	ADP
ejpam-841	788	18	the	the	DET
ejpam-841	788	19	µ-basis	µ-basis	NOUN
ejpam-841	788	20	elements	element	NOUN
ejpam-841	788	21	p	p	X
ejpam-841	788	22	,	,	PUNCT
ejpam-841	788	23	q	q	X
ejpam-841	788	24	,	,	PUNCT
ejpam-841	788	25	and	and	CCONJ
ejpam-841	788	26	a	a	DET
ejpam-841	788	27	projective	projective	ADJ
ejpam-841	788	28	change	change	NOUN
ejpam-841	788	29	of	of	ADP
ejpam-841	788	30	coordinates	coordinate	NOUN
ejpam-841	788	31	so	so	SCONJ
ejpam-841	788	32	that	that	SCONJ
ejpam-841	788	33	p	p	X
ejpam-841	788	34	=	=	X
ejpam-841	788	35	ys−	ys−	PUNCT
ejpam-841	788	36	x	x	SYM
ejpam-841	788	37	t	t	PROPN
ejpam-841	788	38	,	,	PUNCT
ejpam-841	788	39	q	q	NOUN
ejpam-841	788	40	=	=	SYM
ejpam-841	788	41	zs−	zs−	NUM
ejpam-841	788	42	y	y	PROPN
ejpam-841	788	43	t	t	PROPN
ejpam-841	788	44	,	,	PUNCT
ejpam-841	788	45	and	and	CCONJ
ejpam-841	788	46	the	the	DET
ejpam-841	788	47	singular	singular	ADJ
ejpam-841	788	48	point	point	NOUN
ejpam-841	788	49	is	be	AUX
ejpam-841	788	50	located	locate	VERB
ejpam-841	788	51	at	at	ADP
ejpam-841	788	52	(	(	PUNCT
ejpam-841	788	53	0,0,0,1	0,0,0,1	NOUN
ejpam-841	788	54	)	)	PUNCT
ejpam-841	788	55	.	.	PUNCT
ejpam-841	789	1	now	now	ADV
ejpam-841	789	2	the	the	DET
ejpam-841	789	3	axial	axial	ADJ
ejpam-841	789	4	plane	plane	NOUN
ejpam-841	789	5	determined	determine	VERB
ejpam-841	789	6	by	by	ADP
ejpam-841	789	7	the	the	DET
ejpam-841	789	8	axes	axis	NOUN
ejpam-841	789	9	of	of	ADP
ejpam-841	789	10	p	p	PROPN
ejpam-841	789	11	and	and	CCONJ
ejpam-841	789	12	q	q	NOUN
ejpam-841	789	13	is	be	AUX
ejpam-841	789	14	defined	define	VERB
ejpam-841	789	15	by	by	ADP
ejpam-841	789	16	a(x	a(x	NOUN
ejpam-841	789	17	,	,	PUNCT
ejpam-841	789	18	y	y	PROPN
ejpam-841	789	19	,	,	PUNCT
ejpam-841	789	20	z	z	PROPN
ejpam-841	789	21	,	,	PUNCT
ejpam-841	789	22	w	w	PROPN
ejpam-841	789	23	)	)	PUNCT
ejpam-841	789	24	=	=	SYM
ejpam-841	790	1	y	y	PROPN
ejpam-841	790	2	=	=	SYM
ejpam-841	790	3	0	0	X
ejpam-841	790	4	.	.	PUNCT
ejpam-841	791	1	in	in	ADP
ejpam-841	791	2	addition	addition	NOUN
ejpam-841	791	3	,	,	PUNCT
ejpam-841	791	4	notice	notice	VERB
ejpam-841	791	5	that	that	SCONJ
ejpam-841	791	6	p(1,0	p(1,0	NOUN
ejpam-841	791	7	)	)	PUNCT
ejpam-841	791	8	=	=	SYM
ejpam-841	791	9	y	y	PROPN
ejpam-841	791	10	and	and	CCONJ
ejpam-841	791	11	q(0,1	q(0,1	NOUN
ejpam-841	791	12	)	)	PUNCT
ejpam-841	791	13	=	=	VERB
ejpam-841	791	14	−y	−y	PROPN
ejpam-841	791	15	.	.	PUNCT
ejpam-841	792	1	therefore	therefore	ADV
ejpam-841	792	2	,	,	PUNCT
ejpam-841	792	3	since	since	SCONJ
ejpam-841	792	4	the	the	DET
ejpam-841	792	5	µ-basis	µ-basis	NOUN
ejpam-841	792	6	elements	element	NOUN
ejpam-841	792	7	p(s	p(s	NOUN
ejpam-841	792	8	,	,	PUNCT
ejpam-841	792	9	t	t	PROPN
ejpam-841	792	10	)	)	PUNCT
ejpam-841	792	11	and	and	CCONJ
ejpam-841	792	12	q(s	q(s	PROPN
ejpam-841	792	13	,	,	PUNCT
ejpam-841	792	14	t	t	PROPN
ejpam-841	792	15	)	)	PUNCT
ejpam-841	792	16	follow	follow	VERB
ejpam-841	792	17	the	the	DET
ejpam-841	792	18	curve	curve	NOUN
ejpam-841	792	19	f(s	f(s	PROPN
ejpam-841	792	20	,	,	PUNCT
ejpam-841	792	21	t	t	PROPN
ejpam-841	792	22	)	)	PUNCT
ejpam-841	792	23	,	,	PUNCT
ejpam-841	792	24	the	the	DET
ejpam-841	792	25	quartic	quartic	ADJ
ejpam-841	792	26	space	space	NOUN
ejpam-841	792	27	curve	curve	NOUN
ejpam-841	792	28	f(s	f(s	PROPN
ejpam-841	792	29	,	,	PUNCT
ejpam-841	792	30	t	t	PROPN
ejpam-841	792	31	)	)	PUNCT
ejpam-841	792	32	intersects	intersect	VERB
ejpam-841	792	33	the	the	DET
ejpam-841	792	34	axial	axial	ADJ
ejpam-841	792	35	plane	plane	NOUN
ejpam-841	792	36	y	y	NOUN
ejpam-841	792	37	=	=	NOUN
ejpam-841	792	38	0	0	PROPN
ejpam-841	793	1	at	at	ADP
ejpam-841	793	2	the	the	DET
ejpam-841	793	3	two	two	NUM
ejpam-841	793	4	points	point	NOUN
ejpam-841	793	5	f(0,1	f(0,1	NOUN
ejpam-841	793	6	)	)	PUNCT
ejpam-841	793	7	and	and	CCONJ
ejpam-841	793	8	f(1,0	f(1,0	NOUN
ejpam-841	793	9	)	)	PUNCT
ejpam-841	793	10	.	.	PUNCT
ejpam-841	794	1	hence	hence	ADV
ejpam-841	794	2	st	st	PROPN
ejpam-841	794	3	must	must	AUX
ejpam-841	794	4	be	be	AUX
ejpam-841	794	5	a	a	DET
ejpam-841	794	6	factor	factor	NOUN
ejpam-841	794	7	of	of	ADP
ejpam-841	794	8	y	y	PROPN
ejpam-841	794	9	,	,	PUNCT
ejpam-841	794	10	so	so	ADV
ejpam-841	794	11	y	y	NOUN
ejpam-841	794	12	=	=	PUNCT
ejpam-841	794	13	stβ	stβ	PROPN
ejpam-841	794	14	,	,	PUNCT
ejpam-841	794	15	where	where	SCONJ
ejpam-841	794	16	β	β	PROPN
ejpam-841	794	17	is	be	AUX
ejpam-841	794	18	a	a	DET
ejpam-841	794	19	homogenous	homogenous	ADJ
ejpam-841	794	20	form	form	NOUN
ejpam-841	794	21	of	of	ADP
ejpam-841	794	22	degree	degree	NOUN
ejpam-841	794	23	2	2	NUM
ejpam-841	794	24	in	in	ADP
ejpam-841	794	25	s	s	PROPN
ejpam-841	794	26	,	,	PUNCT
ejpam-841	794	27	t.	t.	PROPN
ejpam-841	794	28	moreover	moreover	ADV
ejpam-841	794	29	,	,	PUNCT
ejpam-841	794	30	since	since	SCONJ
ejpam-841	794	31	the	the	DET
ejpam-841	794	32	two	two	NUM
ejpam-841	794	33	µ-basis	µ-basis	NOUN
ejpam-841	794	34	elements	element	NOUN
ejpam-841	794	35	p	p	X
ejpam-841	794	36	=	=	X
ejpam-841	794	37	ys	ys	NOUN
ejpam-841	794	38	−	−	PROPN
ejpam-841	794	39	x	x	SYM
ejpam-841	794	40	t	t	PROPN
ejpam-841	794	41	,	,	PUNCT
ejpam-841	794	42	q	q	NOUN
ejpam-841	795	1	=	=	PUNCT
ejpam-841	795	2	zs	zs	PROPN
ejpam-841	795	3	−	−	PROPN
ejpam-841	795	4	y	y	PROPN
ejpam-841	795	5	t	t	PROPN
ejpam-841	795	6	follow	follow	VERB
ejpam-841	795	7	the	the	DET
ejpam-841	795	8	curve	curve	NOUN
ejpam-841	795	9	f(s	f(s	PROPN
ejpam-841	795	10	,	,	PUNCT
ejpam-841	795	11	t	t	PROPN
ejpam-841	795	12	)	)	PUNCT
ejpam-841	795	13	,	,	PUNCT
ejpam-841	795	14	we	we	PRON
ejpam-841	795	15	conclude	conclude	VERB
ejpam-841	795	16	that	that	SCONJ
ejpam-841	795	17	on	on	ADP
ejpam-841	795	18	the	the	DET
ejpam-841	795	19	curve	curve	NOUN
ejpam-841	795	20	f(s	f(s	PROPN
ejpam-841	795	21	,	,	PUNCT
ejpam-841	795	22	t	t	PROPN
ejpam-841	795	23	):	):	PUNCT
ejpam-841	795	24	x	x	X
ejpam-841	795	25	=	=	SYM
ejpam-841	795	26	s2β	s2β	X
ejpam-841	795	27	,	,	PUNCT
ejpam-841	795	28	y	y	PROPN
ejpam-841	795	29	=	=	PUNCT
ejpam-841	795	30	stβ	stβ	PROPN
ejpam-841	795	31	,	,	PUNCT
ejpam-841	795	32	z	z	PROPN
ejpam-841	795	33	=	=	PUNCT
ejpam-841	795	34	t2β	t2β	NOUN
ejpam-841	795	35	,	,	PUNCT
ejpam-841	795	36	where	where	SCONJ
ejpam-841	795	37	β	β	PROPN
ejpam-841	795	38	is	be	AUX
ejpam-841	795	39	a	a	DET
ejpam-841	795	40	homogeneous	homogeneous	ADJ
ejpam-841	795	41	form	form	NOUN
ejpam-841	795	42	in	in	ADP
ejpam-841	795	43	s	s	PROPN
ejpam-841	795	44	,	,	PUNCT
ejpam-841	795	45	t	t	PROPN
ejpam-841	795	46	of	of	ADP
ejpam-841	795	47	degree	degree	NOUN
ejpam-841	795	48	2	2	NUM
ejpam-841	795	49	.	.	PUNCT
ejpam-841	796	1	(	(	PUNCT
ejpam-841	796	2	13	13	NUM
ejpam-841	796	3	)	)	PUNCT
ejpam-841	796	4	in	in	ADP
ejpam-841	796	5	fact	fact	NOUN
ejpam-841	796	6	,	,	PUNCT
ejpam-841	796	7	the	the	DET
ejpam-841	796	8	roots	root	NOUN
ejpam-841	796	9	of	of	ADP
ejpam-841	796	10	β(s	β(s	PROPN
ejpam-841	796	11	,	,	PUNCT
ejpam-841	796	12	t	t	PROPN
ejpam-841	796	13	)	)	PUNCT
ejpam-841	796	14	=	=	SYM
ejpam-841	797	1	0	0	NUM
ejpam-841	797	2	are	be	AUX
ejpam-841	797	3	the	the	DET
ejpam-841	797	4	two	two	NUM
ejpam-841	797	5	parameters	parameter	NOUN
ejpam-841	797	6	corresponding	correspond	VERB
ejpam-841	797	7	to	to	ADP
ejpam-841	797	8	the	the	DET
ejpam-841	797	9	singular	singular	ADJ
ejpam-841	797	10	point	point	NOUN
ejpam-841	797	11	(	(	PUNCT
ejpam-841	797	12	0,0,0,1	0,0,0,1	NOUN
ejpam-841	797	13	)	)	PUNCT
ejpam-841	797	14	on	on	ADP
ejpam-841	797	15	the	the	DET
ejpam-841	797	16	curve	curve	NOUN
ejpam-841	797	17	f(s	f(s	PROPN
ejpam-841	797	18	,	,	PUNCT
ejpam-841	797	19	t	t	PROPN
ejpam-841	797	20	)	)	PUNCT
ejpam-841	797	21	.	.	PUNCT
ejpam-841	798	1	we	we	PRON
ejpam-841	798	2	will	will	AUX
ejpam-841	798	3	study	study	VERB
ejpam-841	798	4	the	the	DET
ejpam-841	798	5	generator	generator	NOUN
ejpam-841	798	6	det(m(p	det(m(p	PROPN
ejpam-841	798	7	,	,	PUNCT
ejpam-841	798	8	q	q	NOUN
ejpam-841	798	9	,	,	PUNCT
ejpam-841	798	10	r))/a(x	r))/a(x	VERB
ejpam-841	798	11	,	,	PUNCT
ejpam-841	798	12	y	y	PROPN
ejpam-841	798	13	,	,	PUNCT
ejpam-841	798	14	z	z	PROPN
ejpam-841	798	15	,	,	PUNCT
ejpam-841	798	16	w	w	PROPN
ejpam-841	798	17	)	)	PUNCT
ejpam-841	798	18	by	by	ADP
ejpam-841	798	19	investigating	investigate	VERB
ejpam-841	798	20	the	the	DET
ejpam-841	798	21	roots	root	NOUN
ejpam-841	798	22	of	of	ADP
ejpam-841	798	23	the	the	DET
ejpam-841	798	24	polynomial	polynomial	ADJ
ejpam-841	798	25	β(s	β(s	PROPN
ejpam-841	798	26	,	,	PUNCT
ejpam-841	798	27	t	t	PROPN
ejpam-841	798	28	)	)	PUNCT
ejpam-841	798	29	=	=	SYM
ejpam-841	798	30	0	0	NUM
ejpam-841	798	31	in	in	ADP
ejpam-841	798	32	the	the	DET
ejpam-841	798	33	following	following	ADJ
ejpam-841	798	34	three	three	NUM
ejpam-841	798	35	cases	case	NOUN
ejpam-841	798	36	.	.	PUNCT
ejpam-841	799	1	case	case	NOUN
ejpam-841	799	2	1	1	NUM
ejpam-841	799	3	:	:	PUNCT
ejpam-841	799	4	neither	neither	CCONJ
ejpam-841	799	5	(	(	PUNCT
ejpam-841	799	6	0,1	0,1	NUM
ejpam-841	799	7	)	)	PUNCT
ejpam-841	799	8	nor	nor	CCONJ
ejpam-841	799	9	(	(	PUNCT
ejpam-841	799	10	1,0	1,0	NUM
ejpam-841	799	11	)	)	PUNCT
ejpam-841	799	12	is	be	AUX
ejpam-841	799	13	a	a	DET
ejpam-841	799	14	root	root	NOUN
ejpam-841	799	15	of	of	ADP
ejpam-841	799	16	β(s	β(s	PROPN
ejpam-841	799	17	,	,	PUNCT
ejpam-841	799	18	t	t	PROPN
ejpam-841	799	19	)	)	PUNCT
ejpam-841	799	20	=	=	SYM
ejpam-841	800	1	0	0	X
ejpam-841	800	2	.	.	PUNCT
ejpam-841	800	3	then	then	ADV
ejpam-841	800	4	f(0,1	f(0,1	PROPN
ejpam-841	800	5	)	)	PUNCT
ejpam-841	800	6	and	and	CCONJ
ejpam-841	800	7	f(1,0	f(1,0	NOUN
ejpam-841	800	8	)	)	PUNCT
ejpam-841	800	9	are	be	AUX
ejpam-841	800	10	both	both	DET
ejpam-841	800	11	nonsingular	nonsingular	ADJ
ejpam-841	800	12	points	point	NOUN
ejpam-841	800	13	on	on	ADP
ejpam-841	800	14	the	the	DET
ejpam-841	800	15	curve	curve	NOUN
ejpam-841	800	16	f(s	f(s	PROPN
ejpam-841	800	17	,	,	PUNCT
ejpam-841	800	18	t	t	PROPN
ejpam-841	800	19	)	)	PUNCT
ejpam-841	800	20	.	.	PUNCT
ejpam-841	801	1	setting	set	VERB
ejpam-841	801	2	(	(	PUNCT
ejpam-841	801	3	s1	s1	NOUN
ejpam-841	801	4	,	,	PUNCT
ejpam-841	801	5	t1	t1	NOUN
ejpam-841	801	6	)	)	PUNCT
ejpam-841	801	7	=	=	SYM
ejpam-841	801	8	(	(	PUNCT
ejpam-841	801	9	0,1	0,1	NUM
ejpam-841	801	10	)	)	PUNCT
ejpam-841	801	11	and	and	CCONJ
ejpam-841	801	12	(	(	PUNCT
ejpam-841	801	13	s2	s2	PROPN
ejpam-841	801	14	,	,	PUNCT
ejpam-841	801	15	t2	t2	NOUN
ejpam-841	801	16	)	)	PUNCT
ejpam-841	801	17	=	=	PUNCT
ejpam-841	801	18	(	(	PUNCT
ejpam-841	801	19	1,0	1,0	NUM
ejpam-841	801	20	)	)	PUNCT
ejpam-841	801	21	in	in	ADP
ejpam-841	801	22	equation	equation	NOUN
ejpam-841	801	23	(	(	PUNCT
ejpam-841	801	24	11	11	NUM
ejpam-841	801	25	)	)	PUNCT
ejpam-841	801	26	yields	yield	NOUN
ejpam-841	801	27	:	:	PUNCT
ejpam-841	801	28	det(m(p	det(m(p	NOUN
ejpam-841	801	29	,	,	PUNCT
ejpam-841	801	30	q	q	NOUN
ejpam-841	801	31	,	,	PUNCT
ejpam-841	801	32	r	r	NOUN
ejpam-841	801	33	)	)	PUNCT
ejpam-841	801	34	)	)	PUNCT
ejpam-841	802	1	=	=	SYM
ejpam-841	802	2	det	det	PROPN
ejpam-841	802	3			PROPN
ejpam-841	802	4			NOUN
ejpam-841	802	5			NOUN
ejpam-841	802	6	y	y	PROPN
ejpam-841	802	7	−x	−x	VERB
ejpam-841	802	8	0	0	NUM
ejpam-841	802	9	0	0	NUM
ejpam-841	803	1	−z	−z	NOUN
ejpam-841	803	2	y	y	PROPN
ejpam-841	803	3	r2	r2	PROPN
ejpam-841	803	4	r1	r1	PROPN
ejpam-841	803	5	r0	r0	PROPN
ejpam-841	803	6			PROPN
ejpam-841	803	7			VERB
ejpam-841	803	8			PUNCT
ejpam-841	804	1	=	=	SYM
ejpam-841	804	2	−r2	−r2	PROPN
ejpam-841	804	3	x	x	SYM
ejpam-841	804	4	y	y	PROPN
ejpam-841	804	5	−	−	PROPN
ejpam-841	804	6	r1	r1	PROPN
ejpam-841	804	7	y2	y2	PROPN
ejpam-841	804	8	−	−	PROPN
ejpam-841	804	9	r0	r0	PROPN
ejpam-841	804	10	yz	yz	PROPN
ejpam-841	804	11	;	;	PUNCT
ejpam-841	804	12	j.	j.	PROPN
ejpam-841	804	13	hoffman	hoffman	PROPN
ejpam-841	804	14	,	,	PUNCT
ejpam-841	804	15	h.	h.	PROPN
ejpam-841	804	16	wang	wang	PROPN
ejpam-841	804	17	,	,	PUNCT
ejpam-841	804	18	x.	x.	PROPN
ejpam-841	804	19	jia	jia	PROPN
ejpam-841	804	20	,	,	PUNCT
ejpam-841	804	21	r.	r.	PROPN
ejpam-841	804	22	goldman	goldman	PROPN
ejpam-841	804	23	/	/	SYM
ejpam-841	804	24	eur	eur	PROPN
ejpam-841	804	25	.	.	PUNCT
ejpam-841	805	1	j.	j.	PROPN
ejpam-841	805	2	pure	pure	PROPN
ejpam-841	805	3	appl	appl	PROPN
ejpam-841	805	4	.	.	PROPN
ejpam-841	805	5	math	math	PROPN
ejpam-841	805	6	,	,	PUNCT
ejpam-841	805	7	3	3	NUM
ejpam-841	805	8	(	(	PUNCT
ejpam-841	805	9	2010	2010	NUM
ejpam-841	805	10	)	)	PUNCT
ejpam-841	805	11	,	,	PUNCT
ejpam-841	805	12	602	602	NUM
ejpam-841	805	13	-	-	SYM
ejpam-841	805	14	632	632	NUM
ejpam-841	805	15	629	629	NUM
ejpam-841	805	16	det(m(p	det(m(p	NOUN
ejpam-841	805	17	,	,	PUNCT
ejpam-841	805	18	q	q	NOUN
ejpam-841	805	19	,	,	PUNCT
ejpam-841	805	20	r	r	NOUN
ejpam-841	805	21	)	)	PUNCT
ejpam-841	805	22	)	)	PUNCT
ejpam-841	806	1	a(x	a(x	PROPN
ejpam-841	806	2	,	,	PUNCT
ejpam-841	806	3	y	y	PROPN
ejpam-841	806	4	,	,	PUNCT
ejpam-841	806	5	z	z	PROPN
ejpam-841	806	6	,	,	PUNCT
ejpam-841	806	7	w	w	PROPN
ejpam-841	806	8	)	)	PUNCT
ejpam-841	806	9	=	=	SYM
ejpam-841	806	10	−[r2	−[r2	PUNCT
ejpam-841	806	11	x	x	PUNCT
ejpam-841	807	1	+	+	CCONJ
ejpam-841	807	2	r1	r1	PROPN
ejpam-841	807	3	y	y	PROPN
ejpam-841	807	4	+	+	CCONJ
ejpam-841	807	5	r0z	r0z	PROPN
ejpam-841	807	6	]	]	X
ejpam-841	807	7	=	=	SYM
ejpam-841	807	8	−x[r2	−x[r2	NUM
ejpam-841	807	9	+	+	SYM
ejpam-841	807	10	r1	r1	PROPN
ejpam-841	807	11	(	(	PUNCT
ejpam-841	807	12	y	y	NOUN
ejpam-841	807	13	x	x	PROPN
ejpam-841	807	14	)	)	PUNCT
ejpam-841	808	1	+	+	CCONJ
ejpam-841	808	2	r0	r0	NOUN
ejpam-841	808	3	(	(	PUNCT
ejpam-841	808	4	z	z	NOUN
ejpam-841	808	5	x	x	NOUN
ejpam-841	808	6	)	)	PUNCT
ejpam-841	808	7	]	]	PUNCT
ejpam-841	809	1	=	=	PUNCT
ejpam-841	809	2	−x	−x	PRON
ejpam-841	809	3	r	r	NOUN
ejpam-841	809	4	′1	′1	PROPN
ejpam-841	809	5	(	(	PUNCT
ejpam-841	809	6	y	y	NOUN
ejpam-841	809	7	x	x	PROPN
ejpam-841	809	8	,	,	PUNCT
ejpam-841	809	9	z	z	NOUN
ejpam-841	809	10	x	x	NOUN
ejpam-841	809	11	)	)	PUNCT
ejpam-841	809	12	.	.	PUNCT
ejpam-841	810	1	thus	thus	ADV
ejpam-841	810	2	,	,	PUNCT
ejpam-841	810	3	the	the	DET
ejpam-841	810	4	above	above	ADJ
ejpam-841	810	5	generators	generator	NOUN
ejpam-841	810	6	are	be	AUX
ejpam-841	810	7	the	the	DET
ejpam-841	810	8	same	same	ADJ
ejpam-841	810	9	as	as	ADP
ejpam-841	810	10	the	the	DET
ejpam-841	810	11	generators	generator	NOUN
ejpam-841	810	12	described	describe	VERB
ejpam-841	810	13	in	in	ADP
ejpam-841	810	14	theorem	theorem	ADJ
ejpam-841	810	15	1	1	NUM
ejpam-841	810	16	.	.	PUNCT
ejpam-841	810	17	case	case	NOUN
ejpam-841	810	18	2	2	NUM
ejpam-841	810	19	:	:	PUNCT
ejpam-841	810	20	either	either	CCONJ
ejpam-841	810	21	(	(	PUNCT
ejpam-841	810	22	0,1	0,1	NUM
ejpam-841	810	23	)	)	PUNCT
ejpam-841	810	24	or	or	CCONJ
ejpam-841	810	25	(	(	PUNCT
ejpam-841	810	26	1,0	1,0	NUM
ejpam-841	810	27	)	)	PUNCT
ejpam-841	810	28	is	be	AUX
ejpam-841	810	29	a	a	DET
ejpam-841	810	30	root	root	NOUN
ejpam-841	810	31	of	of	ADP
ejpam-841	810	32	β(s	β(s	PROPN
ejpam-841	810	33	,	,	PUNCT
ejpam-841	810	34	t	t	PROPN
ejpam-841	810	35	)	)	PUNCT
ejpam-841	810	36	=	=	SYM
ejpam-841	810	37	0	0	NUM
ejpam-841	810	38	,	,	PUNCT
ejpam-841	810	39	but	but	CCONJ
ejpam-841	810	40	β(1,1	β(1,1	NOUN
ejpam-841	810	41	)	)	PUNCT
ejpam-841	810	42	6=	6=	ADP
ejpam-841	810	43	0	0	NUM
ejpam-841	810	44	and	and	CCONJ
ejpam-841	810	45	β(1,−1	β(1,−1	PROPN
ejpam-841	810	46	)	)	PUNCT
ejpam-841	810	47	6=	6=	ADP
ejpam-841	810	48	0	0	X
ejpam-841	810	49	.	.	PUNCT
ejpam-841	811	1	then	then	ADV
ejpam-841	811	2	by	by	ADP
ejpam-841	811	3	equation	equation	NOUN
ejpam-841	811	4	(	(	PUNCT
ejpam-841	811	5	13	13	NUM
ejpam-841	811	6	)	)	PUNCT
ejpam-841	811	7	f(s1	f(s1	NOUN
ejpam-841	811	8	,	,	PUNCT
ejpam-841	811	9	t1	t1	NOUN
ejpam-841	811	10	)	)	PUNCT
ejpam-841	811	11	=	=	SYM
ejpam-841	811	12	f(1,−1	f(1,−1	PROPN
ejpam-841	811	13	)	)	PUNCT
ejpam-841	811	14	=	=	SYM
ejpam-841	811	15	(	(	PUNCT
ejpam-841	811	16	1,−1,1,∗	1,−1,1,∗	NOUN
ejpam-841	811	17	)	)	PUNCT
ejpam-841	811	18	and	and	CCONJ
ejpam-841	811	19	f(s2	f(s2	NOUN
ejpam-841	811	20	,	,	PUNCT
ejpam-841	811	21	t2	t2	NOUN
ejpam-841	811	22	)	)	PUNCT
ejpam-841	811	23	=	=	SYM
ejpam-841	811	24	f(1,1	f(1,1	X
ejpam-841	811	25	)	)	PUNCT
ejpam-841	811	26	=	=	SYM
ejpam-841	811	27	(	(	PUNCT
ejpam-841	811	28	1,1,1,∗	1,1,1,∗	NOUN
ejpam-841	811	29	)	)	PUNCT
ejpam-841	811	30	are	be	AUX
ejpam-841	811	31	two	two	NUM
ejpam-841	811	32	distinct	distinct	ADJ
ejpam-841	811	33	non	non	ADJ
ejpam-841	811	34	-	-	ADJ
ejpam-841	811	35	singular	singular	ADJ
ejpam-841	811	36	points	point	NOUN
ejpam-841	811	37	on	on	ADP
ejpam-841	811	38	the	the	DET
ejpam-841	811	39	curve	curve	NOUN
ejpam-841	811	40	f(s	f(s	PROPN
ejpam-841	811	41	,	,	PUNCT
ejpam-841	811	42	t	t	PROPN
ejpam-841	811	43	)	)	PUNCT
ejpam-841	811	44	.	.	PUNCT
ejpam-841	812	1	therefore	therefore	ADV
ejpam-841	812	2	,	,	PUNCT
ejpam-841	812	3	we	we	PRON
ejpam-841	812	4	may	may	AUX
ejpam-841	812	5	choose	choose	VERB
ejpam-841	812	6	a	a	DET
ejpam-841	812	7	new	new	ADJ
ejpam-841	812	8	µ-basis	µ-basis	NOUN
ejpam-841	812	9	p+	p+	NOUN
ejpam-841	812	10	q	q	NOUN
ejpam-841	812	11	=	=	SYM
ejpam-841	812	12	(	(	PUNCT
ejpam-841	812	13	y	y	PROPN
ejpam-841	812	14	+	+	PROPN
ejpam-841	812	15	z)s−	z)s−	PROPN
ejpam-841	812	16	(	(	PUNCT
ejpam-841	812	17	x	x	SYM
ejpam-841	812	18	+	+	X
ejpam-841	812	19	y)t	y)t	NOUN
ejpam-841	812	20	,	,	PUNCT
ejpam-841	812	21	p−	p−	NOUN
ejpam-841	812	22	q	q	NOUN
ejpam-841	812	23	=	=	PUNCT
ejpam-841	812	24	(	(	PUNCT
ejpam-841	812	25	y	y	PROPN
ejpam-841	812	26	−	−	PROPN
ejpam-841	812	27	z)s−	z)s−	PROPN
ejpam-841	812	28	(	(	PUNCT
ejpam-841	812	29	x	x	NOUN
ejpam-841	812	30	−	−	NOUN
ejpam-841	812	31	y)t	y)t	PROPN
ejpam-841	812	32	,	,	PUNCT
ejpam-841	812	33	r.	r.	PROPN
ejpam-841	812	34	now	now	ADV
ejpam-841	812	35	the	the	DET
ejpam-841	812	36	axial	axial	ADJ
ejpam-841	812	37	moving	move	VERB
ejpam-841	812	38	plane	plane	NOUN
ejpam-841	812	39	is	be	AUX
ejpam-841	812	40	a(x	a(x	PROPN
ejpam-841	812	41	,	,	PUNCT
ejpam-841	812	42	y	y	PROPN
ejpam-841	812	43	,	,	PUNCT
ejpam-841	812	44	z	z	PROPN
ejpam-841	812	45	,	,	PUNCT
ejpam-841	812	46	w	w	NOUN
ejpam-841	812	47	)	)	PUNCT
ejpam-841	812	48	=	=	SYM
ejpam-841	812	49	det	det	PROPN
ejpam-841	812	50			PROPN
ejpam-841	812	51			NOUN
ejpam-841	812	52			NOUN
ejpam-841	812	53			NOUN
ejpam-841	812	54			NOUN
ejpam-841	812	55	x	x	PUNCT
ejpam-841	812	56	y	y	PROPN
ejpam-841	812	57	z	z	PROPN
ejpam-841	812	58	w	w	PROPN
ejpam-841	812	59	0	0	NUM
ejpam-841	812	60	0	0	NUM
ejpam-841	812	61	0	0	NUM
ejpam-841	812	62	1	1	NUM
ejpam-841	812	63	1	1	NUM
ejpam-841	812	64	1	1	NUM
ejpam-841	812	65	1	1	NUM
ejpam-841	812	66	∗	∗	NOUN
ejpam-841	812	67	1	1	NUM
ejpam-841	812	68	−1	−1	NOUN
ejpam-841	812	69	1	1	NUM
ejpam-841	812	70	∗	∗	NOUN
ejpam-841	812	71			NOUN
ejpam-841	812	72			NOUN
ejpam-841	812	73			VERB
ejpam-841	812	74			NOUN
ejpam-841	812	75			PUNCT
ejpam-841	813	1	=	=	PUNCT
ejpam-841	814	1	2(x	2(x	NUM
ejpam-841	814	2	−	−	PROPN
ejpam-841	814	3	z	z	X
ejpam-841	814	4	)	)	PUNCT
ejpam-841	814	5	=	=	SYM
ejpam-841	814	6	0	0	X
ejpam-841	814	7	.	.	PUNCT
ejpam-841	815	1	in	in	ADP
ejpam-841	815	2	this	this	DET
ejpam-841	815	3	case	case	NOUN
ejpam-841	815	4	,	,	PUNCT
ejpam-841	815	5	equation	equation	NOUN
ejpam-841	815	6	(	(	PUNCT
ejpam-841	815	7	11	11	NUM
ejpam-841	815	8	)	)	PUNCT
ejpam-841	815	9	again	again	ADV
ejpam-841	815	10	yields	yield	VERB
ejpam-841	815	11	:	:	PUNCT
ejpam-841	815	12	det(m(p	det(m(p	NOUN
ejpam-841	815	13	,	,	PUNCT
ejpam-841	815	14	q	q	NOUN
ejpam-841	815	15	,	,	PUNCT
ejpam-841	815	16	r	r	NOUN
ejpam-841	815	17	)	)	PUNCT
ejpam-841	815	18	)	)	PUNCT
ejpam-841	816	1	=	=	SYM
ejpam-841	816	2	det	det	PROPN
ejpam-841	816	3			PROPN
ejpam-841	816	4			NOUN
ejpam-841	816	5			NOUN
ejpam-841	816	6	−(y	−(y	NOUN
ejpam-841	816	7	+	+	CCONJ
ejpam-841	816	8	z	z	X
ejpam-841	816	9	)	)	PUNCT
ejpam-841	816	10	x	x	SYM
ejpam-841	817	1	−	−	PROPN
ejpam-841	817	2	z	z	NOUN
ejpam-841	817	3	x	x	PROPN
ejpam-841	818	1	+	+	CCONJ
ejpam-841	818	2	y	y	PROPN
ejpam-841	818	3	y	y	NOUN
ejpam-841	818	4	−	−	PROPN
ejpam-841	818	5	z	z	NOUN
ejpam-841	818	6	−x	−x	NOUN
ejpam-841	818	7	+	+	CCONJ
ejpam-841	818	8	z	z	NOUN
ejpam-841	818	9	x	x	SYM
ejpam-841	818	10	−	−	PROPN
ejpam-841	818	11	y	y	PROPN
ejpam-841	818	12	r2	r2	PROPN
ejpam-841	818	13	r1	r1	PROPN
ejpam-841	818	14	r0	r0	PROPN
ejpam-841	818	15			PROPN
ejpam-841	818	16			VERB
ejpam-841	818	17			PUNCT
ejpam-841	819	1	=	=	PUNCT
ejpam-841	820	1	2(x	2(x	NUM
ejpam-841	820	2	−	−	NOUN
ejpam-841	820	3	z)[r2	z)[r2	NUM
ejpam-841	820	4	x	x	PUNCT
ejpam-841	821	1	+	+	CCONJ
ejpam-841	821	2	r1	r1	PROPN
ejpam-841	821	3	y	y	PROPN
ejpam-841	821	4	+	+	CCONJ
ejpam-841	821	5	r0z	r0z	PROPN
ejpam-841	821	6	]	]	X
ejpam-841	821	7	;	;	PUNCT
ejpam-841	821	8	det(m(p	det(m(p	NOUN
ejpam-841	821	9	,	,	PUNCT
ejpam-841	821	10	q	q	NOUN
ejpam-841	821	11	,	,	PUNCT
ejpam-841	821	12	r	r	NOUN
ejpam-841	821	13	)	)	PUNCT
ejpam-841	821	14	a(x	a(x	PROPN
ejpam-841	821	15	,	,	PUNCT
ejpam-841	821	16	y	y	PROPN
ejpam-841	821	17	,	,	PUNCT
ejpam-841	821	18	z	z	PROPN
ejpam-841	821	19	,	,	PUNCT
ejpam-841	821	20	w	w	PROPN
ejpam-841	821	21	)	)	PUNCT
ejpam-841	821	22	=	=	SYM
ejpam-841	821	23	−[r2	−[r2	PUNCT
ejpam-841	821	24	x	x	PUNCT
ejpam-841	822	1	+	+	CCONJ
ejpam-841	822	2	r1	r1	PROPN
ejpam-841	822	3	y	y	PROPN
ejpam-841	822	4	+	+	CCONJ
ejpam-841	822	5	r0z	r0z	PROPN
ejpam-841	822	6	]	]	X
ejpam-841	822	7	=	=	SYM
ejpam-841	822	8	−x[r2	−x[r2	NUM
ejpam-841	822	9	+	+	SYM
ejpam-841	822	10	r1	r1	PROPN
ejpam-841	822	11	(	(	PUNCT
ejpam-841	822	12	y	y	NOUN
ejpam-841	822	13	x	x	PROPN
ejpam-841	822	14	)	)	PUNCT
ejpam-841	823	1	+	+	CCONJ
ejpam-841	823	2	r0	r0	NOUN
ejpam-841	823	3	(	(	PUNCT
ejpam-841	823	4	z	z	NOUN
ejpam-841	823	5	x	x	NOUN
ejpam-841	823	6	)	)	PUNCT
ejpam-841	823	7	]	]	PUNCT
ejpam-841	824	1	=	=	PUNCT
ejpam-841	824	2	−x	−x	PRON
ejpam-841	824	3	r	r	NOUN
ejpam-841	824	4	′1	′1	PROPN
ejpam-841	824	5	(	(	PUNCT
ejpam-841	824	6	y	y	NOUN
ejpam-841	824	7	x	x	PROPN
ejpam-841	824	8	,	,	PUNCT
ejpam-841	824	9	z	z	NOUN
ejpam-841	824	10	x	x	NOUN
ejpam-841	824	11	)	)	PUNCT
ejpam-841	824	12	.	.	PUNCT
ejpam-841	825	1	thus	thus	ADV
ejpam-841	825	2	,	,	PUNCT
ejpam-841	825	3	the	the	DET
ejpam-841	825	4	above	above	ADJ
ejpam-841	825	5	generators	generator	NOUN
ejpam-841	825	6	are	be	AUX
ejpam-841	825	7	the	the	DET
ejpam-841	825	8	same	same	ADJ
ejpam-841	825	9	as	as	ADP
ejpam-841	825	10	the	the	DET
ejpam-841	825	11	generators	generator	NOUN
ejpam-841	825	12	described	describe	VERB
ejpam-841	825	13	in	in	ADP
ejpam-841	825	14	theorem	theorem	ADJ
ejpam-841	825	15	1	1	NUM
ejpam-841	825	16	.	.	PUNCT
ejpam-841	825	17	case	case	NOUN
ejpam-841	825	18	3	3	NUM
ejpam-841	825	19	:	:	PUNCT
ejpam-841	825	20	either	either	CCONJ
ejpam-841	825	21	(	(	PUNCT
ejpam-841	825	22	0,1	0,1	NUM
ejpam-841	825	23	)	)	PUNCT
ejpam-841	825	24	or	or	CCONJ
ejpam-841	825	25	(	(	PUNCT
ejpam-841	825	26	1,0	1,0	NUM
ejpam-841	825	27	)	)	PUNCT
ejpam-841	825	28	and	and	CCONJ
ejpam-841	825	29	either	either	CCONJ
ejpam-841	825	30	(	(	PUNCT
ejpam-841	825	31	1,1	1,1	NUM
ejpam-841	825	32	)	)	PUNCT
ejpam-841	825	33	or	or	CCONJ
ejpam-841	825	34	(	(	PUNCT
ejpam-841	825	35	1,−1	1,−1	NUM
ejpam-841	825	36	)	)	PUNCT
ejpam-841	825	37	is	be	AUX
ejpam-841	825	38	a	a	DET
ejpam-841	825	39	root	root	NOUN
ejpam-841	825	40	of	of	ADP
ejpam-841	825	41	β(s	β(s	PROPN
ejpam-841	825	42	,	,	PUNCT
ejpam-841	825	43	t	t	PROPN
ejpam-841	825	44	)	)	PUNCT
ejpam-841	825	45	=	=	NOUN
ejpam-841	825	46	0	0	X
ejpam-841	825	47	.	.	PUNCT
ejpam-841	825	48	without	without	ADP
ejpam-841	825	49	loss	loss	NOUN
ejpam-841	825	50	of	of	ADP
ejpam-841	825	51	generality	generality	NOUN
ejpam-841	825	52	,	,	PUNCT
ejpam-841	825	53	assume	assume	VERB
ejpam-841	825	54	that	that	SCONJ
ejpam-841	825	55	β(0,1	β(0,1	ADP
ejpam-841	825	56	)	)	PUNCT
ejpam-841	825	57	6=	6=	ADP
ejpam-841	825	58	0	0	NUM
ejpam-841	825	59	and	and	CCONJ
ejpam-841	825	60	β(1,−1	β(1,−1	PROPN
ejpam-841	825	61	)	)	PUNCT
ejpam-841	825	62	6=	6=	ADP
ejpam-841	826	1	0	0	X
ejpam-841	826	2	.	.	PUNCT
ejpam-841	827	1	then	then	ADV
ejpam-841	827	2	f(1,0	f(1,0	NOUN
ejpam-841	827	3	)	)	PUNCT
ejpam-841	827	4	=	=	SYM
ejpam-841	827	5	f(1,1	f(1,1	X
ejpam-841	827	6	)	)	PUNCT
ejpam-841	827	7	=	=	SYM
ejpam-841	827	8	(	(	PUNCT
ejpam-841	827	9	0,0,0,1	0,0,0,1	NOUN
ejpam-841	827	10	)	)	PUNCT
ejpam-841	827	11	is	be	AUX
ejpam-841	827	12	the	the	DET
ejpam-841	827	13	singular	singular	ADJ
ejpam-841	827	14	point	point	NOUN
ejpam-841	827	15	,	,	PUNCT
ejpam-841	827	16	and	and	CCONJ
ejpam-841	827	17	by	by	ADP
ejpam-841	827	18	equation	equation	NOUN
ejpam-841	827	19	(	(	PUNCT
ejpam-841	827	20	13	13	NUM
ejpam-841	827	21	)	)	PUNCT
ejpam-841	827	22	f(s1	f(s1	NOUN
ejpam-841	827	23	,	,	PUNCT
ejpam-841	827	24	t1	t1	NOUN
ejpam-841	827	25	)	)	PUNCT
ejpam-841	827	26	=	=	SYM
ejpam-841	827	27	f(0,1	f(0,1	NOUN
ejpam-841	827	28	)	)	PUNCT
ejpam-841	827	29	=	=	SYM
ejpam-841	827	30	(	(	PUNCT
ejpam-841	827	31	0,0,1,∗	0,0,1,∗	NOUN
ejpam-841	827	32	)	)	PUNCT
ejpam-841	827	33	and	and	CCONJ
ejpam-841	827	34	f(s2	f(s2	NOUN
ejpam-841	827	35	,	,	PUNCT
ejpam-841	827	36	t2	t2	NOUN
ejpam-841	827	37	)	)	PUNCT
ejpam-841	827	38	=	=	SYM
ejpam-841	827	39	f(1,−1	f(1,−1	PROPN
ejpam-841	827	40	)	)	PUNCT
ejpam-841	828	1	=	=	SYM
ejpam-841	828	2	(	(	PUNCT
ejpam-841	828	3	1,−1,1,∗	1,−1,1,∗	NOUN
ejpam-841	828	4	)	)	PUNCT
ejpam-841	828	5	are	be	AUX
ejpam-841	828	6	two	two	NUM
ejpam-841	828	7	distinct	distinct	ADJ
ejpam-841	828	8	non	non	ADJ
ejpam-841	828	9	-	-	ADJ
ejpam-841	828	10	singular	singular	ADJ
ejpam-841	828	11	point	point	NOUN
ejpam-841	828	12	on	on	ADP
ejpam-841	828	13	the	the	DET
ejpam-841	828	14	curve	curve	NOUN
ejpam-841	828	15	f(s	f(s	PROPN
ejpam-841	828	16	,	,	PUNCT
ejpam-841	828	17	t	t	PROPN
ejpam-841	828	18	)	)	PUNCT
ejpam-841	828	19	.	.	PUNCT
ejpam-841	829	1	therefore	therefore	ADV
ejpam-841	829	2	,	,	PUNCT
ejpam-841	829	3	we	we	PRON
ejpam-841	829	4	may	may	AUX
ejpam-841	829	5	choose	choose	VERB
ejpam-841	829	6	a	a	DET
ejpam-841	829	7	new	new	ADJ
ejpam-841	829	8	µ-basis	µ-basis	NOUN
ejpam-841	829	9	p	p	NOUN
ejpam-841	829	10	=	=	X
ejpam-841	829	11	ys−	ys−	PUNCT
ejpam-841	829	12	x	x	SYM
ejpam-841	829	13	t	t	PROPN
ejpam-841	829	14	,	,	PUNCT
ejpam-841	829	15	p+	p+	NOUN
ejpam-841	829	16	q	q	NOUN
ejpam-841	829	17	=	=	SYM
ejpam-841	829	18	(	(	PUNCT
ejpam-841	829	19	y	y	PROPN
ejpam-841	829	20	+	+	PROPN
ejpam-841	829	21	z)s−	z)s−	PROPN
ejpam-841	829	22	(	(	PUNCT
ejpam-841	829	23	x	x	SYM
ejpam-841	829	24	+	+	X
ejpam-841	829	25	y)t	y)t	PROPN
ejpam-841	829	26	,	,	PUNCT
ejpam-841	829	27	r.	r.	PROPN
ejpam-841	829	28	now	now	ADV
ejpam-841	829	29	the	the	DET
ejpam-841	829	30	axial	axial	ADJ
ejpam-841	829	31	moving	move	VERB
ejpam-841	829	32	plane	plane	NOUN
ejpam-841	829	33	is	be	AUX
ejpam-841	829	34	a(x	a(x	PROPN
ejpam-841	829	35	,	,	PUNCT
ejpam-841	829	36	y	y	PROPN
ejpam-841	829	37	,	,	PUNCT
ejpam-841	829	38	z	z	PROPN
ejpam-841	829	39	,	,	PUNCT
ejpam-841	829	40	w	w	NOUN
ejpam-841	829	41	)	)	PUNCT
ejpam-841	829	42	=	=	SYM
ejpam-841	829	43	det	det	PROPN
ejpam-841	829	44			PROPN
ejpam-841	829	45			NOUN
ejpam-841	829	46			NOUN
ejpam-841	829	47			NOUN
ejpam-841	829	48			NOUN
ejpam-841	829	49	x	x	PUNCT
ejpam-841	829	50	y	y	PROPN
ejpam-841	829	51	z	z	PROPN
ejpam-841	829	52	w	w	PROPN
ejpam-841	829	53	0	0	NUM
ejpam-841	829	54	0	0	NUM
ejpam-841	829	55	0	0	NUM
ejpam-841	829	56	1	1	NUM
ejpam-841	829	57	0	0	NUM
ejpam-841	829	58	0	0	NUM
ejpam-841	829	59	1	1	NUM
ejpam-841	829	60	∗	∗	NOUN
ejpam-841	829	61	1	1	NUM
ejpam-841	829	62	−1	−1	NOUN
ejpam-841	829	63	1	1	NUM
ejpam-841	829	64	∗	∗	NOUN
ejpam-841	829	65			NOUN
ejpam-841	829	66			NOUN
ejpam-841	829	67			VERB
ejpam-841	829	68			NOUN
ejpam-841	829	69			PUNCT
ejpam-841	830	1	=	=	PUNCT
ejpam-841	830	2	x	x	PUNCT
ejpam-841	831	1	+	+	NUM
ejpam-841	831	2	y	y	PROPN
ejpam-841	831	3	=	=	SYM
ejpam-841	831	4	0	0	X
ejpam-841	831	5	.	.	PUNCT
ejpam-841	832	1	in	in	ADP
ejpam-841	832	2	this	this	DET
ejpam-841	832	3	case	case	NOUN
ejpam-841	832	4	,	,	PUNCT
ejpam-841	832	5	equation	equation	NOUN
ejpam-841	832	6	(	(	PUNCT
ejpam-841	832	7	11	11	NUM
ejpam-841	832	8	)	)	PUNCT
ejpam-841	832	9	again	again	ADV
ejpam-841	832	10	gives	give	VERB
ejpam-841	832	11	:	:	PUNCT
ejpam-841	832	12	det(m(p	det(m(p	NOUN
ejpam-841	832	13	,	,	PUNCT
ejpam-841	832	14	q	q	NOUN
ejpam-841	832	15	,	,	PUNCT
ejpam-841	832	16	r	r	NOUN
ejpam-841	832	17	)	)	PUNCT
ejpam-841	832	18	)	)	PUNCT
ejpam-841	833	1	=	=	SYM
ejpam-841	833	2	det	det	PROPN
ejpam-841	833	3			PROPN
ejpam-841	833	4			NOUN
ejpam-841	833	5			NOUN
ejpam-841	833	6	y	y	PROPN
ejpam-841	833	7	−x	−x	NOUN
ejpam-841	833	8	0	0	PUNCT
ejpam-841	834	1	−(y	−(y	NOUN
ejpam-841	834	2	+	+	CCONJ
ejpam-841	834	3	z	z	X
ejpam-841	834	4	)	)	PUNCT
ejpam-841	834	5	x	x	SYM
ejpam-841	835	1	−	−	PROPN
ejpam-841	835	2	z	z	NOUN
ejpam-841	835	3	x	x	PUNCT
ejpam-841	836	1	+	+	CCONJ
ejpam-841	836	2	y	y	PROPN
ejpam-841	836	3	r2	r2	PROPN
ejpam-841	836	4	r1	r1	PROPN
ejpam-841	836	5	r0	r0	PROPN
ejpam-841	836	6			PROPN
ejpam-841	836	7			VERB
ejpam-841	836	8			PUNCT
ejpam-841	837	1	=	=	PUNCT
ejpam-841	837	2	−(x	−(x	NOUN
ejpam-841	837	3	+	+	CCONJ
ejpam-841	837	4	y)(r2	y)(r2	NOUN
ejpam-841	837	5	x	x	SYM
ejpam-841	837	6	+	+	CCONJ
ejpam-841	837	7	r1	r1	PROPN
ejpam-841	837	8	y	y	PROPN
ejpam-841	837	9	+	+	CCONJ
ejpam-841	837	10	r0z	r0z	PROPN
ejpam-841	837	11	)	)	PUNCT
ejpam-841	837	12	;	;	PUNCT
ejpam-841	837	13	det(m(p	det(m(p	NOUN
ejpam-841	837	14	,	,	PUNCT
ejpam-841	837	15	q	q	NOUN
ejpam-841	837	16	,	,	PUNCT
ejpam-841	837	17	r	r	NOUN
ejpam-841	837	18	)	)	PUNCT
ejpam-841	837	19	)	)	PUNCT
ejpam-841	837	20	a(x	a(x	PROPN
ejpam-841	837	21	,	,	PUNCT
ejpam-841	837	22	y	y	PROPN
ejpam-841	837	23	,	,	PUNCT
ejpam-841	837	24	z	z	PROPN
ejpam-841	837	25	,	,	PUNCT
ejpam-841	837	26	w	w	PROPN
ejpam-841	837	27	)	)	PUNCT
ejpam-841	837	28	=	=	SYM
ejpam-841	837	29	−[r2	−[r2	PUNCT
ejpam-841	837	30	x	x	PUNCT
ejpam-841	838	1	+	+	CCONJ
ejpam-841	838	2	r1	r1	PROPN
ejpam-841	838	3	y	y	PROPN
ejpam-841	838	4	+	+	CCONJ
ejpam-841	838	5	r0z	r0z	PROPN
ejpam-841	838	6	]	]	X
ejpam-841	838	7	=	=	SYM
ejpam-841	838	8	−x[r2	−x[r2	NUM
ejpam-841	838	9	+	+	SYM
ejpam-841	838	10	r1	r1	PROPN
ejpam-841	838	11	(	(	PUNCT
ejpam-841	838	12	y	y	NOUN
ejpam-841	838	13	x	x	PROPN
ejpam-841	838	14	)	)	PUNCT
ejpam-841	839	1	+	+	CCONJ
ejpam-841	839	2	r0	r0	NOUN
ejpam-841	839	3	(	(	PUNCT
ejpam-841	839	4	z	z	NOUN
ejpam-841	839	5	x	x	NOUN
ejpam-841	839	6	)	)	PUNCT
ejpam-841	839	7	]	]	PUNCT
ejpam-841	840	1	=	=	PUNCT
ejpam-841	840	2	−x	−x	PRON
ejpam-841	840	3	r	r	NOUN
ejpam-841	840	4	′1	′1	PROPN
ejpam-841	840	5	(	(	PUNCT
ejpam-841	840	6	y	y	NOUN
ejpam-841	840	7	x	x	PROPN
ejpam-841	840	8	,	,	PUNCT
ejpam-841	840	9	z	z	NOUN
ejpam-841	840	10	x	x	NOUN
ejpam-841	840	11	)	)	PUNCT
ejpam-841	840	12	.	.	PUNCT
ejpam-841	841	1	thus	thus	ADV
ejpam-841	841	2	,	,	PUNCT
ejpam-841	841	3	the	the	DET
ejpam-841	841	4	above	above	ADJ
ejpam-841	841	5	generators	generator	NOUN
ejpam-841	841	6	are	be	AUX
ejpam-841	841	7	the	the	DET
ejpam-841	841	8	same	same	ADJ
ejpam-841	841	9	as	as	ADP
ejpam-841	841	10	the	the	DET
ejpam-841	841	11	generators	generator	NOUN
ejpam-841	841	12	described	describe	VERB
ejpam-841	841	13	in	in	ADP
ejpam-841	841	14	theorem	theorem	ADJ
ejpam-841	841	15	1	1	NUM
ejpam-841	841	16	.	.	PUNCT
ejpam-841	841	17	references	reference	NOUN
ejpam-841	841	18	630	630	NUM
ejpam-841	841	19	for	for	ADP
ejpam-841	841	20	non	non	ADJ
ejpam-841	841	21	-	-	ADJ
ejpam-841	841	22	singular	singular	ADJ
ejpam-841	841	23	quartic	quartic	ADJ
ejpam-841	841	24	space	space	NOUN
ejpam-841	841	25	curves	curve	NOUN
ejpam-841	841	26	,	,	PUNCT
ejpam-841	841	27	recall	recall	VERB
ejpam-841	841	28	that	that	SCONJ
ejpam-841	841	29	by	by	ADP
ejpam-841	841	30	lemma	lemma	PROPN
ejpam-841	841	31	3	3	NUM
ejpam-841	841	32	there	there	PRON
ejpam-841	841	33	is	be	VERB
ejpam-841	841	34	a	a	DET
ejpam-841	841	35	linear	linear	ADJ
ejpam-841	841	36	transformation	transformation	NOUN
ejpam-841	841	37	on	on	ADP
ejpam-841	841	38	the	the	DET
ejpam-841	841	39	µ-basis	µ-basis	NOUN
ejpam-841	841	40	elements	element	NOUN
ejpam-841	841	41	p	p	X
ejpam-841	841	42	,	,	PUNCT
ejpam-841	841	43	q	q	X
ejpam-841	841	44	,	,	PUNCT
ejpam-841	841	45	and	and	CCONJ
ejpam-841	841	46	a	a	DET
ejpam-841	841	47	projective	projective	ADJ
ejpam-841	841	48	change	change	NOUN
ejpam-841	841	49	of	of	ADP
ejpam-841	841	50	coordinates	coordinate	NOUN
ejpam-841	841	51	so	so	SCONJ
ejpam-841	841	52	that	that	SCONJ
ejpam-841	841	53	p	p	X
ejpam-841	841	54	=	=	X
ejpam-841	841	55	ys−	ys−	PUNCT
ejpam-841	841	56	x	x	SYM
ejpam-841	841	57	t	t	PROPN
ejpam-841	841	58	,	,	PUNCT
ejpam-841	841	59	q	q	X
ejpam-841	841	60	=	=	SYM
ejpam-841	841	61	ws−	ws−	PROPN
ejpam-841	841	62	zt	zt	PROPN
ejpam-841	841	63	.	.	PUNCT
ejpam-841	842	1	then	then	ADV
ejpam-841	842	2	res(p	res(p	PROPN
ejpam-841	842	3	,	,	PUNCT
ejpam-841	842	4	r	r	NOUN
ejpam-841	842	5	)	)	PUNCT
ejpam-841	843	1	=	=	SYM
ejpam-841	843	2	r2	r2	PROPN
ejpam-841	843	3	x2	x2	PROPN
ejpam-841	843	4	+	+	CCONJ
ejpam-841	843	5	r1	r1	NOUN
ejpam-841	843	6	x	x	SYM
ejpam-841	843	7	y	y	PROPN
ejpam-841	843	8	+	+	NUM
ejpam-841	843	9	r0	r0	NOUN
ejpam-841	843	10	y2	y2	NOUN
ejpam-841	843	11	=	=	PUNCT
ejpam-841	844	1	x2[r2	x2[r2	PROPN
ejpam-841	845	1	+	+	CCONJ
ejpam-841	845	2	r1	r1	PROPN
ejpam-841	845	3	(	(	PUNCT
ejpam-841	845	4	y	y	NOUN
ejpam-841	845	5	x	x	PROPN
ejpam-841	845	6	)	)	PUNCT
ejpam-841	845	7	+	+	CCONJ
ejpam-841	845	8	r0	r0	NOUN
ejpam-841	845	9	(	(	PUNCT
ejpam-841	845	10	y	y	NOUN
ejpam-841	845	11	x	x	PROPN
ejpam-841	845	12	)	)	PUNCT
ejpam-841	845	13	2	2	NUM
ejpam-841	845	14	]	]	PUNCT
ejpam-841	845	15	=	=	SYM
ejpam-841	845	16	x2r	x2r	NOUN
ejpam-841	845	17	′2	′2	X
ejpam-841	845	18	(	(	PUNCT
ejpam-841	845	19	y	y	NOUN
ejpam-841	845	20	x	x	X
ejpam-841	845	21	,	,	PUNCT
ejpam-841	845	22	w	w	PROPN
ejpam-841	845	23	z	z	NOUN
ejpam-841	845	24	)	)	PUNCT
ejpam-841	845	25	,	,	PUNCT
ejpam-841	845	26	res(q	res(q	PROPN
ejpam-841	845	27	,	,	PUNCT
ejpam-841	845	28	r	r	NOUN
ejpam-841	845	29	)	)	PUNCT
ejpam-841	845	30	=	=	PUNCT
ejpam-841	845	31	r2z2	r2z2	X
ejpam-841	845	32	+	+	X
ejpam-841	845	33	r1zw	r1zw	PUNCT
ejpam-841	845	34	+	+	PUNCT
ejpam-841	845	35	r0w2	r0w2	NOUN
ejpam-841	845	36	=	=	SYM
ejpam-841	845	37	z2[r2	z2[r2	PROPN
ejpam-841	845	38	+	+	CCONJ
ejpam-841	845	39	r1	r1	PROPN
ejpam-841	845	40	(	(	PUNCT
ejpam-841	845	41	w	w	PROPN
ejpam-841	845	42	z	z	NOUN
ejpam-841	845	43	)	)	PUNCT
ejpam-841	846	1	+	+	CCONJ
ejpam-841	846	2	r0	r0	NOUN
ejpam-841	846	3	(	(	PUNCT
ejpam-841	846	4	w	w	NOUN
ejpam-841	846	5	z	z	NOUN
ejpam-841	846	6	)	)	PUNCT
ejpam-841	846	7	2	2	NUM
ejpam-841	846	8	]	]	PUNCT
ejpam-841	846	9	=	=	SYM
ejpam-841	846	10	z2r	z2r	NUM
ejpam-841	846	11	′0	′0	NOUN
ejpam-841	846	12	(	(	PUNCT
ejpam-841	846	13	y	y	NOUN
ejpam-841	846	14	x	x	X
ejpam-841	846	15	,	,	PUNCT
ejpam-841	846	16	w	w	PROPN
ejpam-841	846	17	z	z	NOUN
ejpam-841	846	18	)	)	PUNCT
ejpam-841	846	19	,	,	PUNCT
ejpam-841	846	20	det(n	det(n	PROPN
ejpam-841	846	21	)	)	PUNCT
ejpam-841	846	22	=	=	SYM
ejpam-841	846	23	r2	r2	PROPN
ejpam-841	846	24	xz+	xz+	PROPN
ejpam-841	846	25	r1	r1	PROPN
ejpam-841	846	26	xw+	xw+	PROPN
ejpam-841	846	27	r0	r0	PROPN
ejpam-841	846	28	yw	yw	NOUN
ejpam-841	846	29	=	=	PUNCT
ejpam-841	847	1	xz[r2	xz[r2	PROPN
ejpam-841	848	1	+	+	NUM
ejpam-841	848	2	r1	r1	PROPN
ejpam-841	848	3	(	(	PUNCT
ejpam-841	848	4	w	w	PROPN
ejpam-841	848	5	z	z	NOUN
ejpam-841	848	6	)	)	PUNCT
ejpam-841	849	1	+	+	CCONJ
ejpam-841	849	2	r0	r0	NOUN
ejpam-841	849	3	(	(	PUNCT
ejpam-841	849	4	w	w	PROPN
ejpam-841	849	5	z	z	NOUN
ejpam-841	849	6	)	)	PUNCT
ejpam-841	849	7	(	(	PUNCT
ejpam-841	849	8	y	y	NOUN
ejpam-841	849	9	x	x	PROPN
ejpam-841	849	10	)	)	PUNCT
ejpam-841	849	11	]	]	PUNCT
ejpam-841	849	12	=	=	PUNCT
ejpam-841	849	13	xzr	xzr	X
ejpam-841	849	14	′1	′1	PROPN
ejpam-841	849	15	(	(	PUNCT
ejpam-841	849	16	y	y	NOUN
ejpam-841	849	17	x	x	PROPN
ejpam-841	849	18	,	,	PUNCT
ejpam-841	849	19	w	w	PROPN
ejpam-841	849	20	z	z	NOUN
ejpam-841	849	21	)	)	PUNCT
ejpam-841	849	22	,	,	PUNCT
ejpam-841	849	23	sylvs	sylvs	NOUN
ejpam-841	849	24	,	,	PUNCT
ejpam-841	849	25	t(p	t(p	PROPN
ejpam-841	849	26	,	,	PUNCT
ejpam-841	849	27	r	r	NOUN
ejpam-841	849	28	)	)	PUNCT
ejpam-841	849	29	=	=	SYM
ejpam-841	850	1	r2	r2	PROPN
ejpam-841	850	2	xs+	xs+	PROPN
ejpam-841	850	3	r1	r1	PROPN
ejpam-841	850	4	x	x	PROPN
ejpam-841	851	1	t	t	PROPN
ejpam-841	851	2	+	+	CCONJ
ejpam-841	851	3	r0	r0	NOUN
ejpam-841	851	4	y	y	PROPN
ejpam-841	851	5	t	t	PROPN
ejpam-841	851	6	=	=	PUNCT
ejpam-841	852	1	x[r2s+	x[r2s+	PROPN
ejpam-841	852	2	r1	r1	PROPN
ejpam-841	852	3	t	t	PROPN
ejpam-841	852	4	+	+	CCONJ
ejpam-841	852	5	r0	r0	PROPN
ejpam-841	852	6	t	t	PROPN
ejpam-841	852	7	(	(	PUNCT
ejpam-841	852	8	y	y	NOUN
ejpam-841	852	9	x	x	PROPN
ejpam-841	852	10	)	)	PUNCT
ejpam-841	852	11	]	]	PUNCT
ejpam-841	853	1	=	=	PUNCT
ejpam-841	853	2	x	x	SYM
ejpam-841	853	3	r	r	NOUN
ejpam-841	853	4	′1,0	′1,0	NOUN
ejpam-841	853	5	(	(	PUNCT
ejpam-841	853	6	y	y	PROPN
ejpam-841	853	7	x	x	X
ejpam-841	853	8	,	,	PUNCT
ejpam-841	853	9	w	w	PROPN
ejpam-841	853	10	z	z	NOUN
ejpam-841	853	11	)	)	PUNCT
ejpam-841	853	12	,	,	PUNCT
ejpam-841	853	13	sylvs	sylvs	PROPN
ejpam-841	853	14	,	,	PUNCT
ejpam-841	853	15	t(q	t(q	PROPN
ejpam-841	853	16	,	,	PUNCT
ejpam-841	853	17	r	r	NOUN
ejpam-841	853	18	)	)	PUNCT
ejpam-841	853	19	=	=	SYM
ejpam-841	853	20	r2zs+	r2zs+	NOUN
ejpam-841	853	21	r1zt	r1zt	VERB
ejpam-841	853	22	+	+	X
ejpam-841	853	23	r0wt	r0wt	NOUN
ejpam-841	853	24	=	=	SYM
ejpam-841	853	25	z[r2s+	z[r2s+	NUM
ejpam-841	853	26	r1	r1	PROPN
ejpam-841	853	27	t	t	PROPN
ejpam-841	853	28	+	+	CCONJ
ejpam-841	853	29	r0	r0	PROPN
ejpam-841	853	30	t	t	PROPN
ejpam-841	853	31	(	(	PUNCT
ejpam-841	853	32	w	w	PROPN
ejpam-841	853	33	z	z	NOUN
ejpam-841	853	34	)	)	PUNCT
ejpam-841	853	35	]	]	PUNCT
ejpam-841	854	1	=	=	PUNCT
ejpam-841	854	2	zr	zr	VERB
ejpam-841	854	3	′0,1	′0,1	ADV
ejpam-841	854	4	(	(	PUNCT
ejpam-841	854	5	y	y	PROPN
ejpam-841	854	6	x	x	X
ejpam-841	854	7	,	,	PUNCT
ejpam-841	854	8	w	w	PROPN
ejpam-841	854	9	z	z	NOUN
ejpam-841	854	10	)	)	PUNCT
ejpam-841	854	11	.	.	PUNCT
ejpam-841	855	1	thus	thus	ADV
ejpam-841	855	2	,	,	PUNCT
ejpam-841	855	3	the	the	DET
ejpam-841	855	4	generators	generator	NOUN
ejpam-841	855	5	listed	list	VERB
ejpam-841	855	6	in	in	ADP
ejpam-841	855	7	the	the	DET
ejpam-841	855	8	statement	statement	NOUN
ejpam-841	855	9	of	of	ADP
ejpam-841	855	10	this	this	DET
ejpam-841	855	11	theorem	theorem	NOUN
ejpam-841	855	12	are	be	AUX
ejpam-841	855	13	exactly	exactly	ADV
ejpam-841	855	14	the	the	DET
ejpam-841	855	15	same	same	ADJ
ejpam-841	855	16	as	as	ADP
ejpam-841	855	17	the	the	DET
ejpam-841	855	18	generators	generator	NOUN
ejpam-841	855	19	listed	list	VERB
ejpam-841	855	20	in	in	ADP
ejpam-841	855	21	the	the	DET
ejpam-841	855	22	statement	statement	NOUN
ejpam-841	855	23	of	of	ADP
ejpam-841	855	24	theorem	theorem	ADJ
ejpam-841	855	25	3	3	X
ejpam-841	855	26	.	.	PUNCT
ejpam-841	855	27	acknowledgements	acknowledgement	NOUN
ejpam-841	855	28	we	we	PRON
ejpam-841	855	29	would	would	AUX
ejpam-841	855	30	like	like	VERB
ejpam-841	855	31	to	to	PART
ejpam-841	855	32	thank	thank	VERB
ejpam-841	855	33	david	david	PROPN
ejpam-841	855	34	cox	cox	PROPN
ejpam-841	855	35	for	for	ADP
ejpam-841	855	36	his	his	PRON
ejpam-841	855	37	helpful	helpful	ADJ
ejpam-841	855	38	suggestions	suggestion	NOUN
ejpam-841	855	39	.	.	PUNCT
ejpam-841	856	1	this	this	DET
ejpam-841	856	2	work	work	NOUN
ejpam-841	856	3	was	be	AUX
ejpam-841	856	4	partially	partially	ADV
ejpam-841	856	5	supported	support	VERB
ejpam-841	856	6	by	by	ADP
ejpam-841	856	7	nsf	nsf	PROPN
ejpam-841	856	8	grant	grant	NOUN
ejpam-841	856	9	ccr-020331	ccr-020331	NOUN
ejpam-841	856	10	,	,	PUNCT
ejpam-841	856	11	and	and	CCONJ
ejpam-841	856	12	by	by	ADP
ejpam-841	856	13	the	the	DET
ejpam-841	856	14	nsf	nsf	PROPN
ejpam-841	856	15	of	of	ADP
ejpam-841	856	16	china	china	PROPN
ejpam-841	856	17	(	(	PUNCT
ejpam-841	856	18	no.60873109	no.60873109	PROPN
ejpam-841	856	19	)	)	PUNCT
ejpam-841	856	20	,	,	PUNCT
ejpam-841	856	21	one	one	NUM
ejpam-841	856	22	hundred	hundred	NUM
ejpam-841	856	23	talent	talent	NOUN
ejpam-841	856	24	project	project	NOUN
ejpam-841	856	25	supported	support	VERB
ejpam-841	856	26	by	by	ADP
ejpam-841	856	27	cas	cas	PROPN
ejpam-841	856	28	and	and	CCONJ
ejpam-841	856	29	the	the	DET
ejpam-841	856	30	111	111	NUM
ejpam-841	856	31	project	project	NOUN
ejpam-841	856	32	(	(	PUNCT
ejpam-841	856	33	no	no	INTJ
ejpam-841	856	34	.	.	PUNCT
ejpam-841	857	1	b07033	b07033	NOUN
ejpam-841	857	2	)	)	PUNCT
ejpam-841	857	3	.	.	PUNCT
ejpam-841	858	1	the	the	DET
ejpam-841	858	2	second	second	ADJ
ejpam-841	858	3	author	author	NOUN
ejpam-841	858	4	would	would	AUX
ejpam-841	858	5	like	like	VERB
ejpam-841	858	6	to	to	PART
ejpam-841	858	7	thank	thank	VERB
ejpam-841	858	8	the	the	DET
ejpam-841	858	9	organizers	organizer	NOUN
ejpam-841	858	10	of	of	ADP
ejpam-841	858	11	the	the	DET
ejpam-841	858	12	special	special	ADJ
ejpam-841	858	13	session	session	NOUN
ejpam-841	858	14	on	on	ADP
ejpam-841	858	15	geometry	geometry	NOUN
ejpam-841	858	16	of	of	ADP
ejpam-841	858	17	varieties	variety	NOUN
ejpam-841	858	18	,	,	PUNCT
ejpam-841	858	19	syzygies	syzygy	NOUN
ejpam-841	858	20	and	and	CCONJ
ejpam-841	858	21	computations	computation	NOUN
ejpam-841	858	22	for	for	ADP
ejpam-841	858	23	giving	give	VERB
ejpam-841	858	24	her	she	PRON
ejpam-841	858	25	the	the	DET
ejpam-841	858	26	opportunity	opportunity	NOUN
ejpam-841	858	27	to	to	PART
ejpam-841	858	28	present	present	VERB
ejpam-841	858	29	this	this	DET
ejpam-841	858	30	paper	paper	NOUN
ejpam-841	858	31	during	during	ADP
ejpam-841	858	32	the	the	DET
ejpam-841	858	33	first	first	ADJ
ejpam-841	858	34	kms	kms	PROPN
ejpam-841	858	35	-	-	PUNCT
ejpam-841	858	36	ams	ams	NOUN
ejpam-841	858	37	joint	joint	ADJ
ejpam-841	858	38	meeting	meeting	NOUN
ejpam-841	858	39	.	.	PUNCT
ejpam-841	859	1	references	reference	NOUN
ejpam-841	859	2	[	[	X
ejpam-841	859	3	1	1	X
ejpam-841	859	4	]	]	PUNCT
ejpam-841	859	5	t.	t.	PROPN
ejpam-841	859	6	c.	c.	PROPN
ejpam-841	859	7	benítez	benítez	PROPN
ejpam-841	859	8	and	and	CCONJ
ejpam-841	859	9	c.	c.	PROPN
ejpam-841	859	10	d’andrea	d’andrea	PROPN
ejpam-841	859	11	,	,	PUNCT
ejpam-841	859	12	minimal	minimal	ADJ
ejpam-841	859	13	generators	generator	NOUN
ejpam-841	859	14	of	of	ADP
ejpam-841	859	15	the	the	DET
ejpam-841	859	16	defining	define	VERB
ejpam-841	859	17	ideal	ideal	NOUN
ejpam-841	859	18	of	of	ADP
ejpam-841	859	19	the	the	DET
ejpam-841	859	20	rees	rees	PROPN
ejpam-841	859	21	algebra	algebra	NOUN
ejpam-841	859	22	associated	associate	VERB
ejpam-841	859	23	to	to	ADP
ejpam-841	859	24	monoid	monoid	NOUN
ejpam-841	859	25	parametrizations	parametrization	NOUN
ejpam-841	859	26	,	,	PUNCT
ejpam-841	859	27	computer	computer	NOUN
ejpam-841	859	28	aided	aid	VERB
ejpam-841	859	29	geometric	geometric	ADJ
ejpam-841	859	30	design	design	NOUN
ejpam-841	859	31	,	,	PUNCT
ejpam-841	859	32	to	to	PART
ejpam-841	859	33	appear	appear	VERB
ejpam-841	859	34	.	.	PUNCT
ejpam-841	860	1	[	[	X
ejpam-841	860	2	2	2	NUM
ejpam-841	860	3	]	]	PUNCT
ejpam-841	860	4	l.	l.	PROPN
ejpam-841	860	5	busé	busé	PROPN
ejpam-841	860	6	,	,	PUNCT
ejpam-841	860	7	j	j	PROPN
ejpam-841	860	8	-	-	PROPN
ejpam-841	860	9	p.	p.	PROPN
ejpam-841	860	10	jouanolou	jouanolou	PROPN
ejpam-841	860	11	,	,	PUNCT
ejpam-841	860	12	on	on	ADP
ejpam-841	860	13	the	the	DET
ejpam-841	860	14	closed	closed	ADJ
ejpam-841	860	15	image	image	NOUN
ejpam-841	860	16	of	of	ADP
ejpam-841	860	17	a	a	DET
ejpam-841	860	18	rational	rational	ADJ
ejpam-841	860	19	map	map	NOUN
ejpam-841	860	20	and	and	CCONJ
ejpam-841	860	21	the	the	DET
ejpam-841	860	22	implicitization	implicitization	NOUN
ejpam-841	860	23	problem	problem	NOUN
ejpam-841	860	24	,	,	PUNCT
ejpam-841	860	25	j.	j.	PROPN
ejpam-841	860	26	algebra	algebra	PROPN
ejpam-841	860	27	265	265	NUM
ejpam-841	860	28	(	(	PUNCT
ejpam-841	860	29	2003	2003	NUM
ejpam-841	860	30	)	)	PUNCT
ejpam-841	860	31	,	,	PUNCT
ejpam-841	860	32	312	312	NUM
ejpam-841	860	33	-	-	SYM
ejpam-841	860	34	357	357	NUM
ejpam-841	860	35	.	.	PUNCT
ejpam-841	861	1	[	[	X
ejpam-841	861	2	3	3	X
ejpam-841	861	3	]	]	X
ejpam-841	861	4	l.	l.	PROPN
ejpam-841	861	5	busé	busé	PROPN
ejpam-841	861	6	,	,	PUNCT
ejpam-841	861	7	m.	m.	NOUN
ejpam-841	861	8	chardin	chardin	PROPN
ejpam-841	861	9	,	,	PUNCT
ejpam-841	861	10	implicitizing	implicitize	VERB
ejpam-841	861	11	rational	rational	ADJ
ejpam-841	861	12	hypersurfaces	hypersurface	NOUN
ejpam-841	861	13	using	use	VERB
ejpam-841	861	14	approximation	approximation	NOUN
ejpam-841	861	15	complexes	complex	NOUN
ejpam-841	861	16	,	,	PUNCT
ejpam-841	861	17	j.	j.	PROPN
ejpam-841	861	18	symb	symb	PROPN
ejpam-841	861	19	.	.	PUNCT
ejpam-841	862	1	comp	comp	PROPN
ejpam-841	862	2	.	.	PUNCT
ejpam-841	863	1	40	40	NUM
ejpam-841	863	2	(	(	PUNCT
ejpam-841	863	3	2005	2005	NUM
ejpam-841	863	4	)	)	PUNCT
ejpam-841	863	5	,	,	PUNCT
ejpam-841	863	6	1150	1150	NUM
ejpam-841	863	7	-	-	SYM
ejpam-841	863	8	1168	1168	NUM
ejpam-841	863	9	.	.	PUNCT
ejpam-841	864	1	[	[	X
ejpam-841	864	2	4	4	NUM
ejpam-841	864	3	]	]	PUNCT
ejpam-841	864	4	l.	l.	PROPN
ejpam-841	864	5	busé	busé	NOUN
ejpam-841	864	6	,	,	PUNCT
ejpam-841	864	7	on	on	ADP
ejpam-841	864	8	the	the	DET
ejpam-841	864	9	equations	equation	NOUN
ejpam-841	864	10	of	of	ADP
ejpam-841	864	11	the	the	DET
ejpam-841	864	12	moving	move	VERB
ejpam-841	864	13	curve	curve	NOUN
ejpam-841	864	14	ideal	ideal	NOUN
ejpam-841	864	15	of	of	ADP
ejpam-841	864	16	a	a	DET
ejpam-841	864	17	rational	rational	ADJ
ejpam-841	864	18	algebraic	algebraic	ADJ
ejpam-841	864	19	plane	plane	NOUN
ejpam-841	864	20	curve	curve	NOUN
ejpam-841	864	21	,	,	PUNCT
ejpam-841	864	22	j.	j.	PROPN
ejpam-841	864	23	algebra	algebra	PROPN
ejpam-841	864	24	,	,	PUNCT
ejpam-841	864	25	321	321	NUM
ejpam-841	864	26	(	(	PUNCT
ejpam-841	864	27	2009	2009	NUM
ejpam-841	864	28	)	)	PUNCT
ejpam-841	864	29	,	,	PUNCT
ejpam-841	864	30	2317–2344	2317–2344	NUM
ejpam-841	864	31	.	.	PUNCT
ejpam-841	865	1	[	[	X
ejpam-841	865	2	5	5	NUM
ejpam-841	865	3	]	]	PUNCT
ejpam-841	865	4	l.	l.	PROPN
ejpam-841	865	5	busé	busé	PROPN
ejpam-841	865	6	,	,	PUNCT
ejpam-841	865	7	m.	m.	NOUN
ejpam-841	865	8	chardin	chardin	PROPN
ejpam-841	865	9	,	,	PUNCT
ejpam-841	865	10	and	and	CCONJ
ejpam-841	865	11	a.	a.	NOUN
ejpam-841	865	12	simis	simis	PROPN
ejpam-841	865	13	,	,	PUNCT
ejpam-841	865	14	elimination	elimination	NOUN
ejpam-841	865	15	and	and	CCONJ
ejpam-841	865	16	nonlinear	nonlinear	ADJ
ejpam-841	865	17	equations	equation	NOUN
ejpam-841	865	18	of	of	ADP
ejpam-841	865	19	rees	rees	PROPN
ejpam-841	865	20	algebra	algebra	PROPN
ejpam-841	865	21	,	,	PUNCT
ejpam-841	865	22	preprint	preprint	NOUN
ejpam-841	865	23	.	.	PUNCT
ejpam-841	866	1	[	[	X
ejpam-841	866	2	6	6	NUM
ejpam-841	866	3	]	]	PUNCT
ejpam-841	866	4	a.	a.	NOUN
ejpam-841	866	5	conca	conca	PROPN
ejpam-841	866	6	,	,	PUNCT
ejpam-841	866	7	j.	j.	PROPN
ejpam-841	866	8	herzog	herzog	PROPN
ejpam-841	866	9	,	,	PUNCT
ejpam-841	866	10	n.	n.	PROPN
ejpam-841	866	11	v.	v.	ADP
ejpam-841	866	12	trung	trung	PROPN
ejpam-841	866	13	and	and	CCONJ
ejpam-841	866	14	g.	g.	PROPN
ejpam-841	866	15	valla	valla	PROPN
ejpam-841	866	16	,	,	PUNCT
ejpam-841	866	17	diagonal	diagonal	ADJ
ejpam-841	866	18	subalgebras	subalgebra	NOUN
ejpam-841	866	19	of	of	ADP
ejpam-841	866	20	bigraded	bigrade	VERB
ejpam-841	866	21	algebras	algebra	NOUN
ejpam-841	866	22	and	and	CCONJ
ejpam-841	866	23	embeddings	embedding	NOUN
ejpam-841	866	24	of	of	ADP
ejpam-841	866	25	blow	blow	NOUN
ejpam-841	866	26	-	-	PUNCT
ejpam-841	866	27	ups	up	NOUN
ejpam-841	866	28	of	of	ADP
ejpam-841	866	29	projective	projective	ADJ
ejpam-841	866	30	spaces	space	NOUN
ejpam-841	866	31	,	,	PUNCT
ejpam-841	866	32	amer	amer	PROPN
ejpam-841	866	33	.	.	PUNCT
ejpam-841	867	1	j.	j.	PROPN
ejpam-841	867	2	math	math	PROPN
ejpam-841	867	3	.	.	PUNCT
ejpam-841	868	1	119	119	NUM
ejpam-841	868	2	(	(	PUNCT
ejpam-841	868	3	1997	1997	NUM
ejpam-841	868	4	)	)	PUNCT
ejpam-841	868	5	,	,	PUNCT
ejpam-841	868	6	859–901	859–901	NUM
ejpam-841	868	7	.	.	PUNCT
ejpam-841	869	1	[	[	X
ejpam-841	869	2	7	7	X
ejpam-841	869	3	]	]	X
ejpam-841	869	4	d.	d.	PROPN
ejpam-841	869	5	cox	cox	PROPN
ejpam-841	869	6	,	,	PUNCT
ejpam-841	869	7	the	the	DET
ejpam-841	869	8	moving	move	VERB
ejpam-841	869	9	curve	curve	NOUN
ejpam-841	869	10	ideal	ideal	NOUN
ejpam-841	869	11	and	and	CCONJ
ejpam-841	869	12	the	the	DET
ejpam-841	869	13	rees	rees	PROPN
ejpam-841	869	14	algebra	algebra	NOUN
ejpam-841	869	15	,	,	PUNCT
ejpam-841	869	16	theoret	theoret	ADJ
ejpam-841	869	17	.	.	PUNCT
ejpam-841	870	1	comput	comput	NOUN
ejpam-841	870	2	.	.	PUNCT
ejpam-841	871	1	sci	sci	PROPN
ejpam-841	871	2	.	.	PROPN
ejpam-841	872	1	392	392	NUM
ejpam-841	872	2	(	(	PUNCT
ejpam-841	872	3	2008	2008	NUM
ejpam-841	872	4	)	)	PUNCT
ejpam-841	872	5	,	,	PUNCT
ejpam-841	872	6	23–36	23–36	NUM
ejpam-841	872	7	.	.	PUNCT
ejpam-841	873	1	references	reference	NOUN
ejpam-841	873	2	631	631	NUM
ejpam-841	874	1	[	[	X
ejpam-841	874	2	8	8	NUM
ejpam-841	874	3	]	]	X
ejpam-841	874	4	d.	d.	PROPN
ejpam-841	874	5	cox	cox	PROPN
ejpam-841	874	6	,	,	PUNCT
ejpam-841	874	7	j.	j.	PROPN
ejpam-841	874	8	w.	w.	PROPN
ejpam-841	874	9	hoffman	hoffman	PROPN
ejpam-841	874	10	and	and	CCONJ
ejpam-841	874	11	h.	h.	PROPN
ejpam-841	874	12	wang	wang	PROPN
ejpam-841	874	13	,	,	PUNCT
ejpam-841	874	14	syzygies	syzygy	NOUN
ejpam-841	874	15	and	and	CCONJ
ejpam-841	874	16	the	the	DET
ejpam-841	874	17	rees	rees	PROPN
ejpam-841	874	18	algebra	algebra	PROPN
ejpam-841	874	19	,	,	PUNCT
ejpam-841	874	20	j.	j.	PROPN
ejpam-841	874	21	pure	pure	PROPN
ejpam-841	874	22	appl	appl	PROPN
ejpam-841	874	23	.	.	PUNCT
ejpam-841	875	1	algebra	algebra	NOUN
ejpam-841	875	2	212	212	NUM
ejpam-841	875	3	(	(	PUNCT
ejpam-841	875	4	2008	2008	NUM
ejpam-841	875	5	)	)	PUNCT
ejpam-841	875	6	,	,	PUNCT
ejpam-841	875	7	1787–1796	1787–1796	NUM
ejpam-841	875	8	.	.	PUNCT
ejpam-841	876	1	[	[	X
ejpam-841	876	2	9	9	NUM
ejpam-841	876	3	]	]	X
ejpam-841	876	4	d.	d.	PROPN
ejpam-841	876	5	cox	cox	PROPN
ejpam-841	876	6	,	,	PUNCT
ejpam-841	876	7	j.	j.	PROPN
ejpam-841	876	8	little	little	PROPN
ejpam-841	876	9	and	and	CCONJ
ejpam-841	876	10	d.	d.	PROPN
ejpam-841	876	11	o’shea	o’shea	PROPN
ejpam-841	876	12	,	,	PUNCT
ejpam-841	876	13	using	use	VERB
ejpam-841	876	14	algebraic	algebraic	ADJ
ejpam-841	876	15	geometry	geometry	NOUN
ejpam-841	876	16	,	,	PUNCT
ejpam-841	876	17	graduate	graduate	NOUN
ejpam-841	876	18	texts	text	NOUN
ejpam-841	876	19	in	in	ADP
ejpam-841	876	20	mathematics	mathematic	NOUN
ejpam-841	876	21	185	185	NUM
ejpam-841	876	22	,	,	PUNCT
ejpam-841	876	23	springer	springer	NOUN
ejpam-841	876	24	,	,	PUNCT
ejpam-841	876	25	new	new	PROPN
ejpam-841	876	26	york	york	PROPN
ejpam-841	876	27	,	,	PUNCT
ejpam-841	876	28	1998	1998	NUM
ejpam-841	876	29	.	.	PUNCT
ejpam-841	877	1	[	[	X
ejpam-841	877	2	10	10	NUM
ejpam-841	877	3	]	]	X
ejpam-841	877	4	d.	d.	PROPN
ejpam-841	877	5	a.	a.	PROPN
ejpam-841	877	6	cox	cox	PROPN
ejpam-841	877	7	,	,	PUNCT
ejpam-841	877	8	t.	t.	PROPN
ejpam-841	877	9	sederberg	sederberg	PROPN
ejpam-841	877	10	and	and	CCONJ
ejpam-841	877	11	f.	f.	PROPN
ejpam-841	877	12	chen	chen	PROPN
ejpam-841	877	13	,	,	PUNCT
ejpam-841	877	14	the	the	DET
ejpam-841	877	15	moving	move	VERB
ejpam-841	877	16	line	line	NOUN
ejpam-841	877	17	ideal	ideal	ADJ
ejpam-841	877	18	basis	basis	NOUN
ejpam-841	877	19	of	of	ADP
ejpam-841	877	20	planar	planar	ADJ
ejpam-841	877	21	rational	rational	ADJ
ejpam-841	877	22	curves	curve	NOUN
ejpam-841	877	23	,	,	PUNCT
ejpam-841	877	24	comput	comput	NOUN
ejpam-841	877	25	.	.	PUNCT
ejpam-841	878	1	aided	aid	VERB
ejpam-841	878	2	geom	geom	NOUN
ejpam-841	878	3	.	.	PUNCT
ejpam-841	879	1	design	design	VERB
ejpam-841	879	2	15	15	NUM
ejpam-841	879	3	(	(	PUNCT
ejpam-841	879	4	1998	1998	NUM
ejpam-841	879	5	)	)	PUNCT
ejpam-841	879	6	,	,	PUNCT
ejpam-841	879	7	803–827	803–827	NUM
ejpam-841	879	8	.	.	PUNCT
ejpam-841	880	1	[	[	X
ejpam-841	880	2	11	11	NUM
ejpam-841	880	3	]	]	PUNCT
ejpam-841	880	4	a.	a.	NOUN
ejpam-841	880	5	geramita	geramita	PROPN
ejpam-841	880	6	,	,	PUNCT
ejpam-841	880	7	a.	a.	PROPN
ejpam-841	880	8	gimilgliano	gimilgliano	PROPN
ejpam-841	880	9	and	and	CCONJ
ejpam-841	880	10	b.	b.	PROPN
ejpam-841	880	11	harbourne	harbourne	PROPN
ejpam-841	880	12	,	,	PUNCT
ejpam-841	880	13	projectively	projectively	ADV
ejpam-841	880	14	normal	normal	ADJ
ejpam-841	880	15	but	but	CCONJ
ejpam-841	880	16	superabundant	superabundant	ADJ
ejpam-841	880	17	embeddings	embedding	NOUN
ejpam-841	880	18	of	of	ADP
ejpam-841	880	19	rational	rational	ADJ
ejpam-841	880	20	surfaces	surface	NOUN
ejpam-841	880	21	in	in	ADP
ejpam-841	880	22	projective	projective	ADJ
ejpam-841	880	23	space	space	NOUN
ejpam-841	880	24	,	,	PUNCT
ejpam-841	880	25	j.	j.	PROPN
ejpam-841	880	26	algebra	algebra	PROPN
ejpam-841	880	27	169	169	NUM
ejpam-841	880	28	(	(	PUNCT
ejpam-841	880	29	1994	1994	NUM
ejpam-841	880	30	)	)	PUNCT
ejpam-841	880	31	,	,	PUNCT
ejpam-841	880	32	791–804	791–804	NUM
ejpam-841	880	33	.	.	PUNCT
ejpam-841	881	1	[	[	X
ejpam-841	881	2	12	12	NUM
ejpam-841	881	3	]	]	PUNCT
ejpam-841	881	4	a.	a.	NOUN
ejpam-841	881	5	gimigliano	gimigliano	PROPN
ejpam-841	881	6	and	and	CCONJ
ejpam-841	881	7	a.	a.	NOUN
ejpam-841	881	8	lorenzini	lorenzini	PROPN
ejpam-841	881	9	,	,	PUNCT
ejpam-841	881	10	on	on	ADP
ejpam-841	881	11	the	the	DET
ejpam-841	881	12	ideal	ideal	NOUN
ejpam-841	881	13	of	of	ADP
ejpam-841	881	14	veronesean	veronesean	ADJ
ejpam-841	881	15	surfaces	surface	NOUN
ejpam-841	881	16	,	,	PUNCT
ejpam-841	881	17	can	can	AUX
ejpam-841	881	18	.	.	PUNCT
ejpam-841	882	1	j.	j.	PROPN
ejpam-841	882	2	math	math	PROPN
ejpam-841	882	3	.	.	PUNCT
ejpam-841	883	1	43	43	NUM
ejpam-841	883	2	(	(	PUNCT
ejpam-841	883	3	1993	1993	NUM
ejpam-841	883	4	)	)	PUNCT
ejpam-841	883	5	,	,	PUNCT
ejpam-841	883	6	758–777	758–777	NUM
ejpam-841	883	7	.	.	PUNCT
ejpam-841	884	1	[	[	X
ejpam-841	884	2	13	13	NUM
ejpam-841	884	3	]	]	PUNCT
ejpam-841	884	4	j.	j.	PROPN
ejpam-841	884	5	harris	harris	PROPN
ejpam-841	884	6	,	,	PUNCT
ejpam-841	884	7	curves	curve	VERB
ejpam-841	884	8	in	in	ADP
ejpam-841	884	9	projective	projective	ADJ
ejpam-841	884	10	space	space	NOUN
ejpam-841	884	11	,	,	PUNCT
ejpam-841	884	12	les	les	X
ejpam-841	884	13	presses	press	NOUN
ejpam-841	884	14	de	de	X
ejpam-841	884	15	l’universite	l’universite	PROPN
ejpam-841	884	16	de	de	X
ejpam-841	884	17	montreal	montreal	PROPN
ejpam-841	884	18	,	,	PUNCT
ejpam-841	884	19	1982	1982	NUM
ejpam-841	884	20	.	.	PUNCT
ejpam-841	885	1	[	[	X
ejpam-841	885	2	14	14	NUM
ejpam-841	885	3	]	]	X
ejpam-841	885	4	r.	r.	PROPN
ejpam-841	885	5	hartshorne	hartshorne	PROPN
ejpam-841	885	6	,	,	PUNCT
ejpam-841	885	7	algebraic	algebraic	ADJ
ejpam-841	885	8	geometry	geometry	NOUN
ejpam-841	885	9	,	,	PUNCT
ejpam-841	885	10	graduate	graduate	NOUN
ejpam-841	885	11	texts	text	NOUN
ejpam-841	885	12	in	in	ADP
ejpam-841	885	13	mathematics	mathematics	PROPN
ejpam-841	885	14	52	52	NUM
ejpam-841	885	15	,	,	PUNCT
ejpam-841	885	16	springer	springer	NOUN
ejpam-841	885	17	,	,	PUNCT
ejpam-841	885	18	new	new	PROPN
ejpam-841	885	19	york	york	PROPN
ejpam-841	885	20	,	,	PUNCT
ejpam-841	885	21	1997	1997	NUM
ejpam-841	885	22	.	.	PUNCT
ejpam-841	886	1	[	[	X
ejpam-841	886	2	15	15	NUM
ejpam-841	886	3	]	]	X
ejpam-841	886	4	j.	j.	PROPN
ejpam-841	886	5	w.	w.	PROPN
ejpam-841	886	6	hoffman	hoffman	PROPN
ejpam-841	886	7	and	and	CCONJ
ejpam-841	886	8	h.	h.	PROPN
ejpam-841	886	9	wang	wang	PROPN
ejpam-841	886	10	,	,	PUNCT
ejpam-841	886	11	defining	define	VERB
ejpam-841	886	12	equations	equation	NOUN
ejpam-841	886	13	of	of	ADP
ejpam-841	886	14	the	the	DET
ejpam-841	886	15	rees	rees	PROPN
ejpam-841	886	16	algebra	algebra	NOUN
ejpam-841	886	17	of	of	ADP
ejpam-841	886	18	certain	certain	ADJ
ejpam-841	886	19	parametric	parametric	ADJ
ejpam-841	886	20	surfaces	surface	NOUN
ejpam-841	886	21	,	,	PUNCT
ejpam-841	886	22	j.	j.	PROPN
ejpam-841	886	23	algebra	algebra	PROPN
ejpam-841	886	24	and	and	CCONJ
ejpam-841	886	25	its	its	PRON
ejpam-841	886	26	applications	application	NOUN
ejpam-841	886	27	,	,	PUNCT
ejpam-841	886	28	to	to	PART
ejpam-841	886	29	appear	appear	VERB
ejpam-841	886	30	.	.	PUNCT
ejpam-841	887	1	[	[	X
ejpam-841	887	2	16	16	NUM
ejpam-841	887	3	]	]	X
ejpam-841	887	4	j.	j.	PROPN
ejpam-841	887	5	hong	hong	PROPN
ejpam-841	887	6	,	,	PUNCT
ejpam-841	887	7	a.	a.	NOUN
ejpam-841	887	8	simis	simis	PROPN
ejpam-841	887	9	and	and	CCONJ
ejpam-841	887	10	w.	w.	PROPN
ejpam-841	887	11	vasconcelos	vasconcelos	PROPN
ejpam-841	887	12	,	,	PUNCT
ejpam-841	887	13	on	on	ADP
ejpam-841	887	14	the	the	DET
ejpam-841	887	15	homology	homology	NOUN
ejpam-841	887	16	of	of	ADP
ejpam-841	887	17	two	two	NUM
ejpam-841	887	18	-	-	PUNCT
ejpam-841	887	19	dimensional	dimensional	ADJ
ejpam-841	887	20	elimination	elimination	NOUN
ejpam-841	887	21	,	,	PUNCT
ejpam-841	887	22	j.	j.	PROPN
ejpam-841	887	23	symb	symb	PROPN
ejpam-841	887	24	.	.	PUNCT
ejpam-841	888	1	comp	comp	PROPN
ejpam-841	888	2	.	.	PUNCT
ejpam-841	889	1	43	43	NUM
ejpam-841	889	2	(	(	PUNCT
ejpam-841	889	3	2008	2008	NUM
ejpam-841	889	4	)	)	PUNCT
ejpam-841	889	5	,	,	PUNCT
ejpam-841	889	6	275–292	275–292	NUM
ejpam-841	889	7	.	.	PUNCT
ejpam-841	890	1	[	[	X
ejpam-841	890	2	17	17	NUM
ejpam-841	890	3	]	]	PUNCT
ejpam-841	890	4	x.	x.	PROPN
ejpam-841	890	5	jia	jia	PROPN
ejpam-841	890	6	,	,	PUNCT
ejpam-841	890	7	h.	h.	PROPN
ejpam-841	890	8	wang	wang	PROPN
ejpam-841	890	9	and	and	CCONJ
ejpam-841	890	10	r.	r.	PROPN
ejpam-841	890	11	goldman	goldman	PROPN
ejpam-841	890	12	,	,	PUNCT
ejpam-841	890	13	set	set	NOUN
ejpam-841	890	14	-	-	PUNCT
ejpam-841	890	15	theoretic	theoretic	NOUN
ejpam-841	890	16	generators	generator	NOUN
ejpam-841	890	17	of	of	ADP
ejpam-841	890	18	rational	rational	ADJ
ejpam-841	890	19	space	space	NOUN
ejpam-841	890	20	curves	curve	NOUN
ejpam-841	890	21	,	,	PUNCT
ejpam-841	890	22	j.	j.	PROPN
ejpam-841	890	23	symb	symb	PROPN
ejpam-841	890	24	.	.	PUNCT
ejpam-841	891	1	comp	comp	PROPN
ejpam-841	891	2	.	.	PUNCT
ejpam-841	892	1	45	45	NUM
ejpam-841	892	2	(	(	PUNCT
ejpam-841	892	3	2010	2010	NUM
ejpam-841	892	4	)	)	PUNCT
ejpam-841	892	5	,	,	PUNCT
ejpam-841	892	6	414	414	NUM
ejpam-841	892	7	-	-	SYM
ejpam-841	892	8	433	433	NUM
ejpam-841	892	9	.	.	PUNCT
ejpam-841	893	1	[	[	X
ejpam-841	893	2	18	18	NUM
ejpam-841	893	3	]	]	PUNCT
ejpam-841	893	4	a.	a.	PROPN
ejpam-841	893	5	r.	r.	PROPN
ejpam-841	893	6	kustin	kustin	PROPN
ejpam-841	893	7	,	,	PUNCT
ejpam-841	893	8	c.	c.	PROPN
ejpam-841	893	9	polini	polini	PROPN
ejpam-841	893	10	,	,	PUNCT
ejpam-841	893	11	and	and	CCONJ
ejpam-841	893	12	b.	b.	PROPN
ejpam-841	893	13	ulrich	ulrich	PROPN
ejpam-841	893	14	,	,	PUNCT
ejpam-841	893	15	rational	rational	ADJ
ejpam-841	893	16	normal	normal	ADJ
ejpam-841	893	17	scrolls	scroll	NOUN
ejpam-841	893	18	and	and	CCONJ
ejpam-841	893	19	the	the	DET
ejpam-841	893	20	defining	define	VERB
ejpam-841	893	21	equations	equation	NOUN
ejpam-841	893	22	of	of	ADP
ejpam-841	893	23	rees	rees	PROPN
ejpam-841	893	24	algebra	algebra	PROPN
ejpam-841	893	25	,	,	PUNCT
ejpam-841	893	26	arxiv:0812.4963v1	arxiv:0812.4963v1	PROPN
ejpam-841	893	27	.	.	PUNCT
ejpam-841	894	1	[	[	X
ejpam-841	894	2	19	19	NUM
ejpam-841	894	3	]	]	PUNCT
ejpam-841	894	4	s.	s.	PROPN
ejpam-841	894	5	morey	morey	PROPN
ejpam-841	894	6	and	and	CCONJ
ejpam-841	894	7	b.	b.	PROPN
ejpam-841	894	8	ulrich	ulrich	PROPN
ejpam-841	894	9	,	,	PUNCT
ejpam-841	894	10	rees	rees	PROPN
ejpam-841	894	11	algebras	algebra	NOUN
ejpam-841	894	12	of	of	ADP
ejpam-841	894	13	ideals	ideal	NOUN
ejpam-841	894	14	with	with	ADP
ejpam-841	894	15	low	low	ADJ
ejpam-841	894	16	codimension	codimension	NOUN
ejpam-841	894	17	,	,	PUNCT
ejpam-841	894	18	proc	proc	NOUN
ejpam-841	894	19	.	.	PUNCT
ejpam-841	895	1	amer	amer	PROPN
ejpam-841	895	2	.	.	PUNCT
ejpam-841	895	3	math	math	PROPN
ejpam-841	895	4	.	.	PUNCT
ejpam-841	896	1	soc	soc	PROPN
ejpam-841	896	2	.	.	PUNCT
ejpam-841	897	1	124	124	NUM
ejpam-841	897	2	(	(	PUNCT
ejpam-841	897	3	1996	1996	NUM
ejpam-841	897	4	)	)	PUNCT
ejpam-841	897	5	,	,	PUNCT
ejpam-841	897	6	3653–3661	3653–3661	NUM
ejpam-841	897	7	.	.	PUNCT
ejpam-841	898	1	[	[	X
ejpam-841	898	2	20	20	NUM
ejpam-841	898	3	]	]	PUNCT
ejpam-841	898	4	t.	t.	PROPN
ejpam-841	898	5	sederberg	sederberg	PROPN
ejpam-841	898	6	and	and	CCONJ
ejpam-841	898	7	f.	f.	PROPN
ejpam-841	898	8	chen	chen	PROPN
ejpam-841	898	9	,	,	PUNCT
ejpam-841	898	10	implicitization	implicitization	NOUN
ejpam-841	898	11	using	use	VERB
ejpam-841	898	12	moving	move	VERB
ejpam-841	898	13	curves	curve	NOUN
ejpam-841	898	14	and	and	CCONJ
ejpam-841	898	15	surfaces	surface	NOUN
ejpam-841	898	16	,	,	PUNCT
ejpam-841	898	17	proceedings	proceeding	NOUN
ejpam-841	898	18	of	of	ADP
ejpam-841	898	19	siggraph	siggraph	NOUN
ejpam-841	898	20	,	,	PUNCT
ejpam-841	898	21	(	(	PUNCT
ejpam-841	898	22	1995	1995	NUM
ejpam-841	898	23	)	)	PUNCT
ejpam-841	898	24	,	,	PUNCT
ejpam-841	898	25	301–308	301–308	NUM
ejpam-841	898	26	.	.	PUNCT
ejpam-841	899	1	[	[	X
ejpam-841	899	2	21	21	NUM
ejpam-841	899	3	]	]	PUNCT
ejpam-841	899	4	t.	t.	PROPN
ejpam-841	899	5	sederberg	sederberg	PROPN
ejpam-841	899	6	,	,	PUNCT
ejpam-841	899	7	r.	r.	PROPN
ejpam-841	899	8	goldman	goldman	PROPN
ejpam-841	899	9	and	and	CCONJ
ejpam-841	899	10	h.	h.	PROPN
ejpam-841	899	11	du	du	PROPN
ejpam-841	899	12	,	,	PUNCT
ejpam-841	899	13	implicitizing	implicitize	VERB
ejpam-841	899	14	rational	rational	ADJ
ejpam-841	899	15	curves	curve	NOUN
ejpam-841	899	16	by	by	ADP
ejpam-841	899	17	the	the	DET
ejpam-841	899	18	method	method	NOUN
ejpam-841	899	19	of	of	ADP
ejpam-841	899	20	moving	move	VERB
ejpam-841	899	21	algebraic	algebraic	ADJ
ejpam-841	899	22	curves	curve	NOUN
ejpam-841	899	23	,	,	PUNCT
ejpam-841	899	24	j.	j.	PROPN
ejpam-841	899	25	symb	symb	PROPN
ejpam-841	899	26	.	.	PUNCT
ejpam-841	900	1	comp	comp	PROPN
ejpam-841	900	2	.	.	PUNCT
ejpam-841	901	1	23	23	NUM
ejpam-841	901	2	(	(	PUNCT
ejpam-841	901	3	1997	1997	NUM
ejpam-841	901	4	)	)	PUNCT
ejpam-841	901	5	,	,	PUNCT
ejpam-841	901	6	153–175	153–175	NUM
ejpam-841	901	7	.	.	PUNCT
ejpam-841	902	1	[	[	X
ejpam-841	902	2	22	22	NUM
ejpam-841	902	3	]	]	PUNCT
ejpam-841	902	4	t.	t.	PROPN
ejpam-841	902	5	sederberg	sederberg	PROPN
ejpam-841	902	6	,	,	PUNCT
ejpam-841	902	7	t.	t.	PROPN
ejpam-841	902	8	saito	saito	PROPN
ejpam-841	902	9	,	,	PUNCT
ejpam-841	902	10	d.	d.	PROPN
ejpam-841	902	11	qi	qi	PROPN
ejpam-841	902	12	and	and	CCONJ
ejpam-841	902	13	k.	k.	PROPN
ejpam-841	902	14	klimaszewski	klimaszewski	PROPN
ejpam-841	902	15	,	,	PUNCT
ejpam-841	902	16	curve	curve	NOUN
ejpam-841	902	17	implicitization	implicitization	NOUN
ejpam-841	902	18	using	use	VERB
ejpam-841	902	19	moving	move	VERB
ejpam-841	902	20	lines	line	NOUN
ejpam-841	902	21	,	,	PUNCT
ejpam-841	902	22	comput	comput	NOUN
ejpam-841	902	23	.	.	PUNCT
ejpam-841	903	1	aided	aid	VERB
ejpam-841	903	2	geom	geom	NOUN
ejpam-841	903	3	.	.	PUNCT
ejpam-841	904	1	design	design	VERB
ejpam-841	904	2	11	11	NUM
ejpam-841	904	3	(	(	PUNCT
ejpam-841	904	4	1994	1994	NUM
ejpam-841	904	5	)	)	PUNCT
ejpam-841	904	6	,	,	PUNCT
ejpam-841	904	7	687–706	687–706	NUM
ejpam-841	904	8	.	.	PUNCT
ejpam-841	905	1	[	[	X
ejpam-841	905	2	23	23	NUM
ejpam-841	905	3	]	]	PUNCT
ejpam-841	905	4	a.	a.	NOUN
ejpam-841	905	5	simis	simis	PROPN
ejpam-841	905	6	,	,	PUNCT
ejpam-841	905	7	n.	n.	PROPN
ejpam-841	905	8	trung	trung	PROPN
ejpam-841	905	9	and	and	CCONJ
ejpam-841	905	10	g.	g.	PROPN
ejpam-841	905	11	valla	valla	PROPN
ejpam-841	905	12	,	,	PUNCT
ejpam-841	905	13	the	the	DET
ejpam-841	905	14	diagonal	diagonal	ADJ
ejpam-841	905	15	subalgebra	subalgebra	NOUN
ejpam-841	905	16	of	of	ADP
ejpam-841	905	17	a	a	DET
ejpam-841	905	18	blow	blow	NOUN
ejpam-841	905	19	-	-	PUNCT
ejpam-841	905	20	up	up	ADP
ejpam-841	905	21	algebra	algebra	NOUN
ejpam-841	905	22	,	,	PUNCT
ejpam-841	905	23	j.	j.	PROPN
ejpam-841	905	24	pure	pure	PROPN
ejpam-841	905	25	appl	appl	PROPN
ejpam-841	905	26	.	.	PUNCT
ejpam-841	906	1	algebra	algebra	NOUN
ejpam-841	906	2	125	125	NUM
ejpam-841	906	3	(	(	PUNCT
ejpam-841	906	4	1998	1998	NUM
ejpam-841	906	5	)	)	PUNCT
ejpam-841	906	6	,	,	PUNCT
ejpam-841	906	7	305–328	305–328	NUM
ejpam-841	906	8	.	.	PUNCT
ejpam-841	907	1	[	[	X
ejpam-841	907	2	24	24	NUM
ejpam-841	907	3	]	]	PUNCT
ejpam-841	907	4	n.	n.	NOUN
ejpam-841	907	5	song	song	NOUN
ejpam-841	907	6	and	and	CCONJ
ejpam-841	907	7	r.	r.	PROPN
ejpam-841	907	8	goldman	goldman	PROPN
ejpam-841	907	9	,	,	PUNCT
ejpam-841	907	10	µ-bases	µ-base	VERB
ejpam-841	907	11	for	for	ADP
ejpam-841	907	12	polynomial	polynomial	ADJ
ejpam-841	907	13	system	system	NOUN
ejpam-841	907	14	in	in	ADP
ejpam-841	907	15	one	one	NUM
ejpam-841	907	16	variable	variable	NOUN
ejpam-841	907	17	,	,	PUNCT
ejpam-841	907	18	comput	comput	NOUN
ejpam-841	907	19	.	.	PUNCT
ejpam-841	908	1	aided	aid	VERB
ejpam-841	908	2	geom	geom	NOUN
ejpam-841	908	3	.	.	PUNCT
ejpam-841	909	1	design	design	VERB
ejpam-841	909	2	26	26	NUM
ejpam-841	909	3	(	(	PUNCT
ejpam-841	909	4	2009	2009	NUM
ejpam-841	909	5	)	)	PUNCT
ejpam-841	909	6	,	,	PUNCT
ejpam-841	909	7	217–230	217–230	NUM
ejpam-841	909	8	.	.	PUNCT
ejpam-841	910	1	references	reference	NOUN
ejpam-841	910	2	632	632	NUM
ejpam-841	911	1	[	[	X
ejpam-841	911	2	25	25	NUM
ejpam-841	911	3	]	]	X
ejpam-841	911	4	w.	w.	PROPN
ejpam-841	911	5	v.	v.	PROPN
ejpam-841	911	6	vasconcelos	vasconcelos	PROPN
ejpam-841	911	7	,	,	PUNCT
ejpam-841	911	8	.	.	PUNCT
ejpam-841	912	1	arithmetic	arithmetic	PROPN
ejpam-841	912	2	of	of	ADP
ejpam-841	912	3	blowup	blowup	ADJ
ejpam-841	912	4	algebras	algebra	NOUN
ejpam-841	912	5	.	.	PUNCT
ejpam-841	913	1	london	london	PROPN
ejpam-841	913	2	mathematical	mathematical	ADJ
ejpam-841	913	3	society	society	NOUN
ejpam-841	913	4	lecture	lecture	NOUN
ejpam-841	913	5	note	note	NOUN
ejpam-841	913	6	series	series	NOUN
ejpam-841	913	7	,	,	PUNCT
ejpam-841	913	8	195	195	NUM
ejpam-841	913	9	.	.	PUNCT
ejpam-841	913	10	cambridge	cambridge	PROPN
ejpam-841	913	11	university	university	PROPN
ejpam-841	913	12	press	press	PROPN
ejpam-841	913	13	,	,	PUNCT
ejpam-841	913	14	cambridge	cambridge	PROPN
ejpam-841	913	15	,	,	PUNCT
ejpam-841	913	16	1994	1994	NUM
ejpam-841	913	17	.	.	PUNCT
ejpam-841	914	1	[	[	X
ejpam-841	914	2	26	26	NUM
ejpam-841	914	3	]	]	X
ejpam-841	914	4	h.	h.	PROPN
ejpam-841	914	5	wang	wang	PROPN
ejpam-841	914	6	,	,	PUNCT
ejpam-841	914	7	x.	x.	PROPN
ejpam-841	914	8	jia	jia	PROPN
ejpam-841	914	9	,	,	PUNCT
ejpam-841	914	10	and	and	CCONJ
ejpam-841	914	11	r.	r.	PROPN
ejpam-841	914	12	goldman	goldman	PROPN
ejpam-841	914	13	,	,	PUNCT
ejpam-841	914	14	axial	axial	ADJ
ejpam-841	914	15	moving	move	VERB
ejpam-841	914	16	planes	plane	NOUN
ejpam-841	914	17	and	and	CCONJ
ejpam-841	914	18	singularities	singularity	NOUN
ejpam-841	914	19	of	of	ADP
ejpam-841	914	20	rational	rational	ADJ
ejpam-841	914	21	space	space	NOUN
ejpam-841	914	22	curves	curve	NOUN
ejpam-841	914	23	,	,	PUNCT
ejpam-841	914	24	comput	comput	NOUN
ejpam-841	914	25	.	.	PUNCT
ejpam-841	914	26	aided	aid	VERB
ejpam-841	914	27	geom	geom	NOUN
ejpam-841	914	28	.	.	PUNCT
ejpam-841	915	1	design	design	VERB
ejpam-841	915	2	26	26	NUM
ejpam-841	915	3	(	(	PUNCT
ejpam-841	915	4	2009	2009	NUM
ejpam-841	915	5	)	)	PUNCT
ejpam-841	915	6	,	,	PUNCT
ejpam-841	915	7	300–316	300–316	NUM
ejpam-841	915	8	.	.	PUNCT
ejpam-841	916	1	[	[	X
ejpam-841	916	2	27	27	NUM
ejpam-841	916	3	]	]	X
ejpam-841	916	4	s.	s.	PROPN
ejpam-841	916	5	xambó	xambó	PROPN
ejpam-841	916	6	,	,	PUNCT
ejpam-841	916	7	scrolls	scroll	NOUN
ejpam-841	916	8	and	and	CCONJ
ejpam-841	916	9	quartics	quartic	NOUN
ejpam-841	916	10	,	,	PUNCT
ejpam-841	916	11	collectanea	collectanea	PROPN
ejpam-841	916	12	mathematica	mathematica	PROPN
ejpam-841	916	13	,	,	PUNCT
ejpam-841	916	14	33	33	NUM
ejpam-841	916	15	(	(	PUNCT
ejpam-841	916	16	1982	1982	NUM
ejpam-841	916	17	)	)	PUNCT
ejpam-841	916	18	,	,	PUNCT
ejpam-841	916	19	89–101	89–101	PROPN
ejpam-841	916	20	.	.	PUNCT