id	sid	tid	token	lemma	pos
ap-4607	1	1	acta	acta	PROPN
ap-4607	1	2	polytechnica	polytechnica	PROPN
ap-4607	1	3	doi:10.14311	doi:10.14311	PROPN
ap-4607	1	4	/	/	SYM
ap-4607	1	5	ap.2017.57.0404	ap.2017.57.0404	PROPN
ap-4607	1	6	acta	acta	PROPN
ap-4607	1	7	polytechnica	polytechnica	PROPN
ap-4607	1	8	57(6):404–411	57(6):404–411	PROPN
ap-4607	1	9	,	,	PUNCT
ap-4607	1	10	2017	2017	NUM
ap-4607	1	11	©	©	PROPN
ap-4607	1	12	czech	czech	PROPN
ap-4607	1	13	technical	technical	PROPN
ap-4607	1	14	university	university	PROPN
ap-4607	1	15	in	in	ADP
ap-4607	1	16	prague	prague	PROPN
ap-4607	1	17	,	,	PUNCT
ap-4607	1	18	2017	2017	NUM
ap-4607	1	19	available	available	ADJ
ap-4607	1	20	online	online	ADV
ap-4607	1	21	at	at	ADP
ap-4607	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4607	1	23	the	the	DET
ap-4607	1	24	analysis	analysis	NOUN
ap-4607	1	25	of	of	ADP
ap-4607	1	26	images	image	NOUN
ap-4607	1	27	in	in	ADP
ap-4607	1	28	n	n	CCONJ
ap-4607	1	29	-	-	PUNCT
ap-4607	1	30	point	point	NOUN
ap-4607	1	31	gravitational	gravitational	ADJ
ap-4607	1	32	lens	lens	NOUN
ap-4607	1	33	by	by	ADP
ap-4607	1	34	methods	method	NOUN
ap-4607	1	35	of	of	ADP
ap-4607	1	36	algebraic	algebraic	PROPN
ap-4607	1	37	geometry	geometry	NOUN
ap-4607	1	38	albert	albert	PROPN
ap-4607	1	39	t.	t.	PROPN
ap-4607	1	40	kotvytskiya	kotvytskiya	PROPN
ap-4607	1	41	,	,	PUNCT
ap-4607	1	42	b,∗	b,∗	PROPN
ap-4607	1	43	,	,	PUNCT
ap-4607	1	44	semen	semen	PROPN
ap-4607	1	45	d.	d.	PROPN
ap-4607	1	46	bronzab	bronzab	PROPN
ap-4607	1	47	,	,	PUNCT
ap-4607	1	48	volodymyr	volodymyr	PROPN
ap-4607	1	49	yu	yu	PROPN
ap-4607	1	50	.	.	PROPN
ap-4607	2	1	shablenkoa	shablenkoa	PROPN
ap-4607	2	2	a	a	DET
ap-4607	2	3	karazin	karazin	PROPN
ap-4607	2	4	kharkov	kharkov	PROPN
ap-4607	2	5	national	national	PROPN
ap-4607	2	6	university	university	PROPN
ap-4607	2	7	,	,	PUNCT
ap-4607	2	8	svobody	svobody	VERB
ap-4607	2	9	square	square	ADJ
ap-4607	2	10	4	4	NUM
ap-4607	2	11	,	,	PUNCT
ap-4607	2	12	kharkiv	kharkiv	PROPN
ap-4607	2	13	,	,	PUNCT
ap-4607	2	14	61022	61022	NUM
ap-4607	2	15	,	,	PUNCT
ap-4607	2	16	ukraine	ukraine	PROPN
ap-4607	2	17	b	b	PROPN
ap-4607	2	18	ukrainian	ukrainian	ADJ
ap-4607	2	19	state	state	PROPN
ap-4607	2	20	university	university	PROPN
ap-4607	2	21	of	of	ADP
ap-4607	2	22	railway	railway	NOUN
ap-4607	2	23	transport	transport	NOUN
ap-4607	2	24	,	,	PUNCT
ap-4607	2	25	feierbakh	feierbakh	PROPN
ap-4607	2	26	square	square	PROPN
ap-4607	2	27	7	7	NUM
ap-4607	2	28	,	,	PUNCT
ap-4607	2	29	61050	61050	NUM
ap-4607	2	30	,	,	PUNCT
ap-4607	2	31	kharkiv	kharkiv	PROPN
ap-4607	2	32	,	,	PUNCT
ap-4607	2	33	ukraine	ukraine	NOUN
ap-4607	2	34	∗	∗	NOUN
ap-4607	2	35	corresponding	correspond	VERB
ap-4607	2	36	author	author	NOUN
ap-4607	2	37	:	:	PUNCT
ap-4607	2	38	kotvytskiy@gmail.com	kotvytskiy@gmail.com	X
ap-4607	3	1	abstract	abstract	ADJ
ap-4607	3	2	.	.	PUNCT
ap-4607	4	1	this	this	DET
ap-4607	4	2	paper	paper	NOUN
ap-4607	4	3	is	be	AUX
ap-4607	4	4	devoted	devote	VERB
ap-4607	4	5	to	to	ADP
ap-4607	4	6	the	the	DET
ap-4607	4	7	study	study	NOUN
ap-4607	4	8	of	of	ADP
ap-4607	4	9	images	image	NOUN
ap-4607	4	10	in	in	ADP
ap-4607	4	11	n	n	PRON
ap-4607	4	12	-point	-point	NOUN
ap-4607	4	13	gravitational	gravitational	ADJ
ap-4607	4	14	lenses	lense	NOUN
ap-4607	4	15	by	by	ADP
ap-4607	4	16	methods	method	NOUN
ap-4607	4	17	of	of	ADP
ap-4607	4	18	algebraic	algebraic	ADJ
ap-4607	4	19	geometry	geometry	NOUN
ap-4607	4	20	.	.	PUNCT
ap-4607	5	1	in	in	ADP
ap-4607	5	2	the	the	DET
ap-4607	5	3	beginning	beginning	NOUN
ap-4607	5	4	,	,	PUNCT
ap-4607	5	5	we	we	PRON
ap-4607	5	6	carefully	carefully	ADV
ap-4607	5	7	define	define	VERB
ap-4607	5	8	images	image	NOUN
ap-4607	5	9	in	in	ADP
ap-4607	5	10	algebraic	algebraic	ADJ
ap-4607	5	11	terms	term	NOUN
ap-4607	5	12	.	.	PUNCT
ap-4607	6	1	based	base	VERB
ap-4607	6	2	on	on	ADP
ap-4607	6	3	the	the	DET
ap-4607	6	4	definition	definition	NOUN
ap-4607	6	5	,	,	PUNCT
ap-4607	6	6	we	we	PRON
ap-4607	6	7	show	show	VERB
ap-4607	6	8	that	that	SCONJ
ap-4607	6	9	in	in	ADP
ap-4607	6	10	this	this	DET
ap-4607	6	11	model	model	NOUN
ap-4607	6	12	of	of	ADP
ap-4607	6	13	gravitational	gravitational	ADJ
ap-4607	6	14	lenses	lense	NOUN
ap-4607	6	15	(	(	PUNCT
ap-4607	6	16	for	for	ADP
ap-4607	6	17	a	a	DET
ap-4607	6	18	point	point	NOUN
ap-4607	6	19	source	source	NOUN
ap-4607	6	20	)	)	PUNCT
ap-4607	6	21	,	,	PUNCT
ap-4607	6	22	the	the	DET
ap-4607	6	23	dimensions	dimension	NOUN
ap-4607	6	24	of	of	ADP
ap-4607	6	25	the	the	DET
ap-4607	6	26	images	image	NOUN
ap-4607	6	27	can	can	AUX
ap-4607	6	28	be	be	AUX
ap-4607	6	29	only	only	ADV
ap-4607	6	30	0	0	NUM
ap-4607	6	31	and	and	CCONJ
ap-4607	6	32	1	1	NUM
ap-4607	6	33	.	.	X
ap-4607	7	1	we	we	PRON
ap-4607	7	2	reduce	reduce	VERB
ap-4607	7	3	it	it	PRON
ap-4607	7	4	to	to	ADP
ap-4607	7	5	the	the	DET
ap-4607	7	6	fundamental	fundamental	ADJ
ap-4607	7	7	problem	problem	NOUN
ap-4607	7	8	of	of	ADP
ap-4607	7	9	classical	classical	ADJ
ap-4607	7	10	algebraic	algebraic	ADJ
ap-4607	7	11	geometry	geometry	NOUN
ap-4607	7	12	the	the	DET
ap-4607	7	13	study	study	NOUN
ap-4607	7	14	of	of	ADP
ap-4607	7	15	solutions	solution	NOUN
ap-4607	7	16	of	of	ADP
ap-4607	7	17	a	a	DET
ap-4607	7	18	polynomial	polynomial	ADJ
ap-4607	7	19	system	system	NOUN
ap-4607	7	20	of	of	ADP
ap-4607	7	21	equations	equation	NOUN
ap-4607	7	22	.	.	PUNCT
ap-4607	8	1	further	far	ADV
ap-4607	8	2	,	,	PUNCT
ap-4607	8	3	we	we	PRON
ap-4607	8	4	use	use	VERB
ap-4607	8	5	well	well	ADV
ap-4607	8	6	-	-	PUNCT
ap-4607	8	7	known	know	VERB
ap-4607	8	8	concepts	concept	NOUN
ap-4607	8	9	and	and	CCONJ
ap-4607	8	10	theorems	theorem	NOUN
ap-4607	8	11	.	.	PUNCT
ap-4607	9	1	we	we	PRON
ap-4607	9	2	adapt	adapt	VERB
ap-4607	9	3	known	know	VERB
ap-4607	9	4	or	or	CCONJ
ap-4607	9	5	prove	prove	VERB
ap-4607	9	6	new	new	ADJ
ap-4607	9	7	assertions	assertion	NOUN
ap-4607	9	8	.	.	PUNCT
ap-4607	10	1	sometimes	sometimes	ADV
ap-4607	10	2	,	,	PUNCT
ap-4607	10	3	these	these	DET
ap-4607	10	4	statements	statement	NOUN
ap-4607	10	5	have	have	VERB
ap-4607	10	6	a	a	DET
ap-4607	10	7	fairly	fairly	ADV
ap-4607	10	8	general	general	ADJ
ap-4607	10	9	form	form	NOUN
ap-4607	10	10	and	and	CCONJ
ap-4607	10	11	can	can	AUX
ap-4607	10	12	be	be	AUX
ap-4607	10	13	applied	apply	VERB
ap-4607	10	14	to	to	ADP
ap-4607	10	15	other	other	ADJ
ap-4607	10	16	problems	problem	NOUN
ap-4607	10	17	of	of	ADP
ap-4607	10	18	algebraic	algebraic	ADJ
ap-4607	10	19	geometry	geometry	NOUN
ap-4607	10	20	.	.	PUNCT
ap-4607	11	1	in	in	ADP
ap-4607	11	2	this	this	DET
ap-4607	11	3	paper	paper	NOUN
ap-4607	11	4	,	,	PUNCT
ap-4607	11	5	the	the	DET
ap-4607	11	6	criterion	criterion	NOUN
ap-4607	11	7	for	for	ADP
ap-4607	11	8	irreducibility	irreducibility	NOUN
ap-4607	11	9	of	of	ADP
ap-4607	11	10	polynomials	polynomial	NOUN
ap-4607	11	11	in	in	ADP
ap-4607	11	12	several	several	ADJ
ap-4607	11	13	variables	variable	NOUN
ap-4607	11	14	over	over	ADP
ap-4607	11	15	the	the	DET
ap-4607	11	16	field	field	NOUN
ap-4607	11	17	of	of	ADP
ap-4607	11	18	complex	complex	ADJ
ap-4607	11	19	numbers	number	NOUN
ap-4607	11	20	is	be	AUX
ap-4607	11	21	effectively	effectively	ADV
ap-4607	11	22	used	use	VERB
ap-4607	11	23	.	.	PUNCT
ap-4607	12	1	in	in	ADP
ap-4607	12	2	this	this	DET
ap-4607	12	3	paper	paper	NOUN
ap-4607	12	4	,	,	PUNCT
ap-4607	12	5	an	an	DET
ap-4607	12	6	algebraic	algebraic	ADJ
ap-4607	12	7	version	version	NOUN
ap-4607	12	8	of	of	ADP
ap-4607	12	9	the	the	DET
ap-4607	12	10	bezout	bezout	NOUN
ap-4607	12	11	theorem	theorem	NOUN
ap-4607	12	12	and	and	CCONJ
ap-4607	12	13	some	some	DET
ap-4607	12	14	other	other	ADJ
ap-4607	12	15	statements	statement	NOUN
ap-4607	12	16	are	be	AUX
ap-4607	12	17	formulated	formulate	VERB
ap-4607	12	18	and	and	CCONJ
ap-4607	12	19	proved	prove	VERB
ap-4607	12	20	.	.	PUNCT
ap-4607	13	1	we	we	PRON
ap-4607	13	2	have	have	AUX
ap-4607	13	3	applied	apply	VERB
ap-4607	13	4	the	the	DET
ap-4607	13	5	theorems	theorem	NOUN
ap-4607	13	6	proved	prove	VERB
ap-4607	13	7	by	by	ADP
ap-4607	13	8	us	we	PRON
ap-4607	13	9	to	to	PART
ap-4607	13	10	study	study	VERB
ap-4607	13	11	the	the	DET
ap-4607	13	12	imaging	imaging	NOUN
ap-4607	13	13	of	of	ADP
ap-4607	13	14	dimensions	dimension	NOUN
ap-4607	13	15	1	1	NUM
ap-4607	13	16	and	and	CCONJ
ap-4607	13	17	0	0	NUM
ap-4607	13	18	.	.	PUNCT
ap-4607	14	1	keywords	keyword	NOUN
ap-4607	14	2	:	:	PUNCT
ap-4607	14	3	gravitational	gravitational	ADJ
ap-4607	14	4	lense	lense	NOUN
ap-4607	14	5	,	,	PUNCT
ap-4607	14	6	images	image	NOUN
ap-4607	14	7	,	,	PUNCT
ap-4607	14	8	algebaric	algebaric	ADJ
ap-4607	14	9	geometry	geometry	NOUN
ap-4607	14	10	,	,	PUNCT
ap-4607	14	11	resultant	resultant	NOUN
ap-4607	14	12	.	.	PUNCT
ap-4607	15	1	1	1	X
ap-4607	15	2	.	.	X
ap-4607	15	3	introduction	introduction	NOUN
ap-4607	15	4	in	in	ADP
ap-4607	15	5	modern	modern	ADJ
ap-4607	15	6	astrophysics	astrophysic	NOUN
ap-4607	15	7	,	,	PUNCT
ap-4607	15	8	gravitational	gravitational	ADJ
ap-4607	15	9	lensing	lensing	NOUN
ap-4607	15	10	has	have	AUX
ap-4607	15	11	been	be	AUX
ap-4607	15	12	transformed	transform	VERB
ap-4607	15	13	from	from	ADP
ap-4607	15	14	an	an	DET
ap-4607	15	15	effect	effect	NOUN
ap-4607	15	16	that	that	PRON
ap-4607	15	17	confirms	confirm	VERB
ap-4607	15	18	the	the	DET
ap-4607	15	19	general	general	ADJ
ap-4607	15	20	theory	theory	NOUN
ap-4607	15	21	of	of	ADP
ap-4607	15	22	relativity	relativity	NOUN
ap-4607	15	23	to	to	ADP
ap-4607	15	24	the	the	DET
ap-4607	15	25	research	research	NOUN
ap-4607	15	26	tool	tool	NOUN
ap-4607	15	27	.	.	PUNCT
ap-4607	16	1	gravitational	gravitational	ADJ
ap-4607	16	2	lensing	lensing	NOUN
ap-4607	16	3	is	be	AUX
ap-4607	16	4	used	use	VERB
ap-4607	16	5	to	to	PART
ap-4607	16	6	study	study	VERB
ap-4607	16	7	both	both	DET
ap-4607	16	8	stellar	stellar	ADJ
ap-4607	16	9	systems	system	NOUN
ap-4607	16	10	and	and	CCONJ
ap-4607	16	11	planets	planet	NOUN
ap-4607	16	12	in	in	ADP
ap-4607	16	13	them	they	PRON
ap-4607	16	14	,	,	PUNCT
ap-4607	16	15	and	and	CCONJ
ap-4607	16	16	galaxies	galaxy	NOUN
ap-4607	16	17	and	and	CCONJ
ap-4607	16	18	systems	system	NOUN
ap-4607	16	19	of	of	ADP
ap-4607	16	20	galaxies	galaxy	NOUN
ap-4607	16	21	.	.	PUNCT
ap-4607	17	1	even	even	ADV
ap-4607	17	2	the	the	DET
ap-4607	17	3	cosmological	cosmological	ADJ
ap-4607	17	4	parameters	parameter	NOUN
ap-4607	17	5	of	of	ADP
ap-4607	17	6	the	the	DET
ap-4607	17	7	entire	entire	ADJ
ap-4607	17	8	metagalaxy	metagalaxy	NOUN
ap-4607	17	9	are	be	AUX
ap-4607	17	10	investigated	investigate	VERB
ap-4607	17	11	.	.	PUNCT
ap-4607	18	1	from	from	ADP
ap-4607	18	2	this	this	DET
ap-4607	18	3	point	point	NOUN
ap-4607	18	4	of	of	ADP
ap-4607	18	5	view	view	NOUN
ap-4607	18	6	,	,	PUNCT
ap-4607	18	7	it	it	PRON
ap-4607	18	8	seems	seem	VERB
ap-4607	18	9	rather	rather	ADV
ap-4607	18	10	strange	strange	ADJ
ap-4607	18	11	that	that	SCONJ
ap-4607	18	12	until	until	ADP
ap-4607	18	13	now	now	ADV
ap-4607	18	14	a	a	DET
ap-4607	18	15	complete	complete	ADJ
ap-4607	18	16	analytical	analytical	ADJ
ap-4607	18	17	description	description	NOUN
ap-4607	18	18	has	have	AUX
ap-4607	18	19	been	be	AUX
ap-4607	18	20	performed	perform	VERB
ap-4607	18	21	only	only	ADV
ap-4607	18	22	for	for	ADP
ap-4607	18	23	the	the	DET
ap-4607	18	24	simplest	simple	ADJ
ap-4607	18	25	lenses	lense	NOUN
ap-4607	18	26	axially	axially	ADV
ap-4607	18	27	symmetric	symmetric	ADJ
ap-4607	18	28	lenses	lense	NOUN
ap-4607	18	29	(	(	PUNCT
ap-4607	18	30	see	see	VERB
ap-4607	18	31	for	for	ADP
ap-4607	18	32	example	example	NOUN
ap-4607	19	1	[	[	X
ap-4607	19	2	1	1	NUM
ap-4607	19	3	]	]	PUNCT
ap-4607	19	4	or	or	CCONJ
ap-4607	19	5	straight	straight	ADV
ap-4607	19	6	infinite	infinite	ADJ
ap-4607	19	7	cosmic	cosmic	ADJ
ap-4607	19	8	strings	string	NOUN
ap-4607	19	9	[	[	X
ap-4607	19	10	2	2	NUM
ap-4607	19	11	]	]	PUNCT
ap-4607	19	12	.	.	PUNCT
ap-4607	20	1	to	to	PART
ap-4607	20	2	analyze	analyze	VERB
ap-4607	20	3	fairly	fairly	ADV
ap-4607	20	4	simple	simple	ADJ
ap-4607	20	5	2	2	NUM
ap-4607	20	6	-	-	PUNCT
ap-4607	20	7	point	point	NOUN
ap-4607	20	8	gravitational	gravitational	ADJ
ap-4607	20	9	lenses	lense	NOUN
ap-4607	20	10	,	,	PUNCT
ap-4607	20	11	only	only	ADV
ap-4607	20	12	approximate	approximate	ADJ
ap-4607	20	13	or	or	CCONJ
ap-4607	20	14	numerical	numerical	ADJ
ap-4607	20	15	methods	method	NOUN
ap-4607	20	16	are	be	AUX
ap-4607	20	17	used	use	VERB
ap-4607	20	18	[	[	PUNCT
ap-4607	20	19	3	3	NUM
ap-4607	20	20	,	,	PUNCT
ap-4607	20	21	4	4	NUM
ap-4607	20	22	]	]	PUNCT
ap-4607	20	23	.	.	PUNCT
ap-4607	21	1	in	in	ADP
ap-4607	21	2	this	this	DET
ap-4607	21	3	paper	paper	NOUN
ap-4607	21	4	,	,	PUNCT
ap-4607	21	5	the	the	DET
ap-4607	21	6	authors	author	NOUN
ap-4607	21	7	continue	continue	VERB
ap-4607	21	8	the	the	DET
ap-4607	21	9	analytic	analytic	ADJ
ap-4607	21	10	study	study	NOUN
ap-4607	21	11	of	of	ADP
ap-4607	21	12	n	n	CCONJ
ap-4607	21	13	-	-	PUNCT
ap-4607	21	14	point	point	NOUN
ap-4607	21	15	gravitational	gravitational	ADJ
ap-4607	21	16	lenses	lense	NOUN
ap-4607	21	17	by	by	ADP
ap-4607	21	18	methods	method	NOUN
ap-4607	21	19	of	of	ADP
ap-4607	21	20	algebraic	algebraic	ADJ
ap-4607	21	21	geometry	geometry	NOUN
ap-4607	21	22	[	[	X
ap-4607	21	23	5–8	5–8	NOUN
ap-4607	21	24	]	]	PUNCT
ap-4607	21	25	.	.	PUNCT
ap-4607	22	1	in	in	ADP
ap-4607	22	2	physics	physics	PROPN
ap-4607	22	3	,	,	PUNCT
ap-4607	22	4	the	the	DET
ap-4607	22	5	concept	concept	NOUN
ap-4607	22	6	of	of	ADP
ap-4607	22	7	“	"	PUNCT
ap-4607	22	8	image	image	NOUN
ap-4607	22	9	in	in	ADP
ap-4607	22	10	a	a	DET
ap-4607	22	11	gravitational	gravitational	ADJ
ap-4607	22	12	lens	len	NOUN
ap-4607	22	13	”	"	PUNCT
ap-4607	22	14	is	be	AUX
ap-4607	22	15	understood	understand	VERB
ap-4607	22	16	intuitively	intuitively	ADV
ap-4607	22	17	and	and	CCONJ
ap-4607	22	18	is	be	AUX
ap-4607	22	19	usually	usually	ADV
ap-4607	22	20	not	not	PART
ap-4607	22	21	determined	determine	VERB
ap-4607	22	22	.	.	PUNCT
ap-4607	23	1	however	however	ADV
ap-4607	23	2	,	,	PUNCT
ap-4607	23	3	the	the	DET
ap-4607	23	4	absence	absence	NOUN
ap-4607	23	5	of	of	ADP
ap-4607	23	6	a	a	DET
ap-4607	23	7	definition	definition	NOUN
ap-4607	23	8	can	can	AUX
ap-4607	23	9	lead	lead	VERB
ap-4607	23	10	to	to	ADP
ap-4607	23	11	ambiguous	ambiguous	ADJ
ap-4607	23	12	understanding	understanding	NOUN
ap-4607	23	13	of	of	ADP
ap-4607	23	14	the	the	DET
ap-4607	23	15	concept	concept	NOUN
ap-4607	23	16	and	and	CCONJ
ap-4607	23	17	a	a	DET
ap-4607	23	18	different	different	ADJ
ap-4607	23	19	interpretation	interpretation	NOUN
ap-4607	23	20	of	of	ADP
ap-4607	23	21	some	some	DET
ap-4607	23	22	results	result	NOUN
ap-4607	23	23	,	,	PUNCT
ap-4607	23	24	for	for	ADP
ap-4607	23	25	example	example	NOUN
ap-4607	23	26	,	,	PUNCT
ap-4607	23	27	the	the	DET
ap-4607	23	28	theorem	theorem	NOUN
ap-4607	23	29	on	on	ADP
ap-4607	23	30	the	the	DET
ap-4607	23	31	odd	odd	ADJ
ap-4607	23	32	number	number	NOUN
ap-4607	23	33	of	of	ADP
ap-4607	23	34	images	image	NOUN
ap-4607	23	35	[	[	X
ap-4607	23	36	9	9	NUM
ap-4607	23	37	,	,	PUNCT
ap-4607	23	38	10	10	NUM
ap-4607	23	39	]	]	PUNCT
ap-4607	23	40	.	.	PUNCT
ap-4607	24	1	on	on	ADP
ap-4607	24	2	the	the	DET
ap-4607	24	3	other	other	ADJ
ap-4607	24	4	hand	hand	NOUN
ap-4607	24	5	,	,	PUNCT
ap-4607	24	6	the	the	DET
ap-4607	24	7	terminology	terminology	NOUN
ap-4607	24	8	developed	develop	VERB
ap-4607	24	9	in	in	ADP
ap-4607	24	10	algebraic	algebraic	ADJ
ap-4607	24	11	geometry	geometry	NOUN
ap-4607	24	12	makes	make	VERB
ap-4607	24	13	it	it	PRON
ap-4607	24	14	possible	possible	ADJ
ap-4607	24	15	to	to	PART
ap-4607	24	16	pinpoint	pinpoint	VERB
ap-4607	24	17	the	the	DET
ap-4607	24	18	concept	concept	NOUN
ap-4607	24	19	of	of	ADP
ap-4607	24	20	an	an	DET
ap-4607	24	21	image	image	NOUN
ap-4607	24	22	in	in	ADP
ap-4607	24	23	a	a	DET
ap-4607	24	24	gravitational	gravitational	ADJ
ap-4607	24	25	lens	len	NOUN
ap-4607	24	26	.	.	PUNCT
ap-4607	25	1	on	on	ADP
ap-4607	25	2	this	this	DET
ap-4607	25	3	basis	basis	NOUN
ap-4607	25	4	it	it	PRON
ap-4607	25	5	is	be	AUX
ap-4607	25	6	possible	possible	ADJ
ap-4607	25	7	to	to	PART
ap-4607	25	8	formulate	formulate	VERB
ap-4607	25	9	a	a	DET
ap-4607	25	10	number	number	NOUN
ap-4607	25	11	of	of	ADP
ap-4607	25	12	statements	statement	NOUN
ap-4607	25	13	.	.	PUNCT
ap-4607	26	1	2	2	X
ap-4607	26	2	.	.	X
ap-4607	26	3	the	the	DET
ap-4607	26	4	physical	physical	ADJ
ap-4607	26	5	formulation	formulation	NOUN
ap-4607	26	6	of	of	ADP
ap-4607	26	7	the	the	DET
ap-4607	26	8	problem	problem	NOUN
ap-4607	26	9	from	from	ADP
ap-4607	26	10	an	an	DET
ap-4607	26	11	algebraic	algebraic	ADJ
ap-4607	26	12	point	point	NOUN
ap-4607	26	13	of	of	ADP
ap-4607	26	14	view	view	NOUN
ap-4607	26	15	for	for	ADP
ap-4607	26	16	the	the	DET
ap-4607	26	17	model	model	NOUN
ap-4607	26	18	of	of	ADP
ap-4607	26	19	a	a	DET
ap-4607	26	20	plane	plane	NOUN
ap-4607	26	21	gravitational	gravitational	ADJ
ap-4607	26	22	lens	len	NOUN
ap-4607	26	23	,	,	PUNCT
ap-4607	26	24	we	we	PRON
ap-4607	26	25	can	can	AUX
ap-4607	26	26	write	write	VERB
ap-4607	26	27	the	the	DET
ap-4607	26	28	equation	equation	NOUN
ap-4607	26	29	that	that	PRON
ap-4607	26	30	connects	connect	VERB
ap-4607	26	31	the	the	DET
ap-4607	26	32	coordinates	coordinate	NOUN
ap-4607	26	33	of	of	ADP
ap-4607	26	34	the	the	DET
ap-4607	26	35	source	source	NOUN
ap-4607	26	36	(	(	PUNCT
ap-4607	26	37	the	the	DET
ap-4607	26	38	radius	radius	NOUN
ap-4607	26	39	vector	vector	NOUN
ap-4607	26	40	~y	~y	NUM
ap-4607	26	41	)	)	PUNCT
ap-4607	26	42	and	and	CCONJ
ap-4607	26	43	the	the	DET
ap-4607	26	44	image	image	NOUN
ap-4607	26	45	coordinates	coordinate	NOUN
ap-4607	26	46	(	(	PUNCT
ap-4607	26	47	radius	radius	NOUN
ap-4607	26	48	vector	vector	NOUN
ap-4607	26	49	~x	~x	NUM
ap-4607	26	50	)	)	PUNCT
ap-4607	26	51	,	,	PUNCT
ap-4607	26	52	see	see	VERB
ap-4607	26	53	[	[	X
ap-4607	26	54	9	9	NUM
ap-4607	26	55	,	,	PUNCT
ap-4607	26	56	10	10	NUM
ap-4607	26	57	]	]	PUNCT
ap-4607	26	58	~y	~y	NUM
ap-4607	26	59	=	=	SYM
ap-4607	26	60	~x−	~x−	NOUN
ap-4607	26	61	~α	~α	NUM
ap-4607	26	62	,	,	PUNCT
ap-4607	26	63	(	(	PUNCT
ap-4607	26	64	1	1	X
ap-4607	26	65	)	)	PUNCT
ap-4607	26	66	where	where	SCONJ
ap-4607	26	67	~α	~α	NUM
ap-4607	26	68	is	be	AUX
ap-4607	26	69	the	the	DET
ap-4607	26	70	total	total	ADJ
ap-4607	26	71	angle	angle	NOUN
ap-4607	26	72	of	of	ADP
ap-4607	26	73	deflection	deflection	NOUN
ap-4607	26	74	of	of	ADP
ap-4607	26	75	the	the	DET
ap-4607	26	76	light	light	ADJ
ap-4607	26	77	beam	beam	NOUN
ap-4607	26	78	in	in	ADP
ap-4607	26	79	the	the	DET
ap-4607	26	80	plane	plane	NOUN
ap-4607	26	81	of	of	ADP
ap-4607	26	82	the	the	DET
ap-4607	26	83	lens	len	NOUN
ap-4607	26	84	.	.	PUNCT
ap-4607	27	1	in	in	ADP
ap-4607	27	2	the	the	DET
ap-4607	27	3	case	case	NOUN
ap-4607	27	4	of	of	ADP
ap-4607	27	5	an	an	DET
ap-4607	27	6	n	n	CCONJ
ap-4607	27	7	-	-	PUNCT
ap-4607	27	8	point	point	NOUN
ap-4607	27	9	gravitational	gravitational	ADJ
ap-4607	27	10	lens	len	NOUN
ap-4607	27	11	,	,	PUNCT
ap-4607	27	12	the	the	DET
ap-4607	27	13	deflection	deflection	NOUN
ap-4607	27	14	angle	angle	NOUN
ap-4607	27	15	is	be	AUX
ap-4607	27	16	determined	determine	VERB
ap-4607	27	17	by	by	ADP
ap-4607	27	18	the	the	DET
ap-4607	27	19	following	follow	VERB
ap-4607	27	20	expression	expression	NOUN
ap-4607	27	21	:	:	PUNCT
ap-4607	27	22	~α	~α	PUNCT
ap-4607	28	1	=	=	SYM
ap-4607	28	2	n∑	n∑	NOUN
ap-4607	28	3	i=1	i=1	PROPN
ap-4607	28	4	mi	mi	PROPN
ap-4607	28	5	~x−~li∣∣~x−~li∣∣2	~x−~li∣∣~x−~li∣∣2	PROPN
ap-4607	28	6	,	,	PUNCT
ap-4607	28	7	(	(	PUNCT
ap-4607	28	8	2	2	X
ap-4607	28	9	)	)	PUNCT
ap-4607	28	10	where	where	SCONJ
ap-4607	28	11	mi	mi	PROPN
ap-4607	28	12	are	be	AUX
ap-4607	28	13	dimensionless	dimensionless	ADJ
ap-4607	28	14	masses	masse	NOUN
ap-4607	28	15	whose	whose	DET
ap-4607	28	16	position	position	NOUN
ap-4607	28	17	in	in	ADP
ap-4607	28	18	the	the	DET
ap-4607	28	19	plane	plane	NOUN
ap-4607	28	20	of	of	ADP
ap-4607	28	21	the	the	DET
ap-4607	28	22	lens	lens	NOUN
ap-4607	28	23	is	be	AUX
ap-4607	28	24	determined	determine	VERB
ap-4607	28	25	by	by	ADP
ap-4607	28	26	the	the	DET
ap-4607	28	27	radius	radius	NOUN
ap-4607	28	28	vectors	vector	NOUN
ap-4607	28	29	~li	~li	NOUN
ap-4607	28	30	.	.	PUNCT
ap-4607	29	1	we	we	PRON
ap-4607	29	2	have	have	VERB
ap-4607	29	3	that	that	PRON
ap-4607	29	4	holds	hold	VERB
ap-4607	29	5	∑n	∑n	PROPN
ap-4607	29	6	i=1	i=1	PROPN
ap-4607	29	7	mi	mi	PROPN
ap-4607	30	1	=	=	NOUN
ap-4607	30	2	1	1	PROPN
ap-4607	30	3	.	.	PUNCT
ap-4607	30	4	equation	equation	NOUN
ap-4607	30	5	(	(	PUNCT
ap-4607	30	6	1	1	NUM
ap-4607	30	7	)	)	PUNCT
ap-4607	30	8	with	with	ADP
ap-4607	30	9	allowance	allowance	NOUN
ap-4607	30	10	for	for	ADP
ap-4607	30	11	(	(	PUNCT
ap-4607	30	12	2	2	NUM
ap-4607	30	13	)	)	PUNCT
ap-4607	30	14	in	in	ADP
ap-4607	30	15	coordinate	coordinate	NOUN
ap-4607	30	16	form	form	NOUN
ap-4607	30	17	has	have	VERB
ap-4607	30	18	the	the	DET
ap-4607	30	19	form	form	PROPN
ap-4607	30	20	(	(	PUNCT
ap-4607	30	21	x1	x1	PROPN
ap-4607	30	22	−	−	PROPN
ap-4607	31	1	n∑	n∑	PROPN
ap-4607	32	1	i=1	i=1	PROPN
ap-4607	33	1	mi	mi	PROPN
ap-4607	34	1	x1	x1	PROPN
ap-4607	34	2	−	−	PROPN
ap-4607	34	3	ai	ai	VERB
ap-4607	34	4	(	(	PUNCT
ap-4607	34	5	x1	x1	ADJ
ap-4607	34	6	−	−	PROPN
ap-4607	34	7	ai)2	ai)2	PROPN
ap-4607	35	1	+	+	CCONJ
ap-4607	36	1	(	(	PUNCT
ap-4607	36	2	x2	x2	INTJ
ap-4607	36	3	−	−	PROPN
ap-4607	36	4	bi)2	bi)2	NOUN
ap-4607	36	5	)	)	PUNCT
ap-4607	36	6	−	−	NOUN
ap-4607	37	1	y1	y1	NOUN
ap-4607	37	2	=	=	SYM
ap-4607	37	3	0	0	NUM
ap-4607	37	4	,	,	PUNCT
ap-4607	37	5	(	(	PUNCT
ap-4607	37	6	x2	x2	INTJ
ap-4607	37	7	−	−	PROPN
ap-4607	37	8	n∑	n∑	PROPN
ap-4607	37	9	i=1	i=1	PROPN
ap-4607	37	10	mi	mi	PROPN
ap-4607	38	1	x2	x2	PROPN
ap-4607	38	2	−	−	PROPN
ap-4607	38	3	bi	bi	NOUN
ap-4607	38	4	(	(	PUNCT
ap-4607	38	5	x1	x1	PROPN
ap-4607	38	6	−	−	PROPN
ap-4607	38	7	ai)2	ai)2	PROPN
ap-4607	38	8	+	+	CCONJ
ap-4607	38	9	(	(	PUNCT
ap-4607	38	10	x2	x2	INTJ
ap-4607	38	11	−	−	PROPN
ap-4607	39	1	bi)2	bi)2	NOUN
ap-4607	39	2	)	)	PUNCT
ap-4607	39	3	−	−	NOUN
ap-4607	40	1	y2	y2	NOUN
ap-4607	40	2	=	=	SYM
ap-4607	40	3	0	0	PROPN
ap-4607	40	4	,	,	PUNCT
ap-4607	40	5	(	(	PUNCT
ap-4607	40	6	3	3	X
ap-4607	40	7	)	)	PUNCT
ap-4607	40	8	where	where	SCONJ
ap-4607	40	9	ai	ai	VERB
ap-4607	40	10	and	and	CCONJ
ap-4607	40	11	bi	bi	NOUN
ap-4607	40	12	are	be	AUX
ap-4607	40	13	the	the	DET
ap-4607	40	14	coordinates	coordinate	NOUN
ap-4607	40	15	of	of	ADP
ap-4607	40	16	the	the	DET
ap-4607	40	17	radius	radius	NOUN
ap-4607	40	18	-	-	PUNCT
ap-4607	40	19	vector	vector	NOUN
ap-4607	40	20	~li	~li	PROPN
ap-4607	40	21	i.e.	i.e.	X
ap-4607	40	22	~li	~li	NOUN
ap-4607	40	23	=	=	SYM
ap-4607	40	24	(	(	PUNCT
ap-4607	40	25	ai	ai	PROPN
ap-4607	40	26	,	,	PUNCT
ap-4607	40	27	bi	bi	NOUN
ap-4607	40	28	)	)	PUNCT
ap-4607	40	29	.	.	PUNCT
ap-4607	41	1	from	from	ADP
ap-4607	41	2	an	an	DET
ap-4607	41	3	algebraic	algebraic	ADJ
ap-4607	41	4	point	point	NOUN
ap-4607	41	5	of	of	ADP
ap-4607	41	6	view	view	NOUN
ap-4607	41	7	,	,	PUNCT
ap-4607	41	8	system	system	NOUN
ap-4607	41	9	(	(	PUNCT
ap-4607	41	10	3	3	X
ap-4607	41	11	)	)	PUNCT
ap-4607	41	12	is	be	AUX
ap-4607	41	13	a	a	DET
ap-4607	41	14	system	system	NOUN
ap-4607	41	15	of	of	ADP
ap-4607	41	16	two	two	NUM
ap-4607	41	17	rational	rational	ADJ
ap-4607	41	18	equations	equation	NOUN
ap-4607	41	19	(	(	PUNCT
ap-4607	41	20	over	over	ADP
ap-4607	41	21	a	a	DET
ap-4607	41	22	field	field	NOUN
ap-4607	41	23	of	of	ADP
ap-4607	41	24	real	real	ADJ
ap-4607	41	25	numbers	number	NOUN
ap-4607	41	26	)	)	PUNCT
ap-4607	41	27	from	from	ADP
ap-4607	41	28	two	two	NUM
ap-4607	41	29	unknowns	unknown	NOUN
ap-4607	41	30	,	,	PUNCT
ap-4607	41	31	which	which	PRON
ap-4607	41	32	are	be	AUX
ap-4607	41	33	given	give	VERB
ap-4607	41	34	in	in	ADP
ap-4607	41	35	cartesian	cartesian	ADJ
ap-4607	41	36	coordinates	coordinate	NOUN
ap-4607	41	37	on	on	ADP
ap-4607	41	38	the	the	DET
ap-4607	41	39	r2	r2	PROPN
ap-4607	41	40	plane	plane	NOUN
ap-4607	41	41	.	.	PUNCT
ap-4607	42	1	the	the	DET
ap-4607	42	2	system	system	NOUN
ap-4607	42	3	(	(	PUNCT
ap-4607	42	4	3	3	X
ap-4607	42	5	)	)	PUNCT
ap-4607	42	6	will	will	AUX
ap-4607	42	7	be	be	AUX
ap-4607	42	8	considered	consider	VERB
ap-4607	42	9	,	,	PUNCT
ap-4607	42	10	just	just	ADV
ap-4607	42	11	above	above	ADP
ap-4607	42	12	the	the	DET
ap-4607	42	13	field	field	NOUN
ap-4607	42	14	of	of	ADP
ap-4607	42	15	complex	complex	ADJ
ap-4607	42	16	numbers	number	NOUN
ap-4607	42	17	c	c	NOUN
ap-4607	42	18	,	,	PUNCT
ap-4607	42	19	while	while	SCONJ
ap-4607	42	20	we	we	PRON
ap-4607	42	21	denote	denote	VERB
ap-4607	42	22	it	it	PRON
ap-4607	42	23	by	by	ADP
ap-4607	42	24	(	(	PUNCT
ap-4607	42	25	3a	3a	NUM
ap-4607	42	26	)	)	PUNCT
ap-4607	42	27	.	.	PUNCT
ap-4607	43	1	the	the	DET
ap-4607	43	2	set	set	NOUN
ap-4607	43	3	of	of	ADP
ap-4607	43	4	solutions	solution	NOUN
ap-4607	43	5	of	of	ADP
ap-4607	43	6	system	system	NOUN
ap-4607	43	7	(	(	PUNCT
ap-4607	43	8	3	3	X
ap-4607	43	9	)	)	PUNCT
ap-4607	43	10	is	be	AUX
ap-4607	43	11	obviously	obviously	ADV
ap-4607	43	12	the	the	DET
ap-4607	43	13	set	set	NOUN
ap-4607	43	14	of	of	ADP
ap-4607	43	15	real	real	ADJ
ap-4607	43	16	solutions	solution	NOUN
ap-4607	43	17	of	of	ADP
ap-4607	43	18	system	system	NOUN
ap-4607	43	19	(	(	PUNCT
ap-4607	43	20	3a	3a	NUM
ap-4607	43	21	)	)	PUNCT
ap-4607	43	22	.	.	PUNCT
ap-4607	44	1	we	we	PRON
ap-4607	44	2	note	note	VERB
ap-4607	44	3	that	that	SCONJ
ap-4607	44	4	all	all	DET
ap-4607	44	5	the	the	DET
ap-4607	44	6	coefficients	coefficient	NOUN
ap-4607	44	7	of	of	ADP
ap-4607	44	8	the	the	DET
ap-4607	44	9	equations	equation	NOUN
ap-4607	44	10	of	of	ADP
ap-4607	44	11	system	system	NOUN
ap-4607	44	12	(	(	PUNCT
ap-4607	44	13	3a	3a	NUM
ap-4607	44	14	)	)	PUNCT
ap-4607	44	15	are	be	AUX
ap-4607	44	16	real	real	ADJ
ap-4607	44	17	.	.	PUNCT
ap-4607	45	1	404	404	NUM
ap-4607	45	2	http://dx.doi.org/10.14311/ap.2017.57.0404	http://dx.doi.org/10.14311/ap.2017.57.0404	NOUN
ap-4607	45	3	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4607	45	4	vol	vol	NOUN
ap-4607	45	5	.	.	PUNCT
ap-4607	46	1	57	57	NUM
ap-4607	46	2	no	no	NOUN
ap-4607	46	3	.	.	PUNCT
ap-4607	47	1	6/2017	6/2017	X
ap-4607	47	2	the	the	DET
ap-4607	47	3	analysis	analysis	NOUN
ap-4607	47	4	of	of	ADP
ap-4607	47	5	images	image	NOUN
ap-4607	47	6	in	in	ADP
ap-4607	47	7	n	n	CCONJ
ap-4607	47	8	-point	-point	NOUN
ap-4607	47	9	gravitational	gravitational	ADJ
ap-4607	47	10	lens	lens	NOUN
ap-4607	47	11	in	in	ADP
ap-4607	47	12	terms	term	NOUN
ap-4607	47	13	of	of	ADP
ap-4607	47	14	algebraic	algebraic	ADJ
ap-4607	47	15	geometry	geometry	NOUN
ap-4607	47	16	,	,	PUNCT
ap-4607	47	17	the	the	DET
ap-4607	47	18	image	image	NOUN
ap-4607	47	19	of	of	ADP
ap-4607	47	20	a	a	DET
ap-4607	47	21	source	source	NOUN
ap-4607	47	22	in	in	ADP
ap-4607	47	23	an	an	DET
ap-4607	47	24	n	n	CCONJ
ap-4607	47	25	-	-	PUNCT
ap-4607	47	26	point	point	NOUN
ap-4607	47	27	gravitational	gravitational	ADJ
ap-4607	47	28	lens	lens	NOUN
ap-4607	47	29	can	can	AUX
ap-4607	47	30	be	be	AUX
ap-4607	47	31	defined	define	VERB
ap-4607	47	32	as	as	SCONJ
ap-4607	47	33	follows	follow	VERB
ap-4607	47	34	:	:	PUNCT
ap-4607	47	35	definition	definition	NOUN
ap-4607	47	36	.	.	PUNCT
ap-4607	48	1	an	an	DET
ap-4607	48	2	image	image	NOUN
ap-4607	48	3	of	of	ADP
ap-4607	48	4	a	a	DET
ap-4607	48	5	point	point	NOUN
ap-4607	48	6	source	source	NOUN
ap-4607	48	7	in	in	ADP
ap-4607	48	8	an	an	DET
ap-4607	48	9	n	n	CCONJ
ap-4607	48	10	-	-	PUNCT
ap-4607	48	11	point	point	NOUN
ap-4607	48	12	gravitational	gravitational	ADJ
ap-4607	48	13	lens	lens	NOUN
ap-4607	48	14	will	will	AUX
ap-4607	48	15	be	be	AUX
ap-4607	48	16	called	call	VERB
ap-4607	48	17	the	the	DET
ap-4607	48	18	real	real	ADJ
ap-4607	48	19	solution	solution	NOUN
ap-4607	48	20	of	of	ADP
ap-4607	48	21	system	system	NOUN
ap-4607	48	22	(	(	PUNCT
ap-4607	48	23	3a	3a	NUM
ap-4607	48	24	)	)	PUNCT
ap-4607	48	25	without	without	ADP
ap-4607	48	26	regard	regard	NOUN
ap-4607	48	27	to	to	ADP
ap-4607	48	28	multiplicity	multiplicity	NOUN
ap-4607	48	29	.	.	PUNCT
ap-4607	49	1	the	the	DET
ap-4607	49	2	set	set	NOUN
ap-4607	49	3	of	of	ADP
ap-4607	49	4	images	image	NOUN
ap-4607	49	5	is	be	AUX
ap-4607	49	6	the	the	DET
ap-4607	49	7	set	set	NOUN
ap-4607	49	8	of	of	ADP
ap-4607	49	9	different	different	ADJ
ap-4607	49	10	real	real	ADJ
ap-4607	49	11	solutions	solution	NOUN
ap-4607	49	12	of	of	ADP
ap-4607	49	13	this	this	DET
ap-4607	49	14	system	system	NOUN
ap-4607	49	15	.	.	PUNCT
ap-4607	50	1	3	3	X
ap-4607	50	2	.	.	X
ap-4607	50	3	reduction	reduction	NOUN
ap-4607	50	4	of	of	ADP
ap-4607	50	5	the	the	DET
ap-4607	50	6	problem	problem	NOUN
ap-4607	50	7	to	to	ADP
ap-4607	50	8	the	the	DET
ap-4607	50	9	fundamental	fundamental	ADJ
ap-4607	50	10	problem	problem	NOUN
ap-4607	50	11	of	of	ADP
ap-4607	50	12	classical	classical	ADJ
ap-4607	50	13	algebraic	algebraic	ADJ
ap-4607	50	14	geometry	geometry	NOUN
ap-4607	50	15	the	the	DET
ap-4607	50	16	main	main	ADJ
ap-4607	50	17	problem	problem	NOUN
ap-4607	50	18	of	of	ADP
ap-4607	50	19	classical	classical	ADJ
ap-4607	50	20	algebraic	algebraic	ADJ
ap-4607	50	21	geometry	geometry	NOUN
ap-4607	50	22	is	be	AUX
ap-4607	50	23	the	the	DET
ap-4607	50	24	problem	problem	NOUN
ap-4607	50	25	of	of	ADP
ap-4607	50	26	studying	study	VERB
ap-4607	50	27	systems	system	NOUN
ap-4607	50	28	of	of	ADP
ap-4607	50	29	polynomial	polynomial	ADJ
ap-4607	50	30	equations	equation	NOUN
ap-4607	50	31	.	.	PUNCT
ap-4607	51	1	let	let	VERB
ap-4607	51	2	us	we	PRON
ap-4607	51	3	investigate	investigate	VERB
ap-4607	51	4	the	the	DET
ap-4607	51	5	set	set	NOUN
ap-4607	51	6	of	of	ADP
ap-4607	51	7	solutions	solution	NOUN
ap-4607	51	8	of	of	ADP
ap-4607	51	9	system	system	NOUN
ap-4607	51	10	(	(	PUNCT
ap-4607	51	11	3a	3a	NUM
ap-4607	51	12	)	)	PUNCT
ap-4607	51	13	.	.	PUNCT
ap-4607	52	1	to	to	PART
ap-4607	52	2	do	do	VERB
ap-4607	52	3	this	this	PRON
ap-4607	52	4	,	,	PUNCT
ap-4607	52	5	we	we	PRON
ap-4607	52	6	transform	transform	VERB
ap-4607	52	7	the	the	DET
ap-4607	52	8	equations	equation	NOUN
ap-4607	52	9	of	of	ADP
ap-4607	52	10	the	the	DET
ap-4607	52	11	system	system	NOUN
ap-4607	52	12	to	to	ADP
ap-4607	52	13	a	a	DET
ap-4607	52	14	polynomial	polynomial	ADJ
ap-4607	52	15	form	form	NOUN
ap-4607	52	16	f1	f1	NOUN
ap-4607	52	17	=	=	PUNCT
ap-4607	52	18	(	(	PUNCT
ap-4607	52	19	x1−	x1−	PROPN
ap-4607	52	20	y1	y1	PROPN
ap-4607	52	21	)	)	PUNCT
ap-4607	52	22	n∏	n∏	PROPN
ap-4607	53	1	i=1	i=1	PROPN
ap-4607	54	1	hi	hi	INTJ
ap-4607	54	2	−	−	PROPN
ap-4607	55	1	n∑	n∑	PROPN
ap-4607	55	2	j=1	j=1	PROPN
ap-4607	55	3	mj(x1−	mj(x1−	PROPN
ap-4607	55	4	aj	aj	PROPN
ap-4607	55	5	)	)	PUNCT
ap-4607	55	6	n∏	n∏	PROPN
ap-4607	55	7	i=1,i6	i=1,i6	NOUN
ap-4607	55	8	=	=	SYM
ap-4607	55	9	j	j	NOUN
ap-4607	55	10	hi	hi	INTJ
ap-4607	55	11	=	=	SYM
ap-4607	55	12	0	0	PROPN
ap-4607	55	13	,	,	PUNCT
ap-4607	55	14	f2	f2	PROPN
ap-4607	55	15	=	=	SYM
ap-4607	55	16	(	(	PUNCT
ap-4607	55	17	x2−	x2−	PROPN
ap-4607	55	18	y2	y2	PROPN
ap-4607	55	19	)	)	PUNCT
ap-4607	55	20	n∏	n∏	PROPN
ap-4607	56	1	i=1	i=1	PROPN
ap-4607	57	1	hi	hi	INTJ
ap-4607	57	2	−	−	PROPN
ap-4607	58	1	n∑	n∑	PROPN
ap-4607	58	2	i=1	i=1	PROPN
ap-4607	59	1	mi(x2−	mi(x2−	PROPN
ap-4607	59	2	bi	bi	PROPN
ap-4607	59	3	)	)	PUNCT
ap-4607	59	4	n∏	n∏	PROPN
ap-4607	59	5	i=1,i6	i=1,i6	NOUN
ap-4607	60	1	=	=	SYM
ap-4607	60	2	j	j	NOUN
ap-4607	60	3	hi	hi	INTJ
ap-4607	60	4	=	=	NOUN
ap-4607	60	5	0	0	PROPN
ap-4607	60	6	,	,	PUNCT
ap-4607	60	7	(	(	PUNCT
ap-4607	60	8	4	4	X
ap-4607	60	9	)	)	PUNCT
ap-4607	60	10	where	where	SCONJ
ap-4607	60	11	hi	hi	ADV
ap-4607	60	12	=	=	SYM
ap-4607	61	1	(	(	PUNCT
ap-4607	61	2	x1	x1	INTJ
ap-4607	61	3	−	−	PROPN
ap-4607	61	4	ai)2	ai)2	PROPN
ap-4607	61	5	+	+	CCONJ
ap-4607	61	6	(	(	PUNCT
ap-4607	61	7	x2	x2	INTJ
ap-4607	61	8	−	−	PROPN
ap-4607	62	1	bi)2	bi)2	PROPN
ap-4607	62	2	,	,	PUNCT
ap-4607	62	3	i	i	PRON
ap-4607	62	4	=	=	NOUN
ap-4607	62	5	1	1	NUM
ap-4607	62	6	,	,	PUNCT
ap-4607	62	7	2	2	NUM
ap-4607	62	8	,	,	PUNCT
ap-4607	62	9	.	.	PUNCT
ap-4607	62	10	.	.	PUNCT
ap-4607	63	1	.	.	PUNCT
ap-4607	64	1	,	,	PUNCT
ap-4607	64	2	n	n	X
ap-4607	64	3	.	.	PUNCT
ap-4607	65	1	the	the	DET
ap-4607	65	2	polynomial	polynomial	ADJ
ap-4607	65	3	form	form	NOUN
ap-4607	65	4	of	of	ADP
ap-4607	65	5	the	the	DET
ap-4607	65	6	equations	equation	NOUN
ap-4607	65	7	in	in	ADP
ap-4607	65	8	system	system	NOUN
ap-4607	65	9	(	(	PUNCT
ap-4607	65	10	4	4	NUM
ap-4607	65	11	)	)	PUNCT
ap-4607	65	12	is	be	AUX
ap-4607	65	13	necessary	necessary	ADJ
ap-4607	65	14	for	for	ADP
ap-4607	65	15	its	its	PRON
ap-4607	65	16	investigation	investigation	NOUN
ap-4607	65	17	by	by	ADP
ap-4607	65	18	methods	method	NOUN
ap-4607	65	19	of	of	ADP
ap-4607	65	20	algebraic	algebraic	ADJ
ap-4607	65	21	geometry	geometry	NOUN
ap-4607	65	22	.	.	PUNCT
ap-4607	66	1	we	we	PRON
ap-4607	66	2	shall	shall	AUX
ap-4607	66	3	consider	consider	VERB
ap-4607	66	4	the	the	DET
ap-4607	66	5	equations	equation	NOUN
ap-4607	66	6	of	of	ADP
ap-4607	66	7	the	the	DET
ap-4607	66	8	system	system	NOUN
ap-4607	66	9	(	(	PUNCT
ap-4607	66	10	4	4	NUM
ap-4607	66	11	)	)	PUNCT
ap-4607	66	12	over	over	ADP
ap-4607	66	13	the	the	DET
ap-4607	66	14	field	field	NOUN
ap-4607	66	15	c	c	NOUN
ap-4607	66	16	of	of	ADP
ap-4607	66	17	complex	complex	ADJ
ap-4607	66	18	numbers	number	NOUN
ap-4607	66	19	in	in	ADP
ap-4607	66	20	the	the	DET
ap-4607	66	21	affine	affine	NOUN
ap-4607	66	22	coordinate	coordinate	NOUN
ap-4607	66	23	system	system	NOUN
ap-4607	66	24	c2	c2	PROPN
ap-4607	66	25	.	.	PUNCT
ap-4607	67	1	the	the	DET
ap-4607	67	2	system	system	NOUN
ap-4607	67	3	(	(	PUNCT
ap-4607	67	4	4	4	X
ap-4607	67	5	)	)	PUNCT
ap-4607	67	6	is	be	AUX
ap-4607	67	7	not	not	PART
ap-4607	67	8	equivalent	equivalent	ADJ
ap-4607	67	9	to	to	ADP
ap-4607	67	10	the	the	DET
ap-4607	67	11	system	system	NOUN
ap-4607	67	12	(	(	PUNCT
ap-4607	67	13	3a	3a	NUM
ap-4607	67	14	)	)	PUNCT
ap-4607	67	15	,	,	PUNCT
ap-4607	67	16	but	but	CCONJ
ap-4607	67	17	it	it	PRON
ap-4607	67	18	follows	follow	VERB
ap-4607	67	19	from	from	ADP
ap-4607	67	20	it	it	PRON
ap-4607	67	21	.	.	PUNCT
ap-4607	68	1	the	the	DET
ap-4607	68	2	set	set	NOUN
ap-4607	68	3	of	of	ADP
ap-4607	68	4	solutions	solution	NOUN
ap-4607	68	5	of	of	ADP
ap-4607	68	6	the	the	DET
ap-4607	68	7	system	system	NOUN
ap-4607	68	8	(	(	PUNCT
ap-4607	68	9	3a	3a	NUM
ap-4607	68	10	)	)	PUNCT
ap-4607	68	11	can	can	AUX
ap-4607	68	12	be	be	AUX
ap-4607	68	13	obtained	obtain	VERB
ap-4607	68	14	from	from	ADP
ap-4607	68	15	the	the	DET
ap-4607	68	16	set	set	NOUN
ap-4607	68	17	of	of	ADP
ap-4607	68	18	solutions	solution	NOUN
ap-4607	68	19	of	of	ADP
ap-4607	68	20	the	the	DET
ap-4607	68	21	system	system	NOUN
ap-4607	68	22	(	(	PUNCT
ap-4607	68	23	4	4	NUM
ap-4607	68	24	)	)	PUNCT
ap-4607	68	25	.	.	PUNCT
ap-4607	69	1	for	for	ADP
ap-4607	69	2	this	this	PRON
ap-4607	69	3	,	,	PUNCT
ap-4607	69	4	by	by	ADP
ap-4607	69	5	removing	remove	VERB
ap-4607	69	6	solutions	solution	NOUN
ap-4607	69	7	from	from	ADP
ap-4607	69	8	it	it	PRON
ap-4607	69	9	in	in	ADP
ap-4607	69	10	which	which	PRON
ap-4607	69	11	the	the	DET
ap-4607	69	12	system	system	NOUN
ap-4607	69	13	(	(	PUNCT
ap-4607	69	14	3a	3a	NUM
ap-4607	69	15	)	)	PUNCT
ap-4607	69	16	is	be	AUX
ap-4607	69	17	not	not	PART
ap-4607	69	18	defined	define	VERB
ap-4607	69	19	.	.	PUNCT
ap-4607	70	1	these	these	DET
ap-4607	70	2	solutions	solution	NOUN
ap-4607	70	3	are	be	AUX
ap-4607	70	4	pairs	pair	NOUN
ap-4607	70	5	of	of	ADP
ap-4607	70	6	numbers	number	NOUN
ap-4607	70	7	that	that	PRON
ap-4607	70	8	are	be	AUX
ap-4607	70	9	the	the	DET
ap-4607	70	10	coordinates	coordinate	NOUN
ap-4607	70	11	of	of	ADP
ap-4607	70	12	the	the	DET
ap-4607	70	13	point	point	NOUN
ap-4607	70	14	masses	masse	NOUN
ap-4607	70	15	.	.	PUNCT
ap-4607	71	1	indeed	indeed	ADV
ap-4607	71	2	,	,	PUNCT
ap-4607	71	3	we	we	PRON
ap-4607	71	4	directly	directly	ADV
ap-4607	71	5	verify	verify	VERB
ap-4607	71	6	that	that	SCONJ
ap-4607	71	7	the	the	DET
ap-4607	71	8	points	point	NOUN
ap-4607	71	9	with	with	ADP
ap-4607	71	10	coordinates	coordinate	NOUN
ap-4607	71	11	(	(	PUNCT
ap-4607	71	12	ai	ai	VERB
ap-4607	71	13	,	,	PUNCT
ap-4607	71	14	bj	bj	NOUN
ap-4607	71	15	)	)	PUNCT
ap-4607	71	16	,	,	PUNCT
ap-4607	71	17	i	i	PRON
ap-4607	71	18	=	=	NOUN
ap-4607	71	19	1	1	NUM
ap-4607	71	20	,	,	PUNCT
ap-4607	71	21	...	...	PUNCT
ap-4607	71	22	,	,	PUNCT
ap-4607	71	23	n	n	CCONJ
ap-4607	71	24	,	,	PUNCT
ap-4607	71	25	is	be	AUX
ap-4607	71	26	a	a	DET
ap-4607	71	27	solution	solution	NOUN
ap-4607	71	28	of	of	ADP
ap-4607	71	29	the	the	DET
ap-4607	71	30	system	system	NOUN
ap-4607	71	31	(	(	PUNCT
ap-4607	71	32	4	4	NUM
ap-4607	71	33	)	)	PUNCT
ap-4607	71	34	,	,	PUNCT
ap-4607	71	35	but	but	CCONJ
ap-4607	71	36	the	the	DET
ap-4607	71	37	system	system	NOUN
ap-4607	71	38	(	(	PUNCT
ap-4607	71	39	3a	3a	NUM
ap-4607	71	40	)	)	PUNCT
ap-4607	71	41	,	,	PUNCT
ap-4607	71	42	in	in	ADP
ap-4607	71	43	these	these	DET
ap-4607	71	44	points	point	NOUN
ap-4607	71	45	is	be	AUX
ap-4607	71	46	not	not	PART
ap-4607	71	47	defined	define	VERB
ap-4607	71	48	.	.	PUNCT
ap-4607	72	1	let	let	VERB
ap-4607	72	2	f1	f1	PROPN
ap-4607	72	3	and	and	CCONJ
ap-4607	72	4	f2	f2	PROPN
ap-4607	72	5	be	be	AUX
ap-4607	72	6	the	the	DET
ap-4607	72	7	left	left	ADJ
ap-4607	72	8	-	-	PUNCT
ap-4607	72	9	hand	hand	NOUN
ap-4607	72	10	sides	side	NOUN
ap-4607	72	11	of	of	ADP
ap-4607	72	12	the	the	DET
ap-4607	72	13	first	first	ADJ
ap-4607	72	14	and	and	CCONJ
ap-4607	72	15	second	second	ADJ
ap-4607	72	16	equations	equation	NOUN
ap-4607	72	17	of	of	ADP
ap-4607	72	18	system	system	NOUN
ap-4607	72	19	(	(	PUNCT
ap-4607	72	20	3a	3a	NUM
ap-4607	72	21	)	)	PUNCT
ap-4607	72	22	,	,	PUNCT
ap-4607	72	23	m(f1	m(f1	PROPN
ap-4607	72	24	,	,	PUNCT
ap-4607	72	25	f2	f2	PROPN
ap-4607	72	26	)	)	PUNCT
ap-4607	72	27	be	be	AUX
ap-4607	72	28	the	the	DET
ap-4607	72	29	solution	solution	NOUN
ap-4607	72	30	set	set	VERB
ap-4607	72	31	of	of	ADP
ap-4607	72	32	system	system	NOUN
ap-4607	72	33	(	(	PUNCT
ap-4607	72	34	3a	3a	NUM
ap-4607	72	35	)	)	PUNCT
ap-4607	72	36	,	,	PUNCT
ap-4607	72	37	v	v	NOUN
ap-4607	72	38	(	(	PUNCT
ap-4607	72	39	f1	f1	NOUN
ap-4607	72	40	,	,	PUNCT
ap-4607	72	41	f2	f2	PROPN
ap-4607	72	42	)	)	PUNCT
ap-4607	72	43	be	be	VERB
ap-4607	72	44	the	the	DET
ap-4607	72	45	solution	solution	NOUN
ap-4607	72	46	set	set	VERB
ap-4607	72	47	of	of	ADP
ap-4607	72	48	system	system	NOUN
ap-4607	72	49	(	(	PUNCT
ap-4607	72	50	4	4	NUM
ap-4607	72	51	)	)	PUNCT
ap-4607	72	52	,	,	PUNCT
ap-4607	72	53	and	and	CCONJ
ap-4607	72	54	rev	rev	PROPN
ap-4607	72	55	(	(	PUNCT
ap-4607	72	56	f1	f1	PROPN
ap-4607	72	57	,	,	PUNCT
ap-4607	72	58	f2	f2	PROPN
ap-4607	72	59	)	)	PUNCT
ap-4607	72	60	⊂	⊂	PROPN
ap-4607	72	61	v	v	X
ap-4607	72	62	(	(	PUNCT
ap-4607	72	63	f1	f1	NOUN
ap-4607	72	64	,	,	PUNCT
ap-4607	72	65	f2	f2	PROPN
ap-4607	72	66	)	)	PUNCT
ap-4607	72	67	the	the	DET
ap-4607	72	68	subset	subset	NOUN
ap-4607	72	69	of	of	ADP
ap-4607	72	70	its	its	PRON
ap-4607	72	71	real	real	ADJ
ap-4607	72	72	solutions	solution	NOUN
ap-4607	72	73	,	,	PUNCT
ap-4607	72	74	then	then	ADV
ap-4607	72	75	we	we	PRON
ap-4607	72	76	have	have	VERB
ap-4607	72	77	m(f1	m(f1	NOUN
ap-4607	72	78	,	,	PUNCT
ap-4607	72	79	f2	f2	PROPN
ap-4607	72	80	)	)	PUNCT
ap-4607	73	1	=	=	PUNCT
ap-4607	73	2	rev	rev	X
ap-4607	73	3	(	(	PUNCT
ap-4607	73	4	f1	f1	PROPN
ap-4607	73	5	,	,	PUNCT
ap-4607	73	6	f2)/{∪(ai	f2)/{∪(ai	PROPN
ap-4607	73	7	,	,	PUNCT
ap-4607	73	8	bi	bi	NOUN
ap-4607	73	9	)	)	PUNCT
ap-4607	73	10	}	}	PUNCT
ap-4607	73	11	.	.	PUNCT
ap-4607	74	1	(	(	PUNCT
ap-4607	74	2	5	5	NUM
ap-4607	74	3	)	)	PUNCT
ap-4607	74	4	from	from	ADP
ap-4607	74	5	the	the	DET
ap-4607	74	6	theorem	theorem	NOUN
ap-4607	74	7	on	on	ADP
ap-4607	74	8	the	the	DET
ap-4607	74	9	structure	structure	NOUN
ap-4607	74	10	of	of	ADP
ap-4607	74	11	the	the	DET
ap-4607	74	12	set	set	NOUN
ap-4607	74	13	of	of	ADP
ap-4607	74	14	solutions	solution	NOUN
ap-4607	74	15	of	of	ADP
ap-4607	74	16	a	a	DET
ap-4607	74	17	system	system	NOUN
ap-4607	74	18	of	of	ADP
ap-4607	74	19	polynomial	polynomial	ADJ
ap-4607	74	20	equations	equation	NOUN
ap-4607	74	21	,	,	PUNCT
ap-4607	74	22	see	see	VERB
ap-4607	74	23	[	[	X
ap-4607	74	24	11	11	NUM
ap-4607	74	25	]	]	PUNCT
ap-4607	74	26	it	it	PRON
ap-4607	74	27	follows	follow	VERB
ap-4607	74	28	that	that	SCONJ
ap-4607	74	29	the	the	DET
ap-4607	74	30	set	set	NOUN
ap-4607	74	31	v	v	NOUN
ap-4607	74	32	(	(	PUNCT
ap-4607	74	33	f1	f1	NOUN
ap-4607	74	34	,	,	PUNCT
ap-4607	74	35	f2	f2	PROPN
ap-4607	74	36	)	)	PUNCT
ap-4607	74	37	can	can	AUX
ap-4607	74	38	be	be	AUX
ap-4607	74	39	represented	represent	VERB
ap-4607	74	40	in	in	ADP
ap-4607	74	41	the	the	DET
ap-4607	74	42	form	form	NOUN
ap-4607	74	43	v	v	NOUN
ap-4607	74	44	(	(	PUNCT
ap-4607	74	45	f1	f1	NOUN
ap-4607	74	46	,	,	PUNCT
ap-4607	74	47	f2	f2	PROPN
ap-4607	74	48	)	)	PUNCT
ap-4607	74	49	=	=	PUNCT
ap-4607	75	1	(	(	PUNCT
ap-4607	75	2	v	v	NUM
ap-4607	75	3	0(f1	0(f1	NOUN
ap-4607	75	4	,	,	PUNCT
ap-4607	75	5	f2	f2	PROPN
ap-4607	75	6	)	)	PUNCT
ap-4607	75	7	)	)	PUNCT
ap-4607	76	1	∪	∪	ADP
ap-4607	76	2	(	(	PUNCT
ap-4607	76	3	v	v	NOUN
ap-4607	76	4	1(f1	1(f1	NUM
ap-4607	76	5	,	,	PUNCT
ap-4607	76	6	f2	f2	PROPN
ap-4607	76	7	)	)	PUNCT
ap-4607	76	8	)	)	PUNCT
ap-4607	76	9	,	,	PUNCT
ap-4607	76	10	(	(	PUNCT
ap-4607	76	11	6	6	NUM
ap-4607	76	12	)	)	PUNCT
ap-4607	76	13	where	where	SCONJ
ap-4607	76	14	v	v	NOUN
ap-4607	76	15	1(f1	1(f1	NUM
ap-4607	76	16	,	,	PUNCT
ap-4607	76	17	f2	f2	PROPN
ap-4607	76	18	)	)	PUNCT
ap-4607	76	19	is	be	AUX
ap-4607	76	20	the	the	DET
ap-4607	76	21	set	set	NOUN
ap-4607	76	22	of	of	ADP
ap-4607	76	23	solutions	solution	NOUN
ap-4607	76	24	depending	depend	VERB
ap-4607	76	25	on	on	ADP
ap-4607	76	26	a	a	DET
ap-4607	76	27	single	single	ADJ
ap-4607	76	28	parameter	parameter	NOUN
ap-4607	76	29	,	,	PUNCT
ap-4607	76	30	and	and	CCONJ
ap-4607	77	1	v	v	ADP
ap-4607	77	2	0(f1	0(f1	NUM
ap-4607	77	3	,	,	PUNCT
ap-4607	77	4	f2	f2	PROPN
ap-4607	77	5	)	)	PUNCT
ap-4607	77	6	is	be	AUX
ap-4607	77	7	the	the	DET
ap-4607	77	8	discrete	discrete	ADJ
ap-4607	77	9	set	set	NOUN
ap-4607	77	10	of	of	ADP
ap-4607	77	11	solutions	solution	NOUN
ap-4607	77	12	of	of	ADP
ap-4607	77	13	system	system	NOUN
ap-4607	77	14	(	(	PUNCT
ap-4607	77	15	3a	3a	NUM
ap-4607	77	16	)	)	PUNCT
ap-4607	77	17	.	.	PUNCT
ap-4607	78	1	the	the	DET
ap-4607	78	2	set	set	NOUN
ap-4607	78	3	v	v	ADP
ap-4607	78	4	0(f1	0(f1	NOUN
ap-4607	78	5	,	,	PUNCT
ap-4607	78	6	f2	f2	PROPN
ap-4607	78	7	)	)	PUNCT
ap-4607	78	8	is	be	AUX
ap-4607	78	9	obviously	obviously	ADV
ap-4607	78	10	discrete	discrete	ADJ
ap-4607	78	11	and	and	CCONJ
ap-4607	78	12	,	,	PUNCT
ap-4607	78	13	moreover	moreover	ADV
ap-4607	78	14	,	,	PUNCT
ap-4607	78	15	finite	finite	PROPN
ap-4607	78	16	.	.	PUNCT
ap-4607	79	1	the	the	DET
ap-4607	79	2	sets	set	NOUN
ap-4607	79	3	have	have	VERB
ap-4607	79	4	dimension	dimension	NOUN
ap-4607	79	5	dim	dim	ADJ
ap-4607	79	6	v	v	ADP
ap-4607	79	7	0(f1	0(f1	NOUN
ap-4607	79	8	,	,	PUNCT
ap-4607	79	9	f2	f2	PROPN
ap-4607	79	10	)	)	PUNCT
ap-4607	79	11	=	=	SYM
ap-4607	79	12	0	0	NUM
ap-4607	79	13	and	and	CCONJ
ap-4607	79	14	dimv	dimv	PROPN
ap-4607	79	15	1(f1	1(f1	PROPN
ap-4607	79	16	,	,	PUNCT
ap-4607	79	17	f2	f2	PROPN
ap-4607	79	18	)	)	PUNCT
ap-4607	79	19	=	=	SYM
ap-4607	80	1	1	1	NUM
ap-4607	80	2	.	.	NOUN
ap-4607	80	3	4	4	NUM
ap-4607	80	4	.	.	X
ap-4607	80	5	study	study	NOUN
ap-4607	80	6	of	of	ADP
ap-4607	80	7	the	the	DET
ap-4607	80	8	set	set	NOUN
ap-4607	80	9	v	v	ADP
ap-4607	80	10	1(f1	1(f1	PROPN
ap-4607	80	11	,	,	PUNCT
ap-4607	80	12	f2	f2	PROPN
ap-4607	80	13	)	)	PUNCT
ap-4607	80	14	(	(	PUNCT
ap-4607	80	15	extended	extend	VERB
ap-4607	80	16	solutions	solution	NOUN
ap-4607	80	17	)	)	PUNCT
ap-4607	80	18	a	a	DET
ap-4607	80	19	number	number	NOUN
ap-4607	80	20	of	of	ADP
ap-4607	80	21	theorems	theorem	NOUN
ap-4607	80	22	,	,	PUNCT
ap-4607	80	23	which	which	PRON
ap-4607	80	24	allow	allow	VERB
ap-4607	80	25	us	we	PRON
ap-4607	80	26	to	to	PART
ap-4607	80	27	determine	determine	VERB
ap-4607	80	28	if	if	SCONJ
ap-4607	80	29	the	the	DET
ap-4607	80	30	set	set	NOUN
ap-4607	80	31	v	v	ADP
ap-4607	80	32	1(f1	1(f1	PROPN
ap-4607	80	33	,	,	PUNCT
ap-4607	80	34	f2	f2	PROPN
ap-4607	80	35	)	)	PUNCT
ap-4607	80	36	is	be	AUX
ap-4607	80	37	empty	empty	ADJ
ap-4607	80	38	,	,	PUNCT
ap-4607	80	39	see	see	VERB
ap-4607	80	40	,	,	PUNCT
ap-4607	80	41	for	for	ADP
ap-4607	80	42	example	example	NOUN
ap-4607	80	43	,	,	PUNCT
ap-4607	80	44	[	[	X
ap-4607	80	45	12	12	NUM
ap-4607	80	46	,	,	PUNCT
ap-4607	80	47	13	13	NUM
ap-4607	80	48	]	]	PUNCT
ap-4607	80	49	.	.	PUNCT
ap-4607	81	1	in	in	ADP
ap-4607	81	2	[	[	X
ap-4607	81	3	5	5	NUM
ap-4607	81	4	]	]	PUNCT
ap-4607	81	5	,	,	PUNCT
ap-4607	81	6	we	we	PRON
ap-4607	81	7	give	give	VERB
ap-4607	81	8	an	an	DET
ap-4607	81	9	algorithm	algorithm	NOUN
ap-4607	81	10	that	that	PRON
ap-4607	81	11	allows	allow	VERB
ap-4607	81	12	us	we	PRON
ap-4607	81	13	to	to	PART
ap-4607	81	14	describe	describe	VERB
ap-4607	81	15	this	this	DET
ap-4607	81	16	set	set	NOUN
ap-4607	81	17	analytically	analytically	ADV
ap-4607	81	18	,	,	PUNCT
ap-4607	81	19	if	if	SCONJ
ap-4607	81	20	it	it	PRON
ap-4607	81	21	is	be	AUX
ap-4607	81	22	not	not	PART
ap-4607	81	23	empty	empty	ADJ
ap-4607	81	24	.	.	PUNCT
ap-4607	82	1	if	if	SCONJ
ap-4607	82	2	the	the	DET
ap-4607	82	3	set	set	NOUN
ap-4607	82	4	v	v	ADP
ap-4607	82	5	1(f1	1(f1	PROPN
ap-4607	82	6	,	,	PUNCT
ap-4607	82	7	f2	f2	PROPN
ap-4607	82	8	)	)	PUNCT
ap-4607	82	9	is	be	AUX
ap-4607	82	10	not	not	PART
ap-4607	82	11	empty	empty	ADJ
ap-4607	82	12	,	,	PUNCT
ap-4607	82	13	then	then	ADV
ap-4607	82	14	the	the	DET
ap-4607	82	15	equations	equation	NOUN
ap-4607	82	16	of	of	ADP
ap-4607	82	17	system	system	NOUN
ap-4607	82	18	(	(	PUNCT
ap-4607	82	19	3	3	X
ap-4607	82	20	)	)	PUNCT
ap-4607	82	21	are	be	AUX
ap-4607	82	22	said	say	VERB
ap-4607	82	23	to	to	PART
ap-4607	82	24	have	have	VERB
ap-4607	82	25	a	a	DET
ap-4607	82	26	common	common	ADJ
ap-4607	82	27	component	component	NOUN
ap-4607	82	28	.	.	PUNCT
ap-4607	83	1	the	the	DET
ap-4607	83	2	equation	equation	NOUN
ap-4607	83	3	of	of	ADP
ap-4607	83	4	the	the	DET
ap-4607	83	5	common	common	ADJ
ap-4607	83	6	component	component	NOUN
ap-4607	83	7	can	can	AUX
ap-4607	83	8	be	be	AUX
ap-4607	83	9	obtained	obtain	VERB
ap-4607	83	10	from	from	ADP
ap-4607	83	11	the	the	DET
ap-4607	83	12	analytical	analytical	ADJ
ap-4607	83	13	description	description	NOUN
ap-4607	83	14	of	of	ADP
ap-4607	83	15	the	the	DET
ap-4607	83	16	set	set	NOUN
ap-4607	83	17	v	v	ADP
ap-4607	83	18	1(f1	1(f1	PROPN
ap-4607	83	19	,	,	PUNCT
ap-4607	83	20	f2	f2	PROPN
ap-4607	83	21	)	)	PUNCT
ap-4607	83	22	.	.	PUNCT
ap-4607	84	1	4.1	4.1	NUM
ap-4607	84	2	.	.	NOUN
ap-4607	84	3	1	1	NUM
ap-4607	84	4	-	-	PUNCT
ap-4607	84	5	point	point	NOUN
ap-4607	84	6	lens	lens	NOUN
ap-4607	84	7	(	(	PUNCT
ap-4607	84	8	schwarzschild	schwarzschild	NOUN
ap-4607	84	9	lens	len	NOUN
ap-4607	84	10	)	)	PUNCT
ap-4607	84	11	we	we	PRON
ap-4607	84	12	apply	apply	VERB
ap-4607	84	13	the	the	DET
ap-4607	84	14	theorems	theorem	NOUN
ap-4607	84	15	presented	present	VERB
ap-4607	84	16	in	in	ADP
ap-4607	84	17	the	the	DET
ap-4607	84	18	appendix	appendix	NOUN
ap-4607	84	19	for	for	ADP
ap-4607	84	20	constructing	construct	VERB
ap-4607	84	21	the	the	DET
ap-4607	84	22	set	set	NOUN
ap-4607	84	23	v	v	ADP
ap-4607	84	24	1(f1	1(f1	PROPN
ap-4607	84	25	,	,	PUNCT
ap-4607	84	26	f2	f2	PROPN
ap-4607	84	27	)	)	PUNCT
ap-4607	84	28	,	,	PUNCT
ap-4607	84	29	in	in	ADP
ap-4607	84	30	the	the	DET
ap-4607	84	31	case	case	NOUN
ap-4607	84	32	of	of	ADP
ap-4607	84	33	a	a	DET
ap-4607	84	34	single	single	ADJ
ap-4607	84	35	-	-	PUNCT
ap-4607	84	36	point	point	NOUN
ap-4607	84	37	gravitational	gravitational	ADJ
ap-4607	84	38	lens	len	NOUN
ap-4607	84	39	.	.	PUNCT
ap-4607	85	1	let	let	VERB
ap-4607	85	2	the	the	DET
ap-4607	85	3	1	1	NUM
ap-4607	85	4	-	-	PUNCT
ap-4607	85	5	point	point	NOUN
ap-4607	85	6	lens	lens	NOUN
ap-4607	85	7	have	have	VERB
ap-4607	85	8	coordinates	coordinate	NOUN
ap-4607	85	9	a1	a1	NOUN
ap-4607	85	10	=	=	SYM
ap-4607	85	11	0	0	NUM
ap-4607	85	12	,	,	PUNCT
ap-4607	85	13	b1	b1	NOUN
ap-4607	85	14	=	=	SYM
ap-4607	86	1	0	0	X
ap-4607	86	2	.	.	PUNCT
ap-4607	87	1	let	let	VERB
ap-4607	87	2	l	l	NOUN
ap-4607	87	3	:	:	PUNCT
ap-4607	87	4	r2	r2	PROPN
ap-4607	87	5	y	y	PROPN
ap-4607	87	6	→	→	SYM
ap-4607	87	7	r2	r2	PROPN
ap-4607	87	8	x	x	PUNCT
ap-4607	87	9	be	be	AUX
ap-4607	87	10	the	the	DET
ap-4607	87	11	transformation	transformation	NOUN
ap-4607	87	12	from	from	ADP
ap-4607	87	13	the	the	DET
ap-4607	87	14	plane	plane	NOUN
ap-4607	87	15	of	of	ADP
ap-4607	87	16	the	the	DET
ap-4607	87	17	source	source	NOUN
ap-4607	87	18	to	to	ADP
ap-4607	87	19	the	the	DET
ap-4607	87	20	plane	plane	NOUN
ap-4607	87	21	of	of	ADP
ap-4607	87	22	the	the	DET
ap-4607	87	23	lens	len	NOUN
ap-4607	87	24	,	,	PUNCT
ap-4607	87	25	determined	determine	VERB
ap-4607	87	26	by	by	ADP
ap-4607	87	27	the	the	DET
ap-4607	87	28	system	system	NOUN
ap-4607	87	29	of	of	ADP
ap-4607	87	30	equations	equations	X
ap-4607	87	31	y1	y1	PROPN
ap-4607	87	32	=	=	PUNCT
ap-4607	88	1	x1	x1	NUM
ap-4607	88	2	−	−	NOUN
ap-4607	89	1	x1	x1	PROPN
ap-4607	89	2	x2	x2	NOUN
ap-4607	89	3	1	1	NUM
ap-4607	90	1	+	+	NUM
ap-4607	90	2	x2	x2	PROPN
ap-4607	90	3	2	2	NUM
ap-4607	90	4	,	,	PUNCT
ap-4607	90	5	y2	y2	NOUN
ap-4607	90	6	=	=	SYM
ap-4607	91	1	x2	x2	PROPN
ap-4607	91	2	−	−	PROPN
ap-4607	92	1	x2	x2	INTJ
ap-4607	92	2	x2	x2	NOUN
ap-4607	92	3	1	1	NUM
ap-4607	93	1	+	+	NUM
ap-4607	93	2	x2	x2	PROPN
ap-4607	93	3	2	2	NUM
ap-4607	93	4	.	.	PUNCT
ap-4607	93	5	(	(	PUNCT
ap-4607	93	6	7	7	X
ap-4607	93	7	)	)	PUNCT
ap-4607	93	8	equations	equation	NOUN
ap-4607	93	9	of	of	ADP
ap-4607	93	10	the	the	DET
ap-4607	93	11	system	system	NOUN
ap-4607	93	12	are	be	AUX
ap-4607	93	13	defined	define	VERB
ap-4607	93	14	for	for	ADP
ap-4607	93	15	all	all	DET
ap-4607	93	16	points	point	NOUN
ap-4607	93	17	such	such	ADJ
ap-4607	93	18	that	that	SCONJ
ap-4607	93	19	x2	x2	PROPN
ap-4607	93	20	1	1	NUM
ap-4607	93	21	+	+	NOUN
ap-4607	93	22	x2	x2	NOUN
ap-4607	93	23	2	2	NUM
ap-4607	93	24	6=	6=	NUM
ap-4607	93	25	0	0	NUM
ap-4607	93	26	,	,	PUNCT
ap-4607	93	27	that	that	ADV
ap-4607	93	28	is	is	ADV
ap-4607	93	29	,	,	PUNCT
ap-4607	93	30	except	except	SCONJ
ap-4607	93	31	for	for	ADP
ap-4607	93	32	the	the	DET
ap-4607	93	33	origin	origin	NOUN
ap-4607	93	34	of	of	ADP
ap-4607	93	35	the	the	DET
ap-4607	93	36	point	point	NOUN
ap-4607	93	37	o(0	o(0	NOUN
ap-4607	93	38	,	,	PUNCT
ap-4607	93	39	0	0	NUM
ap-4607	93	40	)	)	PUNCT
ap-4607	93	41	.	.	PUNCT
ap-4607	94	1	at	at	ADP
ap-4607	94	2	the	the	DET
ap-4607	94	3	origin	origin	NOUN
ap-4607	94	4	,	,	PUNCT
ap-4607	94	5	the	the	DET
ap-4607	94	6	inverse	inverse	NOUN
ap-4607	94	7	mapping	mapping	NOUN
ap-4607	94	8	is	be	AUX
ap-4607	94	9	not	not	PART
ap-4607	94	10	defined	define	VERB
ap-4607	94	11	.	.	PUNCT
ap-4607	95	1	but	but	CCONJ
ap-4607	95	2	if	if	SCONJ
ap-4607	95	3	we	we	PRON
ap-4607	95	4	transform	transform	VERB
ap-4607	95	5	system	system	NOUN
ap-4607	95	6	(	(	PUNCT
ap-4607	95	7	7	7	NUM
ap-4607	95	8	)	)	PUNCT
ap-4607	95	9	to	to	ADP
ap-4607	95	10	polynomial	polynomial	ADJ
ap-4607	95	11	form	form	NOUN
ap-4607	95	12	{	{	PUNCT
ap-4607	95	13	(	(	PUNCT
ap-4607	95	14	x2	x2	NOUN
ap-4607	95	15	1	1	NUM
ap-4607	95	16	+	+	NUM
ap-4607	95	17	x2	x2	PROPN
ap-4607	96	1	2)(x1	2)(x1	NUM
ap-4607	96	2	−	−	NUM
ap-4607	97	1	y1)−	y1)−	NUM
ap-4607	97	2	x1	x1	PROPN
ap-4607	97	3	=	=	SYM
ap-4607	97	4	0	0	PROPN
ap-4607	97	5	,	,	PUNCT
ap-4607	97	6	(	(	PUNCT
ap-4607	97	7	x2	x2	NOUN
ap-4607	97	8	1	1	NUM
ap-4607	97	9	+	+	NUM
ap-4607	97	10	x2	x2	PROPN
ap-4607	97	11	2)(x2	2)(x2	PROPN
ap-4607	98	1	−	−	X
ap-4607	98	2	y1)−	y1)−	NUM
ap-4607	98	3	x2	x2	NOUN
ap-4607	98	4	=	=	SYM
ap-4607	98	5	0	0	NUM
ap-4607	98	6	,	,	PUNCT
ap-4607	98	7	(	(	PUNCT
ap-4607	98	8	8)	8)	NUM
ap-4607	98	9	then	then	ADV
ap-4607	98	10	the	the	DET
ap-4607	98	11	inverse	inverse	ADJ
ap-4607	98	12	transformation	transformation	NOUN
ap-4607	98	13	of	of	ADP
ap-4607	98	14	l−1	l−1	PROPN
ap-4607	98	15	:	:	PUNCT
ap-4607	98	16	r2	r2	PROPN
ap-4607	98	17	x	x	PUNCT
ap-4607	98	18	→	→	SYM
ap-4607	98	19	r2	r2	PROPN
ap-4607	98	20	y	y	PROPN
ap-4607	98	21	is	be	AUX
ap-4607	98	22	completely	completely	ADV
ap-4607	98	23	determined	determine	VERB
ap-4607	98	24	by	by	ADP
ap-4607	98	25	the	the	DET
ap-4607	98	26	system	system	NOUN
ap-4607	98	27	of	of	ADP
ap-4607	98	28	equations	equation	NOUN
ap-4607	98	29	{	{	PUNCT
ap-4607	98	30	x3	x3	NOUN
ap-4607	98	31	1	1	NUM
ap-4607	99	1	+	+	CCONJ
ap-4607	99	2	x1x	x1x	PROPN
ap-4607	99	3	2	2	NUM
ap-4607	99	4	2	2	NUM
ap-4607	99	5	−	−	NOUN
ap-4607	100	1	x2	x2	NOUN
ap-4607	100	2	1y1	1y1	NUM
ap-4607	101	1	−	−	PROPN
ap-4607	102	1	x2	x2	INTJ
ap-4607	103	1	2y1	2y1	NUM
ap-4607	103	2	−	−	NOUN
ap-4607	104	1	x1	x1	NOUN
ap-4607	104	2	=	=	SYM
ap-4607	104	3	0	0	PROPN
ap-4607	104	4	,	,	PUNCT
ap-4607	104	5	x2	x2	PROPN
ap-4607	104	6	1x2	1x2	NUM
ap-4607	105	1	+	+	CCONJ
ap-4607	105	2	x3	x3	ADJ
ap-4607	105	3	2	2	NUM
ap-4607	105	4	−	−	NOUN
ap-4607	105	5	x2	x2	INTJ
ap-4607	105	6	1y2	1y2	NUM
ap-4607	105	7	−	−	NUM
ap-4607	106	1	x2	x2	NOUN
ap-4607	107	1	2y2	2y2	NUM
ap-4607	108	1	−	−	PROPN
ap-4607	109	1	x2	x2	NOUN
ap-4607	110	1	=	=	NOUN
ap-4607	111	1	0	0	PROPN
ap-4607	111	2	.	.	PUNCT
ap-4607	112	1	(	(	PUNCT
ap-4607	112	2	9	9	X
ap-4607	112	3	)	)	PUNCT
ap-4607	112	4	we	we	PRON
ap-4607	112	5	calculate	calculate	VERB
ap-4607	112	6	the	the	DET
ap-4607	112	7	result	result	NOUN
ap-4607	112	8	r1	r1	NOUN
ap-4607	112	9	by	by	ADP
ap-4607	112	10	the	the	DET
ap-4607	112	11	variable	variable	NOUN
ap-4607	112	12	x1	x1	PROPN
ap-4607	112	13	for	for	ADP
ap-4607	112	14	which	which	PRON
ap-4607	112	15	we	we	PRON
ap-4607	112	16	represent	represent	VERB
ap-4607	112	17	the	the	DET
ap-4607	112	18	equation	equation	NOUN
ap-4607	112	19	in	in	ADP
ap-4607	112	20	lexicographic	lexicographic	ADJ
ap-4607	112	21	form	form	NOUN
ap-4607	112	22	{	{	PUNCT
ap-4607	112	23	x3	x3	NOUN
ap-4607	112	24	1	1	NUM
ap-4607	112	25	−	−	NOUN
ap-4607	112	26	y1x	y1x	NOUN
ap-4607	112	27	2	2	NUM
ap-4607	112	28	1	1	NUM
ap-4607	113	1	+	+	CCONJ
ap-4607	113	2	(	(	PUNCT
ap-4607	113	3	x2	x2	INTJ
ap-4607	113	4	2	2	NUM
ap-4607	113	5	−	−	PROPN
ap-4607	113	6	1)x1	1)x1	NUM
ap-4607	113	7	−	−	NOUN
ap-4607	113	8	x2	x2	NOUN
ap-4607	113	9	2y1	2y1	NUM
ap-4607	114	1	=	=	SYM
ap-4607	114	2	0	0	NUM
ap-4607	114	3	,	,	PUNCT
ap-4607	114	4	(	(	PUNCT
ap-4607	114	5	x2	x2	INTJ
ap-4607	114	6	−	−	PROPN
ap-4607	114	7	y2)x2	y2)x2	NOUN
ap-4607	114	8	1	1	NUM
ap-4607	115	1	+	+	CCONJ
ap-4607	115	2	x3	x3	ADJ
ap-4607	115	3	2	2	NUM
ap-4607	115	4	−	−	NOUN
ap-4607	115	5	x2	x2	NOUN
ap-4607	116	1	2y2	2y2	NUM
ap-4607	116	2	−	−	PROPN
ap-4607	117	1	x2	x2	NOUN
ap-4607	117	2	=	=	NOUN
ap-4607	117	3	0	0	PROPN
ap-4607	117	4	.	.	PUNCT
ap-4607	118	1	(	(	PUNCT
ap-4607	118	2	10	10	NUM
ap-4607	118	3	)	)	PUNCT
ap-4607	118	4	result	result	NOUN
ap-4607	118	5	by	by	ADP
ap-4607	118	6	degree	degree	NOUN
ap-4607	118	7	x1	x1	PROPN
ap-4607	118	8	has	have	VERB
ap-4607	118	9	the	the	DET
ap-4607	118	10	form	form	NOUN
ap-4607	118	11	r1	r1	NOUN
ap-4607	118	12	=	=	PUNCT
ap-4607	118	13	∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣	PROPN
ap-4607	118	14	1	1	NUM
ap-4607	118	15	r12	r12	NOUN
ap-4607	118	16	r13	r13	NOUN
ap-4607	118	17	r14	r14	NOUN
ap-4607	118	18	0	0	NUM
ap-4607	118	19	0	0	NUM
ap-4607	118	20	1	1	NUM
ap-4607	118	21	r12	r12	NOUN
ap-4607	118	22	r13	r13	NOUN
ap-4607	118	23	r14	r14	NOUN
ap-4607	118	24	r21	r21	NOUN
ap-4607	118	25	0	0	NUM
ap-4607	118	26	r23	r23	NOUN
ap-4607	118	27	0	0	NUM
ap-4607	118	28	0	0	SYM
ap-4607	118	29	0	0	NUM
ap-4607	118	30	r21	r21	NOUN
ap-4607	118	31	0	0	NUM
ap-4607	118	32	r23	r23	NOUN
ap-4607	118	33	0	0	NUM
ap-4607	118	34	0	0	SYM
ap-4607	118	35	0	0	NUM
ap-4607	118	36	r21	r21	NOUN
ap-4607	118	37	0	0	NUM
ap-4607	118	38	r23	r23	NOUN
ap-4607	118	39	∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣	PROPN
ap-4607	118	40	,	,	PUNCT
ap-4607	118	41	(	(	PUNCT
ap-4607	118	42	11	11	NUM
ap-4607	118	43	)	)	PUNCT
ap-4607	118	44	where	where	SCONJ
ap-4607	118	45	r12	r12	PROPN
ap-4607	118	46	=	=	SYM
ap-4607	118	47	−y1	−y1	PROPN
ap-4607	118	48	,	,	PUNCT
ap-4607	118	49	r13	r13	NOUN
ap-4607	118	50	=	=	SYM
ap-4607	119	1	x2	x2	PROPN
ap-4607	119	2	2	2	NUM
ap-4607	119	3	−	−	NUM
ap-4607	119	4	1	1	NUM
ap-4607	119	5	,	,	PUNCT
ap-4607	119	6	r14	r14	NOUN
ap-4607	119	7	=	=	SYM
ap-4607	119	8	−x2	−x2	NOUN
ap-4607	119	9	2y1	2y1	NUM
ap-4607	119	10	,	,	PUNCT
ap-4607	119	11	r21	r21	NOUN
ap-4607	119	12	=	=	SYM
ap-4607	119	13	x2	x2	PROPN
ap-4607	119	14	−	−	PROPN
ap-4607	119	15	y2	y2	PROPN
ap-4607	119	16	,	,	PUNCT
ap-4607	119	17	r23	r23	NOUN
ap-4607	119	18	=	=	SYM
ap-4607	119	19	x3	x3	NOUN
ap-4607	119	20	2	2	NUM
ap-4607	119	21	−	−	NOUN
ap-4607	120	1	x2	x2	NOUN
ap-4607	120	2	2y2	2y2	NUM
ap-4607	121	1	−	−	NOUN
ap-4607	121	2	x2	x2	NOUN
ap-4607	121	3	.	.	PUNCT
ap-4607	122	1	405	405	NUM
ap-4607	122	2	a.	a.	NOUN
ap-4607	122	3	t.	t.	PROPN
ap-4607	122	4	kotvytskiy	kotvytskiy	PROPN
ap-4607	122	5	,	,	PUNCT
ap-4607	122	6	s.	s.	PROPN
ap-4607	122	7	d.	d.	PROPN
ap-4607	122	8	bronza	bronza	PROPN
ap-4607	122	9	,	,	PUNCT
ap-4607	122	10	v.	v.	PROPN
ap-4607	122	11	yu	yu	PROPN
ap-4607	122	12	.	.	PUNCT
ap-4607	123	1	shablenko	shablenko	PROPN
ap-4607	123	2	acta	acta	PROPN
ap-4607	123	3	polytechnica	polytechnica	PROPN
ap-4607	123	4	we	we	PRON
ap-4607	123	5	have	have	VERB
ap-4607	124	1	r1	r1	PROPN
ap-4607	124	2	=	=	PUNCT
ap-4607	124	3	−x3	−x3	PROPN
ap-4607	124	4	1y	1y	PROPN
ap-4607	124	5	2	2	NUM
ap-4607	124	6	1	1	NUM
ap-4607	124	7	+	+	CCONJ
ap-4607	124	8	x2	x2	PROPN
ap-4607	124	9	2y	2y	NUM
ap-4607	124	10	2	2	NUM
ap-4607	124	11	1y2	1y2	NUM
ap-4607	124	12	+	+	CCONJ
ap-4607	124	13	x2y	x2y	SYM
ap-4607	124	14	2	2	NUM
ap-4607	124	15	1	1	NUM
ap-4607	124	16	−	−	ADP
ap-4607	124	17	x3	x3	ADJ
ap-4607	124	18	2y	2y	NUM
ap-4607	124	19	2	2	NUM
ap-4607	124	20	2	2	NUM
ap-4607	124	21	+	+	CCONJ
ap-4607	124	22	x2	x2	PROPN
ap-4607	124	23	2y	2y	NUM
ap-4607	124	24	3	3	NUM
ap-4607	124	25	2	2	NUM
ap-4607	124	26	.	.	PUNCT
ap-4607	125	1	(	(	PUNCT
ap-4607	125	2	12	12	NUM
ap-4607	125	3	)	)	PUNCT
ap-4607	125	4	in	in	ADP
ap-4607	125	5	order	order	NOUN
ap-4607	125	6	for	for	SCONJ
ap-4607	125	7	the	the	DET
ap-4607	125	8	system	system	NOUN
ap-4607	125	9	equations	equation	NOUN
ap-4607	125	10	(	(	PUNCT
ap-4607	125	11	7	7	NUM
ap-4607	125	12	)	)	PUNCT
ap-4607	125	13	to	to	PART
ap-4607	125	14	have	have	VERB
ap-4607	125	15	a	a	DET
ap-4607	125	16	common	common	ADJ
ap-4607	125	17	component	component	NOUN
ap-4607	125	18	,	,	PUNCT
ap-4607	125	19	we	we	PRON
ap-4607	125	20	need	need	VERB
ap-4607	125	21	r1	r1	PROPN
ap-4607	125	22	≡	≡	PROPN
ap-4607	125	23	0	0	NUM
ap-4607	125	24	.	.	PUNCT
ap-4607	126	1	applying	apply	VERB
ap-4607	126	2	theorem	theorem	ADJ
ap-4607	126	3	5a	5a	NUM
ap-4607	126	4	for	for	ADP
ap-4607	126	5	the	the	DET
ap-4607	126	6	decomposition	decomposition	NOUN
ap-4607	126	7	of	of	ADP
ap-4607	126	8	r1	r1	NOUN
ap-4607	126	9	on	on	ADP
ap-4607	126	10	indecomposable	indecomposable	ADJ
ap-4607	126	11	components	component	NOUN
ap-4607	126	12	,	,	PUNCT
ap-4607	126	13	we	we	PRON
ap-4607	126	14	have	have	VERB
ap-4607	126	15	x2	x2	NUM
ap-4607	126	16	(	(	PUNCT
ap-4607	126	17	−(y2	−(y2	NUM
ap-4607	126	18	1	1	NUM
ap-4607	127	1	+	+	CCONJ
ap-4607	127	2	y2	y2	NOUN
ap-4607	128	1	2)x2	2)x2	ADJ
ap-4607	128	2	2	2	NUM
ap-4607	128	3	+	+	CCONJ
ap-4607	128	4	y2(y2	y2(y2	NUM
ap-4607	128	5	1	1	NUM
ap-4607	128	6	+	+	CCONJ
ap-4607	128	7	y2	y2	NOUN
ap-4607	128	8	2)x2	2)x2	NOUN
ap-4607	128	9	+	+	CCONJ
ap-4607	128	10	y2	y2	NOUN
ap-4607	128	11	1	1	NUM
ap-4607	128	12	)	)	PUNCT
ap-4607	128	13	≡	≡	PROPN
ap-4607	128	14	0	0	NUM
ap-4607	128	15	.	.	PUNCT
ap-4607	129	1	(	(	PUNCT
ap-4607	129	2	13	13	NUM
ap-4607	129	3	)	)	PUNCT
ap-4607	129	4	the	the	DET
ap-4607	129	5	equation	equation	NOUN
ap-4607	129	6	is	be	AUX
ap-4607	129	7	divided	divide	VERB
ap-4607	129	8	into	into	ADP
ap-4607	129	9	two	two	NUM
ap-4607	129	10	equations	equation	NOUN
ap-4607	129	11	x2	x2	PROPN
ap-4607	129	12	≡	≡	PROPN
ap-4607	129	13	0	0	NUM
ap-4607	129	14	,	,	PUNCT
ap-4607	129	15	−(y2	−(y2	DET
ap-4607	129	16	1	1	NUM
ap-4607	129	17	+	+	CCONJ
ap-4607	130	1	y2	y2	NOUN
ap-4607	130	2	2)x2	2)x2	ADJ
ap-4607	130	3	2	2	NUM
ap-4607	130	4	+	+	CCONJ
ap-4607	130	5	y2(y2	y2(y2	NUM
ap-4607	130	6	1	1	NUM
ap-4607	130	7	+	+	CCONJ
ap-4607	130	8	y2	y2	NOUN
ap-4607	131	1	2)x2	2)x2	NOUN
ap-4607	131	2	+	+	CCONJ
ap-4607	131	3	y2	y2	PROPN
ap-4607	131	4	1	1	NUM
ap-4607	131	5	≡	≡	PROPN
ap-4607	131	6	0	0	NUM
ap-4607	131	7	.	.	PUNCT
ap-4607	132	1	(	(	PUNCT
ap-4607	132	2	14	14	NUM
ap-4607	132	3	)	)	PUNCT
ap-4607	132	4	each	each	PRON
ap-4607	132	5	of	of	ADP
ap-4607	132	6	the	the	DET
ap-4607	132	7	equations	equation	NOUN
ap-4607	132	8	is	be	AUX
ap-4607	132	9	considered	consider	VERB
ap-4607	132	10	to	to	PART
ap-4607	132	11	be	be	AUX
ap-4607	132	12	a	a	DET
ap-4607	132	13	polynomial	polynomial	NOUN
ap-4607	132	14	of	of	ADP
ap-4607	132	15	the	the	DET
ap-4607	132	16	variable	variable	ADJ
ap-4607	132	17	x2	x2	NOUN
ap-4607	132	18	.	.	PUNCT
ap-4607	133	1	a	a	DET
ap-4607	133	2	polynomial	polynomial	NOUN
ap-4607	133	3	is	be	AUX
ap-4607	133	4	identically	identically	ADV
ap-4607	133	5	equal	equal	ADJ
ap-4607	133	6	to	to	ADP
ap-4607	133	7	zero	zero	NUM
ap-4607	133	8	if	if	SCONJ
ap-4607	133	9	and	and	CCONJ
ap-4607	133	10	only	only	ADV
ap-4607	133	11	if	if	SCONJ
ap-4607	133	12	all	all	DET
ap-4607	133	13	its	its	PRON
ap-4607	133	14	coefficients	coefficient	NOUN
ap-4607	133	15	are	be	AUX
ap-4607	133	16	equal	equal	ADJ
ap-4607	133	17	to	to	ADP
ap-4607	133	18	zero	zero	NUM
ap-4607	133	19	.	.	PUNCT
ap-4607	134	1	from	from	ADP
ap-4607	134	2	here	here	ADV
ap-4607	134	3	there	there	PRON
ap-4607	134	4	is	be	VERB
ap-4607	134	5	a	a	DET
ap-4607	134	6	system	system	NOUN
ap-4607	134	7	of	of	ADP
ap-4607	134	8	equations	equation	NOUN
ap-4607	134	9	{	{	PUNCT
ap-4607	134	10	y2	y2	NOUN
ap-4607	134	11	1	1	NUM
ap-4607	135	1	+	+	CCONJ
ap-4607	135	2	y2	y2	SYM
ap-4607	135	3	2	2	NUM
ap-4607	135	4	=	=	SYM
ap-4607	135	5	0	0	NUM
ap-4607	135	6	,	,	PUNCT
ap-4607	135	7	y2	y2	NOUN
ap-4607	135	8	1	1	NUM
ap-4607	135	9	=	=	SYM
ap-4607	135	10	0	0	PROPN
ap-4607	135	11	.	.	PUNCT
ap-4607	135	12	.	.	PUNCT
ap-4607	136	1	(	(	PUNCT
ap-4607	136	2	15	15	NUM
ap-4607	136	3	)	)	PUNCT
ap-4607	136	4	next	next	ADV
ap-4607	136	5	we	we	PRON
ap-4607	136	6	have	have	VERB
ap-4607	136	7	that	that	DET
ap-4607	136	8	y1	y1	NOUN
ap-4607	136	9	=	=	SYM
ap-4607	136	10	0	0	NUM
ap-4607	136	11	,	,	PUNCT
ap-4607	136	12	y2	y2	NOUN
ap-4607	136	13	=	=	SYM
ap-4607	136	14	0	0	X
ap-4607	136	15	.	.	X
ap-4607	137	1	substituting	substitute	VERB
ap-4607	137	2	in	in	ADP
ap-4607	137	3	(	(	PUNCT
ap-4607	137	4	7	7	X
ap-4607	137	5	)	)	PUNCT
ap-4607	137	6	we	we	PRON
ap-4607	137	7	have	have	PUNCT
ap-4607	138	1	x1	x1	NUM
ap-4607	138	2	−	−	PROPN
ap-4607	139	1	x1	x1	PROPN
ap-4607	139	2	x2	x2	NOUN
ap-4607	139	3	1	1	NUM
ap-4607	139	4	+	+	NUM
ap-4607	139	5	x22	x22	NOUN
ap-4607	139	6	=	=	SYM
ap-4607	139	7	0	0	PROPN
ap-4607	139	8	,	,	PUNCT
ap-4607	139	9	x2	x2	PROPN
ap-4607	139	10	−	−	PROPN
ap-4607	140	1	x2	x2	INTJ
ap-4607	140	2	x2	x2	NOUN
ap-4607	140	3	1	1	NUM
ap-4607	141	1	+	+	NUM
ap-4607	141	2	x22	x22	NOUN
ap-4607	141	3	=	=	SYM
ap-4607	141	4	0	0	NUM
ap-4607	141	5	⇒	⇒	PROPN
ap-4607	141	6			PUNCT
ap-4607	141	7	x1	x1	PROPN
ap-4607	141	8	(	(	PUNCT
ap-4607	141	9	1−	1−	NUM
ap-4607	141	10	1	1	NUM
ap-4607	141	11	x2	x2	NOUN
ap-4607	141	12	1	1	NUM
ap-4607	141	13	+	+	NUM
ap-4607	141	14	x22	x22	NOUN
ap-4607	141	15	)	)	PUNCT
ap-4607	141	16	=	=	SYM
ap-4607	142	1	0	0	NUM
ap-4607	142	2	,	,	PUNCT
ap-4607	142	3	x2	x2	PROPN
ap-4607	142	4	(	(	PUNCT
ap-4607	142	5	1−	1−	NUM
ap-4607	142	6	1	1	NUM
ap-4607	142	7	x2	x2	NOUN
ap-4607	142	8	1	1	NUM
ap-4607	142	9	+	+	NUM
ap-4607	142	10	x22	x22	NOUN
ap-4607	142	11	)	)	PUNCT
ap-4607	142	12	=	=	SYM
ap-4607	143	1	0	0	X
ap-4607	143	2	.	.	PUNCT
ap-4607	144	1	(	(	PUNCT
ap-4607	144	2	16	16	NUM
ap-4607	144	3	)	)	PUNCT
ap-4607	144	4	the	the	DET
ap-4607	144	5	system	system	NOUN
ap-4607	144	6	(	(	PUNCT
ap-4607	144	7	16	16	NUM
ap-4607	144	8	)	)	PUNCT
ap-4607	144	9	decomposes	decompose	NOUN
ap-4607	144	10	into	into	ADP
ap-4607	144	11	three	three	NUM
ap-4607	144	12	systems	system	NOUN
ap-4607	144	13	and	and	CCONJ
ap-4607	144	14	one	one	NUM
ap-4607	144	15	equation	equation	NOUN
ap-4607	144	16	:	:	PUNCT
ap-4607	144	17	{	{	PUNCT
ap-4607	144	18	x1	x1	PROPN
ap-4607	144	19	=	=	SYM
ap-4607	144	20	0	0	PROPN
ap-4607	144	21	,	,	PUNCT
ap-4607	144	22	x2	x2	NOUN
ap-4607	144	23	=	=	SYM
ap-4607	144	24	0	0	NUM
ap-4607	144	25	,	,	PUNCT
ap-4607	144	26	(	(	PUNCT
ap-4607	144	27	17a)	17a)	NUM
ap-4607	144	28	x1	x1	PROPN
ap-4607	144	29	=	=	SYM
ap-4607	144	30	0	0	NUM
ap-4607	144	31	,	,	PUNCT
ap-4607	144	32	1−	1−	NUM
ap-4607	144	33	1	1	NUM
ap-4607	144	34	x2	x2	NOUN
ap-4607	144	35	1	1	NUM
ap-4607	145	1	+	+	NUM
ap-4607	145	2	x2	x2	PROPN
ap-4607	145	3	2	2	NUM
ap-4607	145	4	=	=	SYM
ap-4607	145	5	0	0	NUM
ap-4607	145	6	,	,	PUNCT
ap-4607	145	7	(	(	PUNCT
ap-4607	145	8	17b	17b	NUM
ap-4607	145	9	)	)	PUNCT
ap-4607	145	10	1−	1−	NUM
ap-4607	145	11	1	1	NUM
ap-4607	145	12	x2	x2	SYM
ap-4607	145	13	1	1	NUM
ap-4607	145	14	+	+	CCONJ
ap-4607	145	15	x2	x2	PROPN
ap-4607	145	16	2	2	NUM
ap-4607	145	17	=	=	SYM
ap-4607	145	18	0	0	NUM
ap-4607	145	19	,	,	PUNCT
ap-4607	145	20	x2	x2	NOUN
ap-4607	145	21	=	=	SYM
ap-4607	145	22	0	0	NUM
ap-4607	145	23	,	,	PUNCT
ap-4607	145	24	(	(	PUNCT
ap-4607	145	25	17c	17c	NUM
ap-4607	145	26	)	)	PUNCT
ap-4607	145	27	1−	1−	NUM
ap-4607	145	28	1	1	NUM
ap-4607	145	29	x2	x2	SYM
ap-4607	145	30	1	1	NUM
ap-4607	145	31	+	+	NUM
ap-4607	145	32	x22	x22	NOUN
ap-4607	145	33	=	=	SYM
ap-4607	145	34	0	0	PROPN
ap-4607	145	35	.	.	PUNCT
ap-4607	146	1	(	(	PUNCT
ap-4607	146	2	18	18	NUM
ap-4607	146	3	)	)	PUNCT
ap-4607	146	4	system	system	NOUN
ap-4607	146	5	(	(	PUNCT
ap-4607	146	6	17a	17a	X
ap-4607	146	7	)	)	PUNCT
ap-4607	146	8	has	have	VERB
ap-4607	146	9	a	a	DET
ap-4607	146	10	solution	solution	NOUN
ap-4607	146	11	x1	x1	NOUN
ap-4607	146	12	=	=	SYM
ap-4607	146	13	0	0	PROPN
ap-4607	146	14	,	,	PUNCT
ap-4607	146	15	x2	x2	NOUN
ap-4607	146	16	=	=	NOUN
ap-4607	146	17	0	0	PUNCT
ap-4607	147	1	but	but	CCONJ
ap-4607	147	2	this	this	DET
ap-4607	147	3	solution	solution	NOUN
ap-4607	147	4	is	be	AUX
ap-4607	147	5	not	not	PART
ap-4607	147	6	a	a	DET
ap-4607	147	7	solution	solution	NOUN
ap-4607	147	8	of	of	ADP
ap-4607	147	9	system	system	NOUN
ap-4607	147	10	(	(	PUNCT
ap-4607	147	11	7	7	NUM
ap-4607	147	12	)	)	PUNCT
ap-4607	147	13	,	,	PUNCT
ap-4607	147	14	since	since	SCONJ
ap-4607	147	15	the	the	DET
ap-4607	147	16	system	system	NOUN
ap-4607	147	17	equations	equation	NOUN
ap-4607	147	18	at	at	ADP
ap-4607	147	19	the	the	DET
ap-4607	147	20	point	point	NOUN
ap-4607	147	21	o(0	o(0	NOUN
ap-4607	147	22	,	,	PUNCT
ap-4607	147	23	0	0	NUM
ap-4607	147	24	)	)	PUNCT
ap-4607	147	25	are	be	AUX
ap-4607	147	26	not	not	PART
ap-4607	147	27	defined	define	VERB
ap-4607	147	28	.	.	PUNCT
ap-4607	148	1	the	the	DET
ap-4607	148	2	system	system	NOUN
ap-4607	148	3	(	(	PUNCT
ap-4607	148	4	17b	17b	NUM
ap-4607	148	5	)	)	PUNCT
ap-4607	148	6	has	have	VERB
ap-4607	148	7	two	two	NUM
ap-4607	148	8	solutions	solution	NOUN
ap-4607	148	9	x1	x1	NOUN
ap-4607	148	10	=	=	SYM
ap-4607	148	11	0	0	NUM
ap-4607	148	12	,	,	PUNCT
ap-4607	148	13	x2	x2	PROPN
ap-4607	148	14	=	=	PUNCT
ap-4607	148	15	±1	±1	VERB
ap-4607	148	16	.	.	PUNCT
ap-4607	149	1	the	the	DET
ap-4607	149	2	system	system	NOUN
ap-4607	149	3	(	(	PUNCT
ap-4607	149	4	17c	17c	NUM
ap-4607	149	5	)	)	PUNCT
ap-4607	149	6	has	have	VERB
ap-4607	149	7	two	two	NUM
ap-4607	149	8	solutions	solution	NOUN
ap-4607	149	9	x1	x1	NOUN
ap-4607	149	10	=	=	SYM
ap-4607	149	11	±1	±1	VERB
ap-4607	149	12	,	,	PUNCT
ap-4607	149	13	x2	x2	PROPN
ap-4607	149	14	=	=	NOUN
ap-4607	149	15	0	0	X
ap-4607	149	16	.	.	PUNCT
ap-4607	150	1	we	we	PRON
ap-4607	150	2	have	have	VERB
ap-4607	150	3	the	the	DET
ap-4607	150	4	transformation	transformation	NOUN
ap-4607	150	5	of	of	ADP
ap-4607	150	6	equation	equation	NOUN
ap-4607	150	7	(	(	PUNCT
ap-4607	150	8	18	18	NUM
ap-4607	150	9	)	)	PUNCT
ap-4607	150	10	into	into	ADP
ap-4607	150	11	x2	x2	PROPN
ap-4607	150	12	1	1	NUM
ap-4607	151	1	+	+	CCONJ
ap-4607	151	2	x2	x2	NOUN
ap-4607	151	3	2	2	NUM
ap-4607	151	4	−	−	NOUN
ap-4607	151	5	1	1	NUM
ap-4607	151	6	=	=	SYM
ap-4607	151	7	0	0	NUM
ap-4607	151	8	.	.	PUNCT
ap-4607	151	9	(	(	PUNCT
ap-4607	151	10	19	19	NUM
ap-4607	151	11	)	)	PUNCT
ap-4607	151	12	equation	equation	NOUN
ap-4607	151	13	(	(	PUNCT
ap-4607	151	14	19	19	NUM
ap-4607	151	15	)	)	PUNCT
ap-4607	151	16	is	be	AUX
ap-4607	151	17	the	the	DET
ap-4607	151	18	equation	equation	NOUN
ap-4607	151	19	of	of	ADP
ap-4607	151	20	an	an	DET
ap-4607	151	21	individual	individual	ADJ
ap-4607	151	22	circle	circle	NOUN
ap-4607	151	23	in	in	ADP
ap-4607	151	24	the	the	DET
ap-4607	151	25	plane	plane	NOUN
ap-4607	151	26	x	x	PUNCT
ap-4607	151	27	with	with	ADP
ap-4607	151	28	a	a	DET
ap-4607	151	29	center	center	NOUN
ap-4607	151	30	at	at	ADP
ap-4607	151	31	the	the	DET
ap-4607	151	32	point	point	NOUN
ap-4607	151	33	o(0	o(0	NOUN
ap-4607	151	34	,	,	PUNCT
ap-4607	151	35	0	0	NUM
ap-4607	151	36	)	)	PUNCT
ap-4607	151	37	.	.	PUNCT
ap-4607	152	1	the	the	DET
ap-4607	152	2	solution	solution	NOUN
ap-4607	152	3	of	of	ADP
ap-4607	152	4	systems	system	NOUN
ap-4607	152	5	(	(	PUNCT
ap-4607	152	6	17b	17b	NUM
ap-4607	152	7	)	)	PUNCT
ap-4607	152	8	and	and	CCONJ
ap-4607	152	9	(	(	PUNCT
ap-4607	152	10	17c	17c	NUM
ap-4607	152	11	)	)	PUNCT
ap-4607	152	12	satisfies	satisfie	NOUN
ap-4607	152	13	equation	equation	NOUN
ap-4607	152	14	(	(	PUNCT
ap-4607	152	15	19	19	NUM
ap-4607	152	16	)	)	PUNCT
ap-4607	152	17	.	.	PUNCT
ap-4607	153	1	the	the	DET
ap-4607	153	2	solution	solution	NOUN
ap-4607	153	3	of	of	ADP
ap-4607	153	4	system	system	NOUN
ap-4607	153	5	(	(	PUNCT
ap-4607	153	6	8)	8)	NUM
ap-4607	153	7	is	be	AUX
ap-4607	153	8	the	the	DET
ap-4607	153	9	coordinates	coordinate	NOUN
ap-4607	153	10	of	of	ADP
ap-4607	153	11	the	the	DET
ap-4607	153	12	points	point	NOUN
ap-4607	153	13	of	of	ADP
ap-4607	153	14	single	single	ADJ
ap-4607	153	15	circle	circle	NOUN
ap-4607	153	16	with	with	ADP
ap-4607	153	17	center	center	NOUN
ap-4607	153	18	at	at	ADP
ap-4607	153	19	the	the	DET
ap-4607	153	20	point	point	NOUN
ap-4607	153	21	o(0	o(0	NOUN
ap-4607	153	22	,	,	PUNCT
ap-4607	153	23	0	0	NUM
ap-4607	153	24	)	)	PUNCT
ap-4607	153	25	.	.	PUNCT
ap-4607	154	1	equation	equation	NOUN
ap-4607	154	2	(	(	PUNCT
ap-4607	154	3	19	19	NUM
ap-4607	154	4	)	)	PUNCT
ap-4607	154	5	is	be	AUX
ap-4607	154	6	the	the	DET
ap-4607	154	7	equation	equation	NOUN
ap-4607	154	8	of	of	ADP
ap-4607	154	9	the	the	DET
ap-4607	154	10	general	general	ADJ
ap-4607	154	11	component	component	NOUN
ap-4607	154	12	,	,	PUNCT
ap-4607	154	13	hence	hence	ADV
ap-4607	154	14	the	the	DET
ap-4607	154	15	set	set	NOUN
ap-4607	154	16	v	v	ADP
ap-4607	154	17	1(f1	1(f1	PROPN
ap-4607	154	18	,	,	PUNCT
ap-4607	154	19	f2	f2	PROPN
ap-4607	154	20	)	)	PUNCT
ap-4607	154	21	=	=	PUNCT
ap-4607	155	1	{	{	PUNCT
ap-4607	155	2	x1	x1	PROPN
ap-4607	155	3	,	,	PUNCT
ap-4607	155	4	x2	x2	PROPN
ap-4607	156	1	|	|	ADV
ap-4607	156	2	x2	x2	NOUN
ap-4607	156	3	1	1	NUM
ap-4607	157	1	+	+	NUM
ap-4607	157	2	x2	x2	NOUN
ap-4607	157	3	2	2	NUM
ap-4607	157	4	−	−	NOUN
ap-4607	157	5	1	1	NUM
ap-4607	157	6	=	=	NOUN
ap-4607	157	7	0	0	NUM
ap-4607	157	8	}	}	PUNCT
ap-4607	157	9	.	.	PUNCT
ap-4607	158	1	(	(	PUNCT
ap-4607	158	2	20	20	NUM
ap-4607	158	3	)	)	PUNCT
ap-4607	158	4	in	in	ADP
ap-4607	158	5	the	the	DET
ap-4607	158	6	same	same	ADJ
ap-4607	158	7	way	way	NOUN
ap-4607	158	8	,	,	PUNCT
ap-4607	158	9	we	we	PRON
ap-4607	158	10	compute	compute	VERB
ap-4607	158	11	the	the	DET
ap-4607	158	12	resultant	resultant	NOUN
ap-4607	158	13	r2	r2	NOUN
ap-4607	158	14	.	.	PUNCT
ap-4607	159	1	by	by	ADP
ap-4607	159	2	virtue	virtue	NOUN
ap-4607	159	3	of	of	ADP
ap-4607	159	4	the	the	DET
ap-4607	159	5	symmetry	symmetry	NOUN
ap-4607	159	6	of	of	ADP
ap-4607	159	7	variables	variable	NOUN
ap-4607	159	8	,	,	PUNCT
ap-4607	159	9	we	we	PRON
ap-4607	159	10	have	have	VERB
ap-4607	159	11	the	the	DET
ap-4607	159	12	same	same	ADJ
ap-4607	159	13	solution	solution	NOUN
ap-4607	159	14	.	.	PUNCT
ap-4607	160	1	4.2	4.2	NUM
ap-4607	160	2	.	.	X
ap-4607	160	3	2	2	NUM
ap-4607	160	4	-	-	PUNCT
ap-4607	160	5	point	point	NOUN
ap-4607	160	6	lens	lens	NOUN
ap-4607	160	7	we	we	PRON
ap-4607	160	8	research	research	VERB
ap-4607	160	9	a	a	DET
ap-4607	160	10	two	two	NUM
ap-4607	160	11	-	-	PUNCT
ap-4607	160	12	point	point	NOUN
ap-4607	160	13	gravitational	gravitational	ADJ
ap-4607	160	14	lens	lens	NOUN
ap-4607	160	15	with	with	ADP
ap-4607	160	16	equal	equal	ADJ
ap-4607	160	17	masses	masse	NOUN
ap-4607	160	18	m1	m1	NOUN
ap-4607	160	19	=	=	SYM
ap-4607	160	20	m2	m2	PROPN
ap-4607	160	21	=	=	SYM
ap-4607	160	22	1	1	NUM
ap-4607	160	23	2	2	NUM
ap-4607	160	24	.	.	PUNCT
ap-4607	161	1	the	the	DET
ap-4607	161	2	masses	masse	NOUN
ap-4607	161	3	are	be	AUX
ap-4607	161	4	on	on	ADP
ap-4607	161	5	the	the	DET
ap-4607	161	6	abscissa	abscissa	NOUN
ap-4607	161	7	at	at	ADP
ap-4607	161	8	a	a	DET
ap-4607	161	9	distance	distance	NOUN
ap-4607	161	10	a	a	PRON
ap-4607	161	11	from	from	ADP
ap-4607	161	12	the	the	DET
ap-4607	161	13	origin	origin	NOUN
ap-4607	161	14	of	of	ADP
ap-4607	161	15	coordinates	coordinate	NOUN
ap-4607	161	16	.	.	PUNCT
ap-4607	162	1	in	in	ADP
ap-4607	162	2	this	this	DET
ap-4607	162	3	case	case	NOUN
ap-4607	162	4	,	,	PUNCT
ap-4607	162	5	system	system	NOUN
ap-4607	162	6	(	(	PUNCT
ap-4607	162	7	3	3	X
ap-4607	162	8	)	)	PUNCT
ap-4607	162	9	looks	look	VERB
ap-4607	162	10	like	like	ADP
ap-4607	162	11	this:	this:	NOUN
ap-4607	162	12	y1	y1	NOUN
ap-4607	162	13	=	=	PUNCT
ap-4607	163	1	x1	x1	NUM
ap-4607	163	2	−	−	NUM
ap-4607	163	3	1	1	NUM
ap-4607	163	4	2	2	NUM
ap-4607	163	5	x1	x1	NOUN
ap-4607	163	6	−	−	PROPN
ap-4607	164	1	a	a	DET
ap-4607	164	2	(	(	PUNCT
ap-4607	164	3	x1	x1	NOUN
ap-4607	164	4	−	−	PROPN
ap-4607	164	5	a)2	a)2	PROPN
ap-4607	164	6	+	+	CCONJ
ap-4607	164	7	x2	x2	PROPN
ap-4607	164	8	2	2	NUM
ap-4607	164	9	−	−	NOUN
ap-4607	164	10	1	1	NUM
ap-4607	164	11	2	2	NUM
ap-4607	164	12	x1	x1	NOUN
ap-4607	164	13	+	+	X
ap-4607	164	14	a	a	DET
ap-4607	164	15	(	(	PUNCT
ap-4607	164	16	x1	x1	PROPN
ap-4607	164	17	+	+	PROPN
ap-4607	164	18	a)2	a)2	PROPN
ap-4607	164	19	+	+	CCONJ
ap-4607	164	20	x2	x2	PROPN
ap-4607	164	21	2	2	NUM
ap-4607	164	22	,	,	PUNCT
ap-4607	164	23	y2	y2	NOUN
ap-4607	164	24	=	=	SYM
ap-4607	165	1	x2	x2	NUM
ap-4607	165	2	−	−	NOUN
ap-4607	165	3	1	1	NUM
ap-4607	165	4	2	2	NUM
ap-4607	165	5	x2	x2	NOUN
ap-4607	165	6	(	(	PUNCT
ap-4607	165	7	x1	x1	NOUN
ap-4607	165	8	−	−	PROPN
ap-4607	165	9	a)2	a)2	PROPN
ap-4607	166	1	+	+	CCONJ
ap-4607	166	2	x2	x2	PROPN
ap-4607	166	3	2	2	NUM
ap-4607	166	4	−	−	NOUN
ap-4607	166	5	1	1	NUM
ap-4607	166	6	2	2	NUM
ap-4607	166	7	x2	x2	NOUN
ap-4607	166	8	(	(	PUNCT
ap-4607	166	9	x1	x1	PROPN
ap-4607	166	10	+	+	PROPN
ap-4607	166	11	a)2	a)2	PROPN
ap-4607	166	12	+	+	CCONJ
ap-4607	166	13	x2	x2	PROPN
ap-4607	166	14	2	2	NUM
ap-4607	166	15	.	.	PUNCT
ap-4607	167	1	(	(	PUNCT
ap-4607	167	2	21	21	NUM
ap-4607	167	3	)	)	PUNCT
ap-4607	167	4	we	we	PRON
ap-4607	167	5	transform	transform	VERB
ap-4607	167	6	the	the	DET
ap-4607	167	7	equation	equation	NOUN
ap-4607	167	8	of	of	ADP
ap-4607	167	9	system	system	NOUN
ap-4607	167	10	(	(	PUNCT
ap-4607	167	11	21	21	NUM
ap-4607	167	12	)	)	PUNCT
ap-4607	167	13	into	into	ADP
ap-4607	167	14	a	a	DET
ap-4607	167	15	polynomial	polynomial	ADJ
ap-4607	167	16	form	form	NOUN
ap-4607	167	17	,	,	PUNCT
ap-4607	167	18	and	and	CCONJ
ap-4607	167	19	represent	represent	VERB
ap-4607	167	20	the	the	DET
ap-4607	167	21	obtained	obtain	VERB
ap-4607	167	22	polynomials	polynomial	NOUN
ap-4607	167	23	f1	f1	NOUN
ap-4607	167	24	and	and	CCONJ
ap-4607	167	25	f2	f2	PROPN
ap-4607	167	26	in	in	ADP
ap-4607	167	27	lexicographic	lexicographic	ADJ
ap-4607	167	28	form	form	NOUN
ap-4607	167	29	with	with	ADP
ap-4607	167	30	increasing	increase	VERB
ap-4607	167	31	degrees	degree	NOUN
ap-4607	167	32	of	of	ADP
ap-4607	167	33	variable	variable	ADJ
ap-4607	167	34	x1:	x1:	NOUN
ap-4607	167	35	f1	f1	NOUN
ap-4607	167	36	=	=	PUNCT
ap-4607	168	1	−a2(a2	−a2(a2	PROPN
ap-4607	168	2	+	+	CCONJ
ap-4607	168	3	2x2	2x2	NUM
ap-4607	168	4	2y1	2y1	NUM
ap-4607	168	5	)	)	PUNCT
ap-4607	169	1	+	+	CCONJ
ap-4607	169	2	(	(	PUNCT
ap-4607	169	3	a2	a2	PROPN
ap-4607	169	4	−	−	PROPN
ap-4607	169	5	x2	x2	NOUN
ap-4607	169	6	2	2	NUM
ap-4607	169	7	+	+	CCONJ
ap-4607	169	8	(	(	PUNCT
ap-4607	169	9	a2	a2	PROPN
ap-4607	169	10	+	+	CCONJ
ap-4607	169	11	x2	x2	PROPN
ap-4607	169	12	2	2	NUM
ap-4607	169	13	)	)	PUNCT
ap-4607	169	14	)	)	PUNCT
ap-4607	170	1	x1	x1	PROPN
ap-4607	171	1	+	+	PUNCT
ap-4607	172	1	2y1(a2	2y1(a2	NUM
ap-4607	172	2	−	−	NOUN
ap-4607	173	1	x2	x2	NOUN
ap-4607	173	2	2)x2	2)x2	NUM
ap-4607	173	3	1	1	NUM
ap-4607	173	4	+	+	CCONJ
ap-4607	173	5	(	(	PUNCT
ap-4607	173	6	2x2	2x2	NUM
ap-4607	173	7	2	2	NUM
ap-4607	173	8	−	−	NOUN
ap-4607	173	9	1−	1−	NUM
ap-4607	173	10	2a2)x3	2a2)x3	NUM
ap-4607	173	11	1	1	NUM
ap-4607	173	12	+	+	CCONJ
ap-4607	173	13	y1x	y1x	NOUN
ap-4607	173	14	4	4	NUM
ap-4607	173	15	1	1	NUM
ap-4607	173	16	+	+	NUM
ap-4607	173	17	x5	x5	NOUN
ap-4607	173	18	1	1	NUM
ap-4607	173	19	,	,	PUNCT
ap-4607	173	20	f2	f2	ADV
ap-4607	173	21	=	=	PUNCT
ap-4607	173	22	(	(	PUNCT
ap-4607	173	23	−a2x2	−a2x2	X
ap-4607	173	24	−	−	PUNCT
ap-4607	173	25	x3	x3	ADJ
ap-4607	173	26	2	2	NUM
ap-4607	173	27	+	+	CCONJ
ap-4607	173	28	(	(	PUNCT
ap-4607	173	29	a2	a2	PROPN
ap-4607	173	30	+	+	CCONJ
ap-4607	173	31	x2	x2	PROPN
ap-4607	173	32	2)2(x2	2)2(x2	NUM
ap-4607	173	33	−	−	PROPN
ap-4607	173	34	y2	y2	PROPN
ap-4607	173	35	)	)	PUNCT
ap-4607	173	36	)	)	PUNCT
ap-4607	174	1	−	−	PROPN
ap-4607	175	1	(	(	PUNCT
ap-4607	175	2	x2	x2	NOUN
ap-4607	175	3	+	+	CCONJ
ap-4607	176	1	2(a2	2(a2	NUM
ap-4607	176	2	−	−	NOUN
ap-4607	176	3	x2	x2	PROPN
ap-4607	176	4	2)(x2	2)(x2	PROPN
ap-4607	176	5	−	−	PROPN
ap-4607	176	6	y2	y2	PROPN
ap-4607	176	7	)	)	PUNCT
ap-4607	176	8	)	)	PUNCT
ap-4607	177	1	x2	x2	NOUN
ap-4607	177	2	1	1	NUM
ap-4607	178	1	+	+	CCONJ
ap-4607	178	2	(	(	PUNCT
ap-4607	178	3	x2	x2	INTJ
ap-4607	178	4	−	−	PROPN
ap-4607	178	5	y2)x4	y2)x4	PROPN
ap-4607	178	6	1	1	NUM
ap-4607	178	7	,	,	PUNCT
ap-4607	178	8	(	(	PUNCT
ap-4607	178	9	22	22	NUM
ap-4607	178	10	)	)	PUNCT
ap-4607	178	11	we	we	PRON
ap-4607	178	12	will	will	AUX
ap-4607	178	13	remove	remove	VERB
ap-4607	178	14	from	from	ADP
ap-4607	178	15	the	the	DET
ap-4607	178	16	system	system	NOUN
ap-4607	178	17	the	the	DET
ap-4607	178	18	variable	variable	NOUN
ap-4607	178	19	x1	x1	PROPN
ap-4607	178	20	,	,	PUNCT
ap-4607	178	21	using	use	VERB
ap-4607	178	22	the	the	DET
ap-4607	178	23	resultant	resultant	NOUN
ap-4607	178	24	r1	r1	NOUN
ap-4607	178	25	=	=	SYM
ap-4607	178	26	r(f1	r(f1	NOUN
ap-4607	178	27	,	,	PUNCT
ap-4607	178	28	f2	f2	PROPN
ap-4607	178	29	)	)	PUNCT
ap-4607	178	30	.	.	PUNCT
ap-4607	179	1	sylvester	sylvest	ADJ
ap-4607	179	2	matrix	matrix	NOUN
ap-4607	179	3	s1	s1	NOUN
ap-4607	179	4	=	=	SYM
ap-4607	179	5	s(f1	s(f1	PROPN
ap-4607	179	6	,	,	PUNCT
ap-4607	179	7	f2	f2	PROPN
ap-4607	179	8	)	)	PUNCT
ap-4607	179	9	has	have	VERB
ap-4607	179	10	order	order	NOUN
ap-4607	179	11	(	(	PUNCT
ap-4607	179	12	for	for	ADP
ap-4607	179	13	x1	x1	NUM
ap-4607	179	14	)	)	PUNCT
ap-4607	179	15	degf1	degf1	NOUN
ap-4607	180	1	+	+	CCONJ
ap-4607	180	2	degf2	degf2	NOUN
ap-4607	180	3	=	=	PUNCT
ap-4607	180	4	9	9	X
ap-4607	180	5	.	.	PUNCT
ap-4607	180	6	because	because	SCONJ
ap-4607	180	7	r1	r1	PROPN
ap-4607	180	8	=	=	SYM
ap-4607	180	9	dets1	dets1	PROPN
ap-4607	180	10	,	,	PUNCT
ap-4607	180	11	we	we	PRON
ap-4607	180	12	have	have	VERB
ap-4607	180	13	r1	r1	PROPN
ap-4607	180	14	=	=	PUNCT
ap-4607	180	15	4a4x2	4a4x2	PROPN
ap-4607	181	1	2(a2	2(a2	NUM
ap-4607	182	1	+	+	NUM
ap-4607	182	2	x2	x2	PROPN
ap-4607	182	3	2	2	NUM
ap-4607	182	4	)	)	PUNCT
ap-4607	183	1	(	(	PUNCT
ap-4607	183	2	−a2y3	−a2y3	PROPN
ap-4607	183	3	2	2	NUM
ap-4607	183	4	+	+	CCONJ
ap-4607	183	5	(	(	PUNCT
ap-4607	183	6	a2y2	a2y2	PROPN
ap-4607	183	7	2	2	NUM
ap-4607	183	8	−	−	NOUN
ap-4607	183	9	y2	y2	NOUN
ap-4607	183	10	2	2	NUM
ap-4607	183	11	−	−	NOUN
ap-4607	183	12	4a4y2	4a4y2	NOUN
ap-4607	183	13	2	2	NUM
ap-4607	183	14	−	−	PROPN
ap-4607	183	15	4a2y4	4a2y4	NOUN
ap-4607	184	1	2)x2	2)x2	NUM
ap-4607	184	2	+	+	CCONJ
ap-4607	184	3	(	(	PUNCT
ap-4607	184	4	−4a2y2	−4a2y2	X
ap-4607	184	5	+	+	CCONJ
ap-4607	184	6	4a4y2	4a4y2	PROPN
ap-4607	184	7	−	−	PROPN
ap-4607	184	8	4a6y2	4a6y2	NUM
ap-4607	184	9	−	−	PROPN
ap-4607	185	1	y2	y2	INTJ
ap-4607	185	2	1y	1y	NUM
ap-4607	185	3	2	2	NUM
ap-4607	185	4	−	−	PROPN
ap-4607	185	5	4a2y2	4a2y2	X
ap-4607	185	6	1y2	1y2	NUM
ap-4607	185	7	+	+	CCONJ
ap-4607	185	8	8a4y2	8a4y2	NOUN
ap-4607	185	9	1y2	1y2	NUM
ap-4607	185	10	−	−	PROPN
ap-4607	185	11	4a2y2	4a2y2	NOUN
ap-4607	185	12	1y2	1y2	NUM
ap-4607	185	13	−	−	NUM
ap-4607	185	14	5y3	5y3	NUM
ap-4607	185	15	2	2	NUM
ap-4607	185	16	+	+	CCONJ
ap-4607	185	17	4a2y3	4a2y3	NUM
ap-4607	185	18	2	2	NUM
ap-4607	185	19	−	−	NOUN
ap-4607	185	20	8a4y3	8a4y3	NOUN
ap-4607	185	21	2	2	NUM
ap-4607	185	22	−	−	PROPN
ap-4607	185	23	4a2y5	4a2y5	NUM
ap-4607	185	24	2)x2	2)x2	NUM
ap-4607	185	25	2	2	NUM
ap-4607	185	26	+	+	CCONJ
ap-4607	185	27	(	(	PUNCT
ap-4607	185	28	−4a4	−4a4	NUM
ap-4607	185	29	+	+	NOUN
ap-4607	185	30	4a6	4a6	NUM
ap-4607	186	1	+	+	CCONJ
ap-4607	186	2	y2	y2	NOUN
ap-4607	186	3	1	1	NUM
ap-4607	186	4	+	+	NUM
ap-4607	186	5	4a2y2	4a2y2	NUM
ap-4607	186	6	1	1	NUM
ap-4607	186	7	−	−	NOUN
ap-4607	186	8	8a4y2	8a4y2	NOUN
ap-4607	186	9	1	1	NUM
ap-4607	186	10	+	+	NUM
ap-4607	186	11	4a2y4	4a2y4	NUM
ap-4607	186	12	1	1	NUM
ap-4607	187	1	+	+	CCONJ
ap-4607	187	2	y2	y2	NOUN
ap-4607	188	1	2	2	NUM
ap-4607	188	2	−	−	PROPN
ap-4607	188	3	12a2y2	12a2y2	NUM
ap-4607	188	4	2	2	NUM
ap-4607	188	5	+	+	SYM
ap-4607	188	6	8a4y2	8a4y2	NOUN
ap-4607	188	7	2	2	NUM
ap-4607	188	8	−	−	NUM
ap-4607	188	9	8y2	8y2	NUM
ap-4607	188	10	1y	1y	NUM
ap-4607	188	11	2	2	NUM
ap-4607	188	12	2	2	NUM
ap-4607	188	13	−	−	NOUN
ap-4607	188	14	8y4	8y4	NUM
ap-4607	188	15	2	2	NUM
ap-4607	188	16	+	+	NUM
ap-4607	188	17	4a2y4	4a2y4	NUM
ap-4607	189	1	2)x3	2)x3	NUM
ap-4607	189	2	2	2	NUM
ap-4607	189	3	−	−	PROPN
ap-4607	189	4	4(a4y2	4(a4y2	NOUN
ap-4607	189	5	−	−	NOUN
ap-4607	189	6	a2y2	a2y2	X
ap-4607	189	7	−	−	PROPN
ap-4607	189	8	y2	y2	PROPN
ap-4607	189	9	1y2	1y2	NUM
ap-4607	189	10	−	−	PROPN
ap-4607	189	11	2a2y2	2a2y2	NUM
ap-4607	189	12	1y2	1y2	NUM
ap-4607	190	1	+	+	CCONJ
ap-4607	190	2	y4	y4	ADJ
ap-4607	190	3	1y2	1y2	NUM
ap-4607	191	1	−	−	NOUN
ap-4607	191	2	y3	y3	NOUN
ap-4607	191	3	2	2	NUM
ap-4607	191	4	+	+	CCONJ
ap-4607	191	5	2a2y3	2a2y3	NUM
ap-4607	191	6	2	2	NUM
ap-4607	191	7	+	+	NUM
ap-4607	191	8	2y2	2y2	NUM
ap-4607	191	9	1y	1y	NOUN
ap-4607	191	10	3	3	NUM
ap-4607	191	11	2	2	NUM
ap-4607	191	12	+	+	NUM
ap-4607	191	13	y5	y5	ADJ
ap-4607	191	14	2)x4	2)x4	NUM
ap-4607	191	15	2	2	NUM
ap-4607	191	16	+	+	NUM
ap-4607	191	17	4(a4	4(a4	NUM
ap-4607	191	18	−	−	NOUN
ap-4607	191	19	2a2y2	2a2y2	NUM
ap-4607	191	20	1	1	NUM
ap-4607	191	21	+	+	CCONJ
ap-4607	191	22	y4	y4	ADJ
ap-4607	191	23	1	1	NUM
ap-4607	191	24	+	+	NUM
ap-4607	191	25	2a2y2	2a2y2	NUM
ap-4607	191	26	2	2	NUM
ap-4607	191	27	+	+	CCONJ
ap-4607	191	28	2y2	2y2	NUM
ap-4607	191	29	1y	1y	NOUN
ap-4607	191	30	2	2	NUM
ap-4607	191	31	2	2	NUM
ap-4607	191	32	+	+	CCONJ
ap-4607	191	33	y4	y4	ADJ
ap-4607	191	34	2)x5	2)x5	NOUN
ap-4607	191	35	2	2	NUM
ap-4607	191	36	)	)	PUNCT
ap-4607	191	37	.	.	PUNCT
ap-4607	192	1	(	(	PUNCT
ap-4607	192	2	23	23	NUM
ap-4607	192	3	)	)	PUNCT
ap-4607	192	4	406	406	NUM
ap-4607	192	5	vol	vol	NOUN
ap-4607	192	6	.	.	PUNCT
ap-4607	192	7	57	57	NUM
ap-4607	192	8	no	no	NOUN
ap-4607	192	9	.	.	PUNCT
ap-4607	193	1	6/2017	6/2017	X
ap-4607	193	2	the	the	DET
ap-4607	193	3	analysis	analysis	NOUN
ap-4607	193	4	of	of	ADP
ap-4607	193	5	images	image	NOUN
ap-4607	193	6	in	in	ADP
ap-4607	193	7	n	n	CCONJ
ap-4607	193	8	-point	-point	NOUN
ap-4607	193	9	gravitational	gravitational	ADJ
ap-4607	193	10	lens	lens	NOUN
ap-4607	193	11	in	in	ADP
ap-4607	193	12	order	order	NOUN
ap-4607	193	13	for	for	SCONJ
ap-4607	193	14	the	the	DET
ap-4607	193	15	system	system	NOUN
ap-4607	193	16	equation	equation	NOUN
ap-4607	193	17	to	to	PART
ap-4607	193	18	have	have	VERB
ap-4607	193	19	a	a	DET
ap-4607	193	20	common	common	ADJ
ap-4607	193	21	component	component	NOUN
ap-4607	193	22	,	,	PUNCT
ap-4607	193	23	it	it	PRON
ap-4607	193	24	is	be	AUX
ap-4607	193	25	sufficient	sufficient	ADJ
ap-4607	193	26	that	that	SCONJ
ap-4607	193	27	the	the	DET
ap-4607	193	28	objects	object	NOUN
ap-4607	193	29	r1	r1	PROPN
ap-4607	193	30	≡	≡	PROPN
ap-4607	193	31	0	0	PUNCT
ap-4607	193	32	.	.	PUNCT
ap-4607	194	1	we	we	PRON
ap-4607	194	2	have	have	VERB
ap-4607	194	3	that	that	SCONJ
ap-4607	194	4	the	the	DET
ap-4607	194	5	equation	equation	NOUN
ap-4607	194	6	decomposes	decompose	VERB
ap-4607	194	7	into	into	ADP
ap-4607	194	8	three	three	NUM
ap-4607	194	9	simple	simple	ADJ
ap-4607	194	10	equations	equation	NOUN
ap-4607	194	11	and	and	CCONJ
ap-4607	194	12	one	one	NUM
ap-4607	194	13	non	non	ADJ
ap-4607	194	14	-	-	ADJ
ap-4607	194	15	trivial	trivial	ADJ
ap-4607	194	16	equation	equation	NOUN
ap-4607	194	17	.	.	PUNCT
ap-4607	195	1	from	from	ADP
ap-4607	195	2	the	the	DET
ap-4607	195	3	trivial	trivial	ADJ
ap-4607	195	4	equation	equation	NOUN
ap-4607	195	5	a4	a4	NOUN
ap-4607	195	6	≡	≡	PROPN
ap-4607	195	7	0	0	NUM
ap-4607	195	8	,	,	PUNCT
ap-4607	195	9	x2	x2	PROPN
ap-4607	195	10	2	2	NUM
ap-4607	195	11	≡	≡	PROPN
ap-4607	195	12	0	0	NUM
ap-4607	195	13	,	,	PUNCT
ap-4607	195	14	(	(	PUNCT
ap-4607	195	15	a2+x2	a2+x2	NOUN
ap-4607	195	16	2	2	NUM
ap-4607	195	17	)	)	PUNCT
ap-4607	195	18	≡	≡	PROPN
ap-4607	195	19	0	0	NUM
ap-4607	196	1	it	it	PRON
ap-4607	196	2	follows	follow	VERB
ap-4607	196	3	that	that	SCONJ
ap-4607	196	4	their	their	PRON
ap-4607	196	5	solutions	solution	NOUN
ap-4607	196	6	are	be	AUX
ap-4607	196	7	reduced	reduce	VERB
ap-4607	196	8	to	to	ADP
ap-4607	196	9	1	1	NUM
ap-4607	196	10	-	-	PUNCT
ap-4607	196	11	lens	len	NOUN
ap-4607	196	12	,	,	PUNCT
ap-4607	196	13	or	or	CCONJ
ap-4607	196	14	incommensurate	incommensurate	ADJ
ap-4607	196	15	.	.	PUNCT
ap-4607	197	1	we	we	PRON
ap-4607	197	2	have	have	VERB
ap-4607	197	3	a	a	DET
ap-4607	197	4	nontrivial	nontrivial	ADJ
ap-4607	197	5	equation	equation	NOUN
ap-4607	197	6	−	−	NOUN
ap-4607	197	7	a2y3	a2y3	NOUN
ap-4607	197	8	2	2	NUM
ap-4607	197	9	+	+	CCONJ
ap-4607	197	10	(	(	PUNCT
ap-4607	197	11	a2y2	a2y2	PROPN
ap-4607	197	12	2	2	NUM
ap-4607	197	13	−	−	NOUN
ap-4607	198	1	y2	y2	NOUN
ap-4607	198	2	2	2	NUM
ap-4607	198	3	−	−	NOUN
ap-4607	198	4	4a4y2	4a4y2	NOUN
ap-4607	198	5	2	2	NUM
ap-4607	198	6	−	−	PROPN
ap-4607	198	7	4a2y4	4a2y4	NOUN
ap-4607	199	1	2)x2	2)x2	NUM
ap-4607	199	2	+	+	NOUN
ap-4607	199	3	(	(	PUNCT
ap-4607	199	4	−4a2y2	−4a2y2	X
ap-4607	199	5	+4a4y2−4a6y2−y2	+4a4y2−4a6y2−y2	X
ap-4607	199	6	1y	1y	NUM
ap-4607	199	7	2	2	NUM
ap-4607	199	8	2−4a2y2	2−4a2y2	NUM
ap-4607	199	9	1y2	1y2	NUM
ap-4607	199	10	+8a4y2	+8a4y2	ADP
ap-4607	199	11	1y2	1y2	NUM
ap-4607	199	12	−	−	PROPN
ap-4607	199	13	4a2y2	4a2y2	NUM
ap-4607	199	14	1y2	1y2	NUM
ap-4607	199	15	−	−	NUM
ap-4607	199	16	5y3	5y3	NUM
ap-4607	199	17	2	2	NUM
ap-4607	199	18	+	+	CCONJ
ap-4607	199	19	4a2y3	4a2y3	NUM
ap-4607	199	20	2	2	NUM
ap-4607	199	21	−	−	NOUN
ap-4607	199	22	8a4y3	8a4y3	NOUN
ap-4607	199	23	2	2	NUM
ap-4607	199	24	−	−	PROPN
ap-4607	199	25	4a2y5	4a2y5	NUM
ap-4607	199	26	2)x2	2)x2	NUM
ap-4607	199	27	2	2	NUM
ap-4607	199	28	+	+	CCONJ
ap-4607	199	29	(	(	PUNCT
ap-4607	199	30	−4a4	−4a4	NUM
ap-4607	199	31	+	+	NOUN
ap-4607	199	32	4a6	4a6	NUM
ap-4607	200	1	+	+	CCONJ
ap-4607	200	2	y2	y2	NOUN
ap-4607	200	3	1	1	NUM
ap-4607	200	4	+	+	NUM
ap-4607	200	5	4a2y2	4a2y2	NUM
ap-4607	200	6	1	1	NUM
ap-4607	200	7	−	−	NOUN
ap-4607	200	8	8a4y2	8a4y2	NOUN
ap-4607	200	9	1	1	NUM
ap-4607	200	10	+	+	NUM
ap-4607	200	11	4a2y4	4a2y4	NUM
ap-4607	200	12	1	1	NUM
ap-4607	201	1	+	+	CCONJ
ap-4607	201	2	y2	y2	NOUN
ap-4607	202	1	2	2	NUM
ap-4607	202	2	−	−	PROPN
ap-4607	202	3	12a2y2	12a2y2	NUM
ap-4607	202	4	2	2	NUM
ap-4607	202	5	+	+	SYM
ap-4607	202	6	8a4y2	8a4y2	NOUN
ap-4607	202	7	2	2	NUM
ap-4607	202	8	−	−	NUM
ap-4607	202	9	8y2	8y2	NUM
ap-4607	202	10	1y	1y	NUM
ap-4607	202	11	2	2	NUM
ap-4607	202	12	2	2	NUM
ap-4607	202	13	−	−	NOUN
ap-4607	202	14	8y4	8y4	NUM
ap-4607	202	15	2	2	NUM
ap-4607	202	16	+	+	NUM
ap-4607	202	17	4a2y4	4a2y4	NUM
ap-4607	203	1	2)x3	2)x3	NUM
ap-4607	203	2	2	2	NUM
ap-4607	203	3	−	−	PROPN
ap-4607	203	4	4(a4y2	4(a4y2	NOUN
ap-4607	203	5	−	−	NOUN
ap-4607	203	6	a2y2	a2y2	X
ap-4607	203	7	−	−	PROPN
ap-4607	203	8	y2	y2	PROPN
ap-4607	203	9	1y2	1y2	NUM
ap-4607	203	10	−	−	PROPN
ap-4607	203	11	2a2y2	2a2y2	NUM
ap-4607	203	12	1y2	1y2	NUM
ap-4607	204	1	+	+	CCONJ
ap-4607	204	2	y4	y4	ADJ
ap-4607	204	3	1y2	1y2	NUM
ap-4607	205	1	−	−	NOUN
ap-4607	205	2	y3	y3	NOUN
ap-4607	205	3	2	2	NUM
ap-4607	205	4	+	+	CCONJ
ap-4607	205	5	2a2y3	2a2y3	NUM
ap-4607	205	6	2	2	NUM
ap-4607	205	7	+	+	NUM
ap-4607	205	8	2y2	2y2	NUM
ap-4607	205	9	1y	1y	NOUN
ap-4607	205	10	3	3	NUM
ap-4607	205	11	2	2	NUM
ap-4607	205	12	+	+	NUM
ap-4607	205	13	y5	y5	ADJ
ap-4607	205	14	2)x4	2)x4	NUM
ap-4607	205	15	2	2	NUM
ap-4607	205	16	+	+	NUM
ap-4607	205	17	4(a4	4(a4	NUM
ap-4607	205	18	−	−	NOUN
ap-4607	205	19	2a2y2	2a2y2	NUM
ap-4607	205	20	1	1	NUM
ap-4607	205	21	+	+	CCONJ
ap-4607	205	22	y4	y4	ADJ
ap-4607	205	23	1	1	NUM
ap-4607	205	24	+	+	NUM
ap-4607	205	25	2a2y2	2a2y2	NUM
ap-4607	205	26	2	2	NUM
ap-4607	205	27	+	+	CCONJ
ap-4607	205	28	2y2	2y2	NUM
ap-4607	205	29	1y	1y	NOUN
ap-4607	205	30	2	2	NUM
ap-4607	205	31	2	2	NUM
ap-4607	205	32	+	+	CCONJ
ap-4607	205	33	y4	y4	ADJ
ap-4607	205	34	2)x5	2)x5	ADJ
ap-4607	205	35	2	2	NUM
ap-4607	205	36	=	=	SYM
ap-4607	205	37	0	0	NUM
ap-4607	205	38	.	.	PUNCT
ap-4607	206	1	(	(	PUNCT
ap-4607	206	2	24	24	NUM
ap-4607	206	3	)	)	PUNCT
ap-4607	206	4	we	we	PRON
ap-4607	206	5	equate	equate	VERB
ap-4607	206	6	all	all	DET
ap-4607	206	7	coefficients	coefficient	NOUN
ap-4607	206	8	to	to	ADP
ap-4607	206	9	zero	zero	NUM
ap-4607	206	10	,	,	PUNCT
ap-4607	206	11	and	and	CCONJ
ap-4607	206	12	have	have	VERB
ap-4607	206	13	a	a	DET
ap-4607	206	14	system	system	NOUN
ap-4607	206	15	of	of	ADP
ap-4607	206	16	equations	equations	PROPN
ap-4607	206	17	−a2y3	−a2y3	PROPN
ap-4607	206	18	2	2	NUM
ap-4607	206	19	=	=	SYM
ap-4607	206	20	0	0	NUM
ap-4607	206	21	,	,	PUNCT
ap-4607	206	22	a2y2	a2y2	X
ap-4607	207	1	2	2	NUM
ap-4607	207	2	−	−	NOUN
ap-4607	207	3	y2	y2	NOUN
ap-4607	207	4	2	2	NUM
ap-4607	207	5	−	−	NOUN
ap-4607	207	6	4a4y2	4a4y2	NOUN
ap-4607	207	7	2	2	NUM
ap-4607	207	8	−	−	PROPN
ap-4607	207	9	4a2y4	4a2y4	NOUN
ap-4607	207	10	2	2	NUM
ap-4607	207	11	=	=	SYM
ap-4607	207	12	0	0	NUM
ap-4607	207	13	,	,	PUNCT
ap-4607	207	14	−4a2y2	−4a2y2	X
ap-4607	207	15	+	+	CCONJ
ap-4607	207	16	4a4y2	4a4y2	PROPN
ap-4607	207	17	−	−	PROPN
ap-4607	207	18	4a6y2	4a6y2	NUM
ap-4607	207	19	−	−	PROPN
ap-4607	208	1	y2	y2	INTJ
ap-4607	208	2	1y	1y	NUM
ap-4607	208	3	2	2	NUM
ap-4607	208	4	2	2	NUM
ap-4607	208	5	−4a2y2	−4a2y2	NOUN
ap-4607	208	6	1y2	1y2	NUM
ap-4607	208	7	−	−	NOUN
ap-4607	208	8	4a2y5	4a2y5	NUM
ap-4607	208	9	2	2	NUM
ap-4607	208	10	−	−	PROPN
ap-4607	208	11	4a2y2	4a2y2	NOUN
ap-4607	208	12	1y2	1y2	NUM
ap-4607	208	13	−	−	NUM
ap-4607	208	14	5y3	5y3	NUM
ap-4607	208	15	2	2	NUM
ap-4607	208	16	+	+	CCONJ
ap-4607	208	17	4a2y3	4a2y3	NUM
ap-4607	208	18	2	2	NUM
ap-4607	208	19	−	−	NOUN
ap-4607	208	20	8a4y3	8a4y3	NOUN
ap-4607	208	21	2	2	NUM
ap-4607	208	22	+	+	SYM
ap-4607	208	23	8a4y2	8a4y2	NOUN
ap-4607	208	24	1y2	1y2	NUM
ap-4607	208	25	=	=	SYM
ap-4607	208	26	0	0	NUM
ap-4607	208	27	,	,	PUNCT
ap-4607	208	28	−4a4	−4a4	NUM
ap-4607	208	29	+	+	NOUN
ap-4607	208	30	4a6	4a6	NUM
ap-4607	209	1	+	+	CCONJ
ap-4607	209	2	y2	y2	NOUN
ap-4607	209	3	1	1	NUM
ap-4607	209	4	+	+	NUM
ap-4607	209	5	4a2y2	4a2y2	NUM
ap-4607	209	6	1	1	NUM
ap-4607	209	7	−	−	NOUN
ap-4607	209	8	8a4y2	8a4y2	NOUN
ap-4607	209	9	1	1	NUM
ap-4607	209	10	+	+	NUM
ap-4607	209	11	4a2y4	4a2y4	NUM
ap-4607	209	12	1	1	NUM
ap-4607	210	1	+	+	CCONJ
ap-4607	210	2	y2	y2	NOUN
ap-4607	211	1	2	2	NUM
ap-4607	211	2	−	−	PROPN
ap-4607	211	3	12a2y2	12a2y2	NUM
ap-4607	211	4	2	2	NUM
ap-4607	211	5	+	+	SYM
ap-4607	211	6	8a4y2	8a4y2	NOUN
ap-4607	211	7	2	2	NUM
ap-4607	211	8	−	−	NUM
ap-4607	211	9	8y2	8y2	NUM
ap-4607	211	10	1y	1y	NUM
ap-4607	211	11	2	2	NUM
ap-4607	211	12	2	2	NUM
ap-4607	211	13	−	−	NOUN
ap-4607	211	14	8y4	8y4	NUM
ap-4607	211	15	2	2	NUM
ap-4607	211	16	+	+	NUM
ap-4607	211	17	4a2y4	4a2y4	X
ap-4607	211	18	2	2	NUM
ap-4607	211	19	=	=	SYM
ap-4607	211	20	0	0	NUM
ap-4607	211	21	,	,	PUNCT
ap-4607	211	22	a4y2	a4y2	ADP
ap-4607	211	23	−	−	NOUN
ap-4607	211	24	a2y2	a2y2	X
ap-4607	211	25	−	−	PROPN
ap-4607	211	26	y2	y2	PROPN
ap-4607	211	27	1y2	1y2	NUM
ap-4607	211	28	−	−	PROPN
ap-4607	211	29	2a2y2	2a2y2	NUM
ap-4607	211	30	1y2	1y2	NUM
ap-4607	212	1	+	+	CCONJ
ap-4607	212	2	y4	y4	ADJ
ap-4607	212	3	1y2	1y2	NUM
ap-4607	213	1	−	−	NOUN
ap-4607	213	2	y3	y3	NOUN
ap-4607	213	3	2	2	NUM
ap-4607	213	4	+	+	CCONJ
ap-4607	213	5	2a2y3	2a2y3	NUM
ap-4607	213	6	2	2	NUM
ap-4607	213	7	+	+	NUM
ap-4607	213	8	2y2	2y2	NUM
ap-4607	213	9	1y	1y	NOUN
ap-4607	213	10	3	3	NUM
ap-4607	213	11	2	2	NUM
ap-4607	213	12	+	+	NUM
ap-4607	213	13	y5	y5	NOUN
ap-4607	213	14	2	2	NUM
ap-4607	213	15	=	=	SYM
ap-4607	213	16	0	0	NUM
ap-4607	213	17	,	,	PUNCT
ap-4607	213	18	a4	a4	NOUN
ap-4607	213	19	−	−	PROPN
ap-4607	213	20	2a2y2	2a2y2	NUM
ap-4607	213	21	1	1	NUM
ap-4607	213	22	+	+	CCONJ
ap-4607	213	23	y4	y4	ADJ
ap-4607	213	24	1	1	NUM
ap-4607	213	25	+	+	NUM
ap-4607	213	26	2a2y2	2a2y2	NUM
ap-4607	213	27	2	2	NUM
ap-4607	213	28	+	+	CCONJ
ap-4607	213	29	2y2	2y2	NUM
ap-4607	213	30	1y	1y	NOUN
ap-4607	213	31	2	2	NUM
ap-4607	213	32	2	2	NUM
ap-4607	213	33	+	+	CCONJ
ap-4607	213	34	y4	y4	ADJ
ap-4607	213	35	2	2	NUM
ap-4607	213	36	=	=	SYM
ap-4607	213	37	0	0	NUM
ap-4607	213	38	.	.	PUNCT
ap-4607	214	1	(	(	PUNCT
ap-4607	214	2	25	25	NUM
ap-4607	214	3	)	)	PUNCT
ap-4607	214	4	we	we	PRON
ap-4607	214	5	have	have	VERB
ap-4607	214	6	a	a	NOUN
ap-4607	214	7	=	=	SYM
ap-4607	214	8	0	0	NUM
ap-4607	214	9	,	,	PUNCT
ap-4607	214	10	y2	y2	NOUN
ap-4607	214	11	=	=	SYM
ap-4607	214	12	0	0	NUM
ap-4607	214	13	,	,	PUNCT
ap-4607	214	14	−y2	−y2	PROPN
ap-4607	214	15	1y	1y	PROPN
ap-4607	214	16	2	2	NUM
ap-4607	214	17	2	2	NUM
ap-4607	214	18	−	−	NUM
ap-4607	214	19	5y3	5y3	NUM
ap-4607	214	20	2	2	NUM
ap-4607	214	21	=	=	SYM
ap-4607	214	22	0	0	NUM
ap-4607	214	23	,	,	PUNCT
ap-4607	214	24	−y2	−y2	PROPN
ap-4607	214	25	1y2	1y2	NUM
ap-4607	215	1	+	+	CCONJ
ap-4607	215	2	y4	y4	PROPN
ap-4607	215	3	1y2	1y2	NUM
ap-4607	215	4	−	−	NOUN
ap-4607	215	5	y3	y3	NOUN
ap-4607	215	6	2	2	NUM
ap-4607	215	7	+	+	CCONJ
ap-4607	215	8	2y2	2y2	NUM
ap-4607	215	9	1y	1y	NOUN
ap-4607	215	10	3	3	NUM
ap-4607	215	11	2	2	NUM
ap-4607	215	12	+	+	NUM
ap-4607	215	13	y5	y5	NOUN
ap-4607	215	14	2	2	NUM
ap-4607	215	15	=	=	SYM
ap-4607	215	16	0	0	NUM
ap-4607	215	17	,	,	PUNCT
ap-4607	215	18	y4	y4	ADJ
ap-4607	215	19	1	1	NUM
ap-4607	215	20	+	+	CCONJ
ap-4607	215	21	2y2	2y2	NUM
ap-4607	215	22	1y	1y	NOUN
ap-4607	215	23	2	2	NUM
ap-4607	215	24	2	2	NUM
ap-4607	215	25	+	+	CCONJ
ap-4607	215	26	y4	y4	ADJ
ap-4607	215	27	2	2	NUM
ap-4607	215	28	=	=	SYM
ap-4607	215	29	0	0	NUM
ap-4607	215	30	.	.	PUNCT
ap-4607	216	1	(	(	PUNCT
ap-4607	216	2	26	26	NUM
ap-4607	216	3	)	)	PUNCT
ap-4607	216	4	the	the	DET
ap-4607	216	5	system	system	NOUN
ap-4607	216	6	has	have	VERB
ap-4607	216	7	one	one	NUM
ap-4607	216	8	solution	solution	NOUN
ap-4607	216	9	a	a	DET
ap-4607	216	10	=	=	SYM
ap-4607	216	11	0	0	NUM
ap-4607	216	12	,	,	PUNCT
ap-4607	216	13	y1	y1	NOUN
ap-4607	216	14	=	=	SYM
ap-4607	216	15	0	0	NUM
ap-4607	216	16	,	,	PUNCT
ap-4607	216	17	y2	y2	NOUN
ap-4607	216	18	=	=	SYM
ap-4607	217	1	0	0	X
ap-4607	217	2	.	.	PUNCT
ap-4607	218	1	hence	hence	ADV
ap-4607	218	2	this	this	DET
ap-4607	218	3	solution	solution	NOUN
ap-4607	218	4	reduces	reduce	VERB
ap-4607	218	5	the	the	DET
ap-4607	218	6	2	2	NUM
ap-4607	218	7	-	-	PUNCT
ap-4607	218	8	point	point	NOUN
ap-4607	218	9	gravity	gravity	NOUN
ap-4607	218	10	lens	lens	NOUN
ap-4607	218	11	to	to	ADP
ap-4607	218	12	1	1	NUM
ap-4607	218	13	-	-	PUNCT
ap-4607	218	14	point	point	NOUN
ap-4607	218	15	.	.	PUNCT
ap-4607	219	1	similarly	similarly	ADV
ap-4607	219	2	we	we	PRON
ap-4607	219	3	calculate	calculate	VERB
ap-4607	219	4	the	the	DET
ap-4607	219	5	resultant	resultant	NOUN
ap-4607	219	6	r2	r2	NOUN
ap-4607	219	7	as	as	ADP
ap-4607	219	8	r2	r2	PROPN
ap-4607	219	9	=	=	PUNCT
ap-4607	220	1	4a4(a−	4a4(a−	NUM
ap-4607	221	1	x1)x2	x1)x2	NUM
ap-4607	221	2	1(a+	1(a+	PROPN
ap-4607	221	3	x1	x1	PROPN
ap-4607	221	4	)	)	PUNCT
ap-4607	222	1	(	(	PUNCT
ap-4607	222	2	−a2y3	−a2y3	PROPN
ap-4607	222	3	1	1	NUM
ap-4607	222	4	+	+	NUM
ap-4607	222	5	+	+	CCONJ
ap-4607	222	6	(	(	PUNCT
ap-4607	222	7	1	1	NUM
ap-4607	222	8	+	+	NUM
ap-4607	222	9	a2	a2	PROPN
ap-4607	222	10	+	+	CCONJ
ap-4607	222	11	4a4	4a4	NUM
ap-4607	222	12	−	−	NOUN
ap-4607	222	13	4a2y2	4a2y2	NOUN
ap-4607	222	14	1	1	NUM
ap-4607	222	15	−	−	PROPN
ap-4607	222	16	4a2y2	4a2y2	X
ap-4607	222	17	2)y2	2)y2	NUM
ap-4607	222	18	1x1	1x1	NUM
ap-4607	222	19	+	+	CCONJ
ap-4607	222	20	(	(	PUNCT
ap-4607	222	21	−4a2	−4a2	PROPN
ap-4607	222	22	+	+	NUM
ap-4607	222	23	4a4	4a4	NUM
ap-4607	222	24	−	−	NOUN
ap-4607	222	25	4a6	4a6	NUM
ap-4607	223	1	+	+	CCONJ
ap-4607	223	2	5y2	5y2	NUM
ap-4607	223	3	1	1	NUM
ap-4607	223	4	+	+	NUM
ap-4607	223	5	4a2y2	4a2y2	NUM
ap-4607	223	6	1	1	NUM
ap-4607	223	7	+	+	NOUN
ap-4607	223	8	8a4y2	8a4y2	NUM
ap-4607	223	9	1	1	NUM
ap-4607	223	10	−	−	PROPN
ap-4607	223	11	4a2y4	4a2y4	NOUN
ap-4607	223	12	1	1	NUM
ap-4607	224	1	+	+	CCONJ
ap-4607	224	2	y2	y2	NOUN
ap-4607	224	3	2	2	NUM
ap-4607	224	4	−	−	PROPN
ap-4607	224	5	4a2y2	4a2y2	PROPN
ap-4607	224	6	2	2	NUM
ap-4607	224	7	−	−	NOUN
ap-4607	224	8	8a4y2	8a4y2	NOUN
ap-4607	224	9	2	2	NUM
ap-4607	224	10	−	−	NOUN
ap-4607	224	11	8a2y2	8a2y2	NOUN
ap-4607	224	12	1y	1y	NOUN
ap-4607	224	13	2	2	NUM
ap-4607	224	14	2	2	NUM
ap-4607	224	15	−	−	PROPN
ap-4607	225	1	4a2y4	4a2y4	NOUN
ap-4607	226	1	2)y1x	2)y1x	NOUN
ap-4607	226	2	2	2	NUM
ap-4607	226	3	1	1	NUM
ap-4607	226	4	+	+	CCONJ
ap-4607	226	5	(	(	PUNCT
ap-4607	226	6	4a4	4a4	NUM
ap-4607	227	1	+	+	CCONJ
ap-4607	227	2	4a6	4a6	NUM
ap-4607	227	3	+	+	CCONJ
ap-4607	227	4	(	(	PUNCT
ap-4607	227	5	−1−	−1−	NOUN
ap-4607	227	6	12a2	12a2	NUM
ap-4607	227	7	−	−	NUM
ap-4607	227	8	8a4	8a4	NUM
ap-4607	227	9	+	+	CCONJ
ap-4607	227	10	8y2	8y2	NUM
ap-4607	227	11	1	1	NUM
ap-4607	227	12	+	+	NUM
ap-4607	227	13	4a2y2	4a2y2	NOUN
ap-4607	227	14	1)y2	1)y2	NUM
ap-4607	227	15	1	1	NUM
ap-4607	227	16	+	+	CCONJ
ap-4607	227	17	(	(	PUNCT
ap-4607	227	18	−1	−1	NOUN
ap-4607	227	19	+	+	CCONJ
ap-4607	227	20	4a2	4a2	NUM
ap-4607	227	21	+	+	NUM
ap-4607	227	22	8a4	8a4	NUM
ap-4607	227	23	+	+	CCONJ
ap-4607	227	24	8y2	8y2	NUM
ap-4607	227	25	1	1	NUM
ap-4607	227	26	+	+	NUM
ap-4607	227	27	8a2y2	8a2y2	NOUN
ap-4607	227	28	1	1	NUM
ap-4607	227	29	+	+	NUM
ap-4607	227	30	4a2y2	4a2y2	NUM
ap-4607	227	31	2)y2	2)y2	NUM
ap-4607	227	32	2	2	NUM
ap-4607	227	33	)	)	PUNCT
ap-4607	227	34	x3	x3	NOUN
ap-4607	227	35	1	1	NUM
ap-4607	227	36	+	+	CCONJ
ap-4607	227	37	(	(	PUNCT
ap-4607	227	38	4a2y2	4a2y2	X
ap-4607	227	39	1	1	NUM
ap-4607	227	40	+	+	CCONJ
ap-4607	227	41	a4y1	a4y1	ADV
ap-4607	227	42	−	−	PROPN
ap-4607	227	43	y3	y3	NOUN
ap-4607	227	44	1	1	NUM
ap-4607	227	45	−	−	NOUN
ap-4607	227	46	2a2y3	2a2y3	NUM
ap-4607	227	47	1	1	NUM
ap-4607	227	48	+	+	NUM
ap-4607	227	49	y5	y5	NOUN
ap-4607	227	50	1	1	NUM
ap-4607	227	51	−	−	NOUN
ap-4607	227	52	y1y	y1y	PROPN
ap-4607	227	53	2	2	NUM
ap-4607	227	54	2	2	NUM
ap-4607	227	55	+	+	CCONJ
ap-4607	227	56	2a2y1y	2a2y1y	NUM
ap-4607	227	57	2	2	NUM
ap-4607	227	58	2	2	NUM
ap-4607	227	59	+	+	SYM
ap-4607	227	60	2y3	2y3	NUM
ap-4607	227	61	1y	1y	NUM
ap-4607	227	62	2	2	NUM
ap-4607	227	63	2	2	NUM
ap-4607	227	64	+	+	CCONJ
ap-4607	227	65	y1y	y1y	PROPN
ap-4607	227	66	4	4	NUM
ap-4607	227	67	2)x4	2)x4	NUM
ap-4607	227	68	1	1	NUM
ap-4607	227	69	−	−	NUM
ap-4607	227	70	4(a4	4(a4	NUM
ap-4607	228	1	−	−	NOUN
ap-4607	228	2	2a2y2	2a2y2	NUM
ap-4607	228	3	1	1	NUM
ap-4607	229	1	+	+	CCONJ
ap-4607	229	2	y4	y4	ADJ
ap-4607	229	3	1	1	NUM
ap-4607	229	4	+	+	NUM
ap-4607	229	5	2a2y2	2a2y2	NUM
ap-4607	229	6	2	2	NUM
ap-4607	229	7	+	+	CCONJ
ap-4607	229	8	2y2	2y2	NUM
ap-4607	229	9	1y	1y	NOUN
ap-4607	229	10	2	2	NUM
ap-4607	229	11	2	2	NUM
ap-4607	229	12	+	+	CCONJ
ap-4607	229	13	y4	y4	ADJ
ap-4607	229	14	2)x5	2)x5	NOUN
ap-4607	229	15	1	1	NUM
ap-4607	229	16	)	)	PUNCT
ap-4607	229	17	.	.	PUNCT
ap-4607	230	1	(	(	PUNCT
ap-4607	230	2	27	27	NUM
ap-4607	230	3	)	)	PUNCT
ap-4607	230	4	we	we	PRON
ap-4607	230	5	have	have	VERB
ap-4607	230	6	that	that	SCONJ
ap-4607	230	7	the	the	DET
ap-4607	230	8	solution	solution	NOUN
ap-4607	230	9	of	of	ADP
ap-4607	230	10	system	system	NOUN
ap-4607	230	11	(	(	PUNCT
ap-4607	230	12	21	21	NUM
ap-4607	230	13	)	)	PUNCT
ap-4607	230	14	reduces	reduce	VERB
ap-4607	230	15	the	the	DET
ap-4607	230	16	2	2	NUM
ap-4607	230	17	-	-	PUNCT
ap-4607	230	18	point	point	NOUN
ap-4607	230	19	gravity	gravity	NOUN
ap-4607	230	20	lens	lens	NOUN
ap-4607	230	21	to	to	ADP
ap-4607	230	22	1	1	NUM
ap-4607	230	23	-	-	PUNCT
ap-4607	230	24	point	point	NOUN
ap-4607	230	25	.	.	PUNCT
ap-4607	231	1	whence	whence	NOUN
ap-4607	231	2	•	•	NOUN
ap-4607	231	3	for	for	ADP
ap-4607	231	4	the	the	DET
ap-4607	231	5	1	1	NUM
ap-4607	231	6	-	-	PUNCT
ap-4607	231	7	point	point	NOUN
ap-4607	231	8	gravitational	gravitational	ADJ
ap-4607	231	9	lens	len	NOUN
ap-4607	231	10	set	set	VERB
ap-4607	231	11	we	we	PRON
ap-4607	231	12	have	have	VERB
ap-4607	231	13	v	v	ADP
ap-4607	231	14	1(f1	1(f1	NUM
ap-4607	231	15	,	,	PUNCT
ap-4607	231	16	f2	f2	PROPN
ap-4607	231	17	)	)	PUNCT
ap-4607	231	18	=	=	PUNCT
ap-4607	232	1	{	{	PUNCT
ap-4607	232	2	x1	x1	PROPN
ap-4607	232	3	,	,	PUNCT
ap-4607	232	4	x2	x2	PROPN
ap-4607	233	1	|	|	ADV
ap-4607	233	2	x2	x2	NOUN
ap-4607	233	3	1	1	NUM
ap-4607	234	1	+	+	NUM
ap-4607	234	2	x2	x2	NOUN
ap-4607	234	3	2	2	NUM
ap-4607	234	4	−	−	NOUN
ap-4607	234	5	1	1	NUM
ap-4607	234	6	=	=	NOUN
ap-4607	234	7	0	0	NUM
ap-4607	234	8	}	}	PUNCT
ap-4607	234	9	;	;	PUNCT
ap-4607	234	10	•	•	ADP
ap-4607	234	11	for	for	ADP
ap-4607	234	12	the	the	DET
ap-4607	234	13	2	2	NUM
ap-4607	234	14	-	-	PUNCT
ap-4607	234	15	point	point	NOUN
ap-4607	234	16	gravitational	gravitational	ADJ
ap-4607	234	17	lens	len	NOUN
ap-4607	234	18	set	set	VERB
ap-4607	234	19	we	we	PRON
ap-4607	234	20	have	have	VERB
ap-4607	234	21	v	v	ADP
ap-4607	234	22	1(f1	1(f1	NUM
ap-4607	234	23	,	,	PUNCT
ap-4607	234	24	f2	f2	PROPN
ap-4607	234	25	)	)	PUNCT
ap-4607	234	26	=	=	PUNCT
ap-4607	234	27	∅.	∅.	ADV
ap-4607	234	28	based	base	VERB
ap-4607	234	29	on	on	ADP
ap-4607	234	30	the	the	DET
ap-4607	234	31	studies	study	NOUN
ap-4607	234	32	we	we	PRON
ap-4607	234	33	have	have	AUX
ap-4607	234	34	carried	carry	VERB
ap-4607	234	35	out	out	ADP
ap-4607	234	36	above	above	ADV
ap-4607	234	37	,	,	PUNCT
ap-4607	234	38	one	one	PRON
ap-4607	234	39	can	can	AUX
ap-4607	234	40	prove	prove	VERB
ap-4607	234	41	that	that	SCONJ
ap-4607	234	42	there	there	PRON
ap-4607	234	43	are	be	VERB
ap-4607	234	44	no	no	DET
ap-4607	234	45	extended	extended	ADJ
ap-4607	234	46	objects	object	NOUN
ap-4607	234	47	for	for	ADP
ap-4607	234	48	n	n	PRON
ap-4607	234	49	-point	-point	NOUN
ap-4607	234	50	gravitational	gravitational	ADJ
ap-4607	234	51	lenses	lense	NOUN
ap-4607	234	52	,	,	PUNCT
ap-4607	234	53	i.e.	i.e.	X
ap-4607	234	54	v	v	ADP
ap-4607	234	55	1(f1	1(f1	NUM
ap-4607	234	56	,	,	PUNCT
ap-4607	234	57	f2	f2	PROPN
ap-4607	234	58	)	)	PUNCT
ap-4607	234	59	=	=	PUNCT
ap-4607	234	60	∅.	∅.	ADP
ap-4607	234	61	the	the	DET
ap-4607	234	62	set	set	NOUN
ap-4607	234	63	m(f1	m(f1	NOUN
ap-4607	234	64	,	,	PUNCT
ap-4607	234	65	f2	f2	PROPN
ap-4607	234	66	)	)	PUNCT
ap-4607	234	67	can	can	AUX
ap-4607	234	68	be	be	AUX
ap-4607	234	69	represented	represent	VERB
ap-4607	234	70	in	in	ADP
ap-4607	234	71	the	the	DET
ap-4607	234	72	form	form	NOUN
ap-4607	234	73	m(f1	m(f1	NOUN
ap-4607	234	74	,	,	PUNCT
ap-4607	234	75	f2	f2	PROPN
ap-4607	234	76	)	)	PUNCT
ap-4607	234	77	=	=	SYM
ap-4607	234	78	m0(f1	m0(f1	NOUN
ap-4607	234	79	,	,	PUNCT
ap-4607	234	80	f2	f2	PROPN
ap-4607	234	81	)	)	PUNCT
ap-4607	234	82	∪m1(f1	∪m1(f1	PROPN
ap-4607	234	83	,	,	PUNCT
ap-4607	234	84	f2	f2	PROPN
ap-4607	234	85	)	)	PUNCT
ap-4607	234	86	,	,	PUNCT
ap-4607	234	87	(	(	PUNCT
ap-4607	234	88	28	28	NUM
ap-4607	234	89	)	)	PUNCT
ap-4607	234	90	where	where	SCONJ
ap-4607	234	91	m0(f1	m0(f1	NOUN
ap-4607	234	92	,	,	PUNCT
ap-4607	234	93	f2	f2	PROPN
ap-4607	234	94	)	)	PUNCT
ap-4607	234	95	=	=	PUNCT
ap-4607	234	96	rev	rev	VERB
ap-4607	234	97	0(f1	0(f1	PROPN
ap-4607	234	98	,	,	PUNCT
ap-4607	234	99	f2)/{∪(ai	f2)/{∪(ai	PROPN
ap-4607	234	100	,	,	PUNCT
ap-4607	234	101	bi	bi	NOUN
ap-4607	234	102	)	)	PUNCT
ap-4607	234	103	}	}	PUNCT
ap-4607	234	104	and	and	CCONJ
ap-4607	234	105	m1(f1	m1(f1	NOUN
ap-4607	234	106	,	,	PUNCT
ap-4607	234	107	f2	f2	PROPN
ap-4607	234	108	)	)	PUNCT
ap-4607	235	1	=	=	VERB
ap-4607	235	2	rev	rev	VERB
ap-4607	235	3	1(f1	1(f1	PROPN
ap-4607	235	4	,	,	PUNCT
ap-4607	235	5	f2)/{∪(ai	f2)/{∪(ai	PROPN
ap-4607	235	6	,	,	PUNCT
ap-4607	235	7	bi	bi	NOUN
ap-4607	235	8	)	)	PUNCT
ap-4607	235	9	}	}	PUNCT
ap-4607	235	10	.	.	PUNCT
ap-4607	236	1	it	it	PRON
ap-4607	236	2	is	be	AUX
ap-4607	236	3	known	know	VERB
ap-4607	236	4	that	that	SCONJ
ap-4607	236	5	the	the	DET
ap-4607	236	6	set	set	NOUN
ap-4607	236	7	m1(f1	m1(f1	NOUN
ap-4607	236	8	,	,	PUNCT
ap-4607	236	9	f2	f2	PROPN
ap-4607	236	10	)	)	PUNCT
ap-4607	236	11	,	,	PUNCT
ap-4607	236	12	for	for	ADP
ap-4607	236	13	a	a	DET
ap-4607	236	14	point	point	NOUN
ap-4607	236	15	source	source	NOUN
ap-4607	236	16	in	in	ADP
ap-4607	236	17	1	1	NUM
ap-4607	236	18	-	-	PUNCT
ap-4607	236	19	point	point	NOUN
ap-4607	236	20	lens	lens	NOUN
ap-4607	236	21	is	be	AUX
ap-4607	236	22	not	not	PART
ap-4607	236	23	empty	empty	ADJ
ap-4607	236	24	,	,	PUNCT
ap-4607	236	25	see	see	VERB
ap-4607	236	26	for	for	ADP
ap-4607	236	27	example	example	NOUN
ap-4607	237	1	[	[	X
ap-4607	237	2	9	9	NUM
ap-4607	237	3	,	,	PUNCT
ap-4607	237	4	10	10	NUM
ap-4607	237	5	,	,	PUNCT
ap-4607	237	6	14	14	NUM
ap-4607	237	7	]	]	PUNCT
ap-4607	237	8	,	,	PUNCT
ap-4607	237	9	coincides	coincide	VERB
ap-4607	237	10	with	with	ADP
ap-4607	237	11	v	v	PROPN
ap-4607	237	12	1(f1	1(f1	NUM
ap-4607	237	13	,	,	PUNCT
ap-4607	237	14	f2	f2	PROPN
ap-4607	237	15	)	)	PUNCT
ap-4607	237	16	,	,	PUNCT
ap-4607	237	17	see	see	VERB
ap-4607	237	18	[	[	X
ap-4607	237	19	5	5	NUM
ap-4607	237	20	]	]	PUNCT
ap-4607	237	21	and	and	CCONJ
ap-4607	237	22	is	be	AUX
ap-4607	237	23	einstein	einstein	PROPN
ap-4607	237	24	ring	ring	NOUN
ap-4607	237	25	.	.	PUNCT
ap-4607	238	1	but	but	CCONJ
ap-4607	238	2	for	for	ADP
ap-4607	238	3	a	a	DET
ap-4607	238	4	point	point	NOUN
ap-4607	238	5	source	source	NOUN
ap-4607	238	6	in	in	ADP
ap-4607	238	7	symmetric	symmetric	ADJ
ap-4607	238	8	2	2	NUM
ap-4607	238	9	-	-	PUNCT
ap-4607	238	10	point	point	NOUN
ap-4607	238	11	lens	len	NOUN
ap-4607	238	12	,	,	PUNCT
ap-4607	238	13	we	we	PRON
ap-4607	238	14	proved	prove	VERB
ap-4607	238	15	[	[	X
ap-4607	238	16	5	5	X
ap-4607	238	17	]	]	PUNCT
ap-4607	238	18	that	that	SCONJ
ap-4607	238	19	the	the	DET
ap-4607	238	20	set	set	NOUN
ap-4607	238	21	m1(f1	m1(f1	PROPN
ap-4607	238	22	,	,	PUNCT
ap-4607	238	23	f2	f2	PROPN
ap-4607	238	24	)	)	PUNCT
ap-4607	238	25	is	be	AUX
ap-4607	238	26	empty	empty	ADJ
ap-4607	238	27	and	and	CCONJ
ap-4607	238	28	put	put	VERB
ap-4607	238	29	forward	forward	ADV
ap-4607	238	30	hypothesis	hypothesis	NOUN
ap-4607	238	31	:	:	PUNCT
ap-4607	238	32	for	for	ADP
ap-4607	238	33	n	n	CCONJ
ap-4607	238	34	-	-	PUNCT
ap-4607	238	35	point	point	NOUN
ap-4607	238	36	lens	lens	NOUN
ap-4607	238	37	this	this	DET
ap-4607	238	38	set	set	NOUN
ap-4607	238	39	is	be	AUX
ap-4607	238	40	empty	empty	ADJ
ap-4607	238	41	for	for	ADP
ap-4607	238	42	n	n	X
ap-4607	238	43	>	>	X
ap-4607	238	44	1	1	NUM
ap-4607	238	45	.	.	X
ap-4607	238	46	5	5	NUM
ap-4607	238	47	.	.	PUNCT
ap-4607	239	1	the	the	DET
ap-4607	239	2	study	study	NOUN
ap-4607	239	3	of	of	ADP
ap-4607	239	4	the	the	DET
ap-4607	239	5	set	set	NOUN
ap-4607	239	6	v	v	ADP
ap-4607	239	7	0(f1	0(f1	NOUN
ap-4607	239	8	,	,	PUNCT
ap-4607	239	9	f2	f2	PROPN
ap-4607	239	10	)	)	PUNCT
ap-4607	239	11	(	(	PUNCT
ap-4607	239	12	point	point	NOUN
ap-4607	239	13	solutions	solution	NOUN
ap-4607	239	14	)	)	PUNCT
ap-4607	239	15	to	to	PART
ap-4607	239	16	research	research	VERB
ap-4607	239	17	the	the	DET
ap-4607	239	18	set	set	NOUN
ap-4607	239	19	of	of	ADP
ap-4607	239	20	solutions	solution	NOUN
ap-4607	239	21	v	v	ADP
ap-4607	239	22	0(f1	0(f1	NOUN
ap-4607	239	23	,	,	PUNCT
ap-4607	239	24	f2	f2	PROPN
ap-4607	239	25	)	)	PUNCT
ap-4607	239	26	of	of	ADP
ap-4607	239	27	system	system	NOUN
ap-4607	239	28	(	(	PUNCT
ap-4607	239	29	3	3	X
ap-4607	239	30	)	)	PUNCT
ap-4607	239	31	we	we	PRON
ap-4607	239	32	use	use	VERB
ap-4607	239	33	the	the	DET
ap-4607	239	34	bezout	bezout	NOUN
ap-4607	239	35	theorem	theorem	NOUN
ap-4607	239	36	,	,	PUNCT
ap-4607	239	37	see	see	VERB
ap-4607	239	38	for	for	ADP
ap-4607	239	39	example	example	NOUN
ap-4607	239	40	[	[	X
ap-4607	239	41	11	11	NUM
ap-4607	239	42	–	–	PUNCT
ap-4607	239	43	13	13	NUM
ap-4607	239	44	,	,	PUNCT
ap-4607	239	45	15	15	NUM
ap-4607	239	46	]	]	PUNCT
ap-4607	239	47	.	.	PUNCT
ap-4607	240	1	in	in	ADP
ap-4607	240	2	most	most	ADJ
ap-4607	240	3	monographs	monograph	NOUN
ap-4607	240	4	,	,	PUNCT
ap-4607	240	5	the	the	DET
ap-4607	240	6	authors	author	NOUN
ap-4607	240	7	formulate	formulate	VERB
ap-4607	240	8	the	the	DET
ap-4607	240	9	bezout	bezout	NOUN
ap-4607	240	10	theorem	theorem	NOUN
ap-4607	240	11	in	in	ADP
ap-4607	240	12	geometric	geometric	ADJ
ap-4607	240	13	terms	term	NOUN
ap-4607	240	14	;	;	PUNCT
ap-4607	240	15	see	see	VERB
ap-4607	240	16	for	for	ADP
ap-4607	240	17	example	example	NOUN
ap-4607	240	18	[	[	X
ap-4607	240	19	11	11	NUM
ap-4607	240	20	,	,	PUNCT
ap-4607	240	21	12	12	NUM
ap-4607	240	22	,	,	PUNCT
ap-4607	240	23	15	15	NUM
ap-4607	240	24	]	]	PUNCT
ap-4607	240	25	.	.	PUNCT
ap-4607	241	1	one	one	NUM
ap-4607	241	2	of	of	ADP
ap-4607	241	3	these	these	DET
ap-4607	241	4	theorems	theorem	NOUN
ap-4607	241	5	is	be	AUX
ap-4607	241	6	quoted	quote	VERB
ap-4607	241	7	in	in	ADP
ap-4607	241	8	appendix	appendix	NOUN
ap-4607	241	9	.	.	PUNCT
ap-4607	242	1	in	in	ADP
ap-4607	242	2	[	[	X
ap-4607	242	3	13	13	NUM
ap-4607	242	4	]	]	SYM
ap-4607	242	5	bezout	bezout	NOUN
ap-4607	242	6	’s	’s	PART
ap-4607	242	7	theorem	theorem	NOUN
ap-4607	242	8	is	be	AUX
ap-4607	242	9	formulated	formulate	VERB
ap-4607	242	10	in	in	ADP
ap-4607	242	11	algebraic	algebraic	ADJ
ap-4607	242	12	terms	term	NOUN
ap-4607	242	13	,	,	PUNCT
ap-4607	242	14	but	but	CCONJ
ap-4607	242	15	for	for	ADP
ap-4607	242	16	equations	equation	NOUN
ap-4607	242	17	given	give	VERB
ap-4607	242	18	in	in	ADP
ap-4607	242	19	affine	affine	NOUN
ap-4607	242	20	coordinates	coordinate	NOUN
ap-4607	242	21	.	.	PUNCT
ap-4607	243	1	this	this	DET
ap-4607	243	2	theorem	theorem	NOUN
ap-4607	243	3	is	be	AUX
ap-4607	243	4	also	also	ADV
ap-4607	243	5	quoted	quote	VERB
ap-4607	243	6	in	in	ADP
ap-4607	243	7	appendix	appendix	NOUN
ap-4607	243	8	.	.	PUNCT
ap-4607	244	1	for	for	ADP
ap-4607	244	2	our	our	PRON
ap-4607	244	3	purposes	purpose	NOUN
ap-4607	244	4	,	,	PUNCT
ap-4607	244	5	we	we	PRON
ap-4607	244	6	formulate	formulate	VERB
ap-4607	244	7	this	this	DET
ap-4607	244	8	theorem	theorem	NOUN
ap-4607	244	9	in	in	ADP
ap-4607	244	10	algebraic	algebraic	ADJ
ap-4607	244	11	terms	term	NOUN
ap-4607	244	12	,	,	PUNCT
ap-4607	244	13	but	but	CCONJ
ap-4607	244	14	for	for	ADP
ap-4607	244	15	functions	function	NOUN
ap-4607	244	16	given	give	VERB
ap-4607	244	17	in	in	ADP
ap-4607	244	18	homogeneous	homogeneous	ADJ
ap-4607	244	19	coordinates	coordinate	NOUN
ap-4607	244	20	.	.	PUNCT
ap-4607	245	1	theorem	theorem	ADJ
ap-4607	245	2	1	1	NUM
ap-4607	245	3	(	(	PUNCT
ap-4607	245	4	bezout	bezout	NOUN
ap-4607	245	5	)	)	PUNCT
ap-4607	245	6	.	.	PUNCT
ap-4607	246	1	let	let	VERB
ap-4607	247	1	g1(x0	g1(x0	VERB
ap-4607	247	2	:	:	PUNCT
ap-4607	247	3	x1	x1	NUM
ap-4607	247	4	:	:	PUNCT
ap-4607	247	5	x2	x2	ADJ
ap-4607	247	6	)	)	PUNCT
ap-4607	247	7	and	and	CCONJ
ap-4607	247	8	g2(x0	g2(x0	NOUN
ap-4607	247	9	:	:	PUNCT
ap-4607	248	1	x1	x1	NUM
ap-4607	248	2	:	:	PUNCT
ap-4607	248	3	x2	x2	X
ap-4607	248	4	)	)	PUNCT
ap-4607	248	5	be	be	AUX
ap-4607	248	6	homogeneous	homogeneous	ADJ
ap-4607	248	7	polynomials	polynomial	NOUN
ap-4607	248	8	,	,	PUNCT
ap-4607	248	9	degg1(x0	degg1(x0	NOUN
ap-4607	248	10	:	:	PUNCT
ap-4607	248	11	x1	x1	NUM
ap-4607	248	12	:	:	PUNCT
ap-4607	248	13	x2	x2	X
ap-4607	248	14	)	)	PUNCT
ap-4607	248	15	=	=	SYM
ap-4607	249	1	n	n	CCONJ
ap-4607	249	2	,	,	PUNCT
ap-4607	249	3	degg2(x0	degg2(x0	PROPN
ap-4607	249	4	:	:	PUNCT
ap-4607	250	1	x1	x1	NUM
ap-4607	250	2	:	:	PUNCT
ap-4607	250	3	x2	x2	X
ap-4607	250	4	)	)	PUNCT
ap-4607	250	5	=	=	SYM
ap-4607	251	1	m	m	NOUN
ap-4607	251	2	and	and	CCONJ
ap-4607	251	3	the	the	DET
ap-4607	251	4	resultant	resultant	NOUN
ap-4607	251	5	r1(g1	r1(g1	VERB
ap-4607	251	6	,	,	PUNCT
ap-4607	251	7	g2	g2	PROPN
ap-4607	251	8	)	)	PUNCT
ap-4607	251	9	,	,	PUNCT
ap-4607	251	10	with	with	ADP
ap-4607	251	11	respect	respect	NOUN
ap-4607	251	12	to	to	ADP
ap-4607	251	13	variable	variable	NOUN
ap-4607	251	14	x1	x1	PROPN
ap-4607	251	15	not	not	PART
ap-4607	251	16	identically	identically	ADV
ap-4607	251	17	equal	equal	ADJ
ap-4607	251	18	to	to	ADP
ap-4607	251	19	zero	zero	NUM
ap-4607	251	20	.	.	PUNCT
ap-4607	252	1	then	then	ADV
ap-4607	252	2	the	the	DET
ap-4607	252	3	resultant	resultant	NOUN
ap-4607	252	4	r1(g1	r1(g1	VERB
ap-4607	252	5	,	,	PUNCT
ap-4607	252	6	g2	g2	PROPN
ap-4607	252	7	)	)	PUNCT
ap-4607	252	8	is	be	AUX
ap-4607	252	9	a	a	DET
ap-4607	252	10	homogeneous	homogeneous	ADJ
ap-4607	252	11	polynomial	polynomial	NOUN
ap-4607	252	12	with	with	ADP
ap-4607	252	13	respect	respect	NOUN
ap-4607	252	14	to	to	ADP
ap-4607	252	15	variables	variable	NOUN
ap-4607	252	16	x0	x0	PROPN
ap-4607	252	17	and	and	CCONJ
ap-4607	252	18	x2	x2	NOUN
ap-4607	252	19	,	,	PUNCT
ap-4607	252	20	and	and	CCONJ
ap-4607	252	21	degr1(g1	degr1(g1	PROPN
ap-4607	252	22	,	,	PUNCT
ap-4607	252	23	g2	g2	PROPN
ap-4607	252	24	)	)	PUNCT
ap-4607	252	25	=	=	SYM
ap-4607	252	26	n	n	PRON
ap-4607	252	27	·	·	PUNCT
ap-4607	252	28	m.	m.	NOUN
ap-4607	252	29	proof	proof	NOUN
ap-4607	252	30	.	.	PUNCT
ap-4607	253	1	the	the	DET
ap-4607	253	2	resultant	resultant	NOUN
ap-4607	253	3	r1(g1	r1(g1	VERB
ap-4607	253	4	,	,	PUNCT
ap-4607	253	5	g2	g2	PROPN
ap-4607	253	6	)	)	PUNCT
ap-4607	253	7	is	be	AUX
ap-4607	253	8	a	a	DET
ap-4607	253	9	polynomial	polynomial	NOUN
ap-4607	253	10	in	in	ADP
ap-4607	253	11	the	the	DET
ap-4607	253	12	variablesx0	variablesx0	NOUN
ap-4607	253	13	andx2	andx2	NOUN
ap-4607	253	14	.	.	PUNCT
ap-4607	254	1	we	we	PRON
ap-4607	254	2	denote	denote	VERB
ap-4607	254	3	it	it	PRON
ap-4607	254	4	by	by	ADP
ap-4607	254	5	f	f	PROPN
ap-4607	254	6	,	,	PUNCT
ap-4607	254	7	and	and	CCONJ
ap-4607	254	8	write	write	VERB
ap-4607	254	9	407	407	NUM
ap-4607	254	10	a.	a.	NOUN
ap-4607	254	11	t.	t.	PROPN
ap-4607	254	12	kotvytskiy	kotvytskiy	PROPN
ap-4607	254	13	,	,	PUNCT
ap-4607	254	14	s.	s.	PROPN
ap-4607	254	15	d.	d.	PROPN
ap-4607	254	16	bronza	bronza	PROPN
ap-4607	254	17	,	,	PUNCT
ap-4607	254	18	v.	v.	PROPN
ap-4607	254	19	yu	yu	PROPN
ap-4607	254	20	.	.	PUNCT
ap-4607	255	1	shablenko	shablenko	PROPN
ap-4607	255	2	acta	acta	PROPN
ap-4607	255	3	polytechnica	polytechnica	PROPN
ap-4607	255	4	f	f	PROPN
ap-4607	256	1	=	=	PUNCT
ap-4607	256	2	r1(g1	r1(g1	PROPN
ap-4607	256	3	,	,	PUNCT
ap-4607	256	4	g2	g2	PROPN
ap-4607	256	5	)	)	PUNCT
ap-4607	256	6	.	.	PUNCT
ap-4607	257	1	let	let	VERB
ap-4607	257	2	us	we	PRON
ap-4607	257	3	prove	prove	VERB
ap-4607	257	4	that	that	SCONJ
ap-4607	257	5	the	the	DET
ap-4607	257	6	polynomial	polynomial	ADJ
ap-4607	257	7	f	f	PROPN
ap-4607	257	8	=	=	SYM
ap-4607	257	9	f	f	PROPN
ap-4607	257	10	(	(	PUNCT
ap-4607	257	11	x0	x0	PROPN
ap-4607	257	12	,	,	PUNCT
ap-4607	257	13	x2	x2	PROPN
ap-4607	257	14	)	)	PUNCT
ap-4607	257	15	is	be	AUX
ap-4607	257	16	homogeneous	homogeneous	ADJ
ap-4607	257	17	and	and	CCONJ
ap-4607	257	18	of	of	ADP
ap-4607	257	19	degree	degree	NOUN
ap-4607	257	20	degf	degf	NOUN
ap-4607	257	21	=	=	SYM
ap-4607	257	22	n	n	PRON
ap-4607	257	23	·	·	PUNCT
ap-4607	257	24	m.	m.	NOUN
ap-4607	257	25	really	really	ADV
ap-4607	257	26	,	,	PUNCT
ap-4607	257	27	we	we	PRON
ap-4607	257	28	have	have	VERB
ap-4607	257	29	f	f	PROPN
ap-4607	257	30	(	(	PUNCT
ap-4607	257	31	tx0	tx0	ADJ
ap-4607	257	32	,	,	PUNCT
ap-4607	257	33	tx2	tx2	NOUN
ap-4607	257	34	)	)	PUNCT
ap-4607	258	1	=	=	SYM
ap-4607	258	2	r1	r1	PROPN
ap-4607	258	3	(	(	PUNCT
ap-4607	258	4	g1(tx0	g1(tx0	X
ap-4607	258	5	:	:	PUNCT
ap-4607	258	6	x1	x1	NUM
ap-4607	258	7	:	:	PUNCT
ap-4607	258	8	tx2	tx2	NOUN
ap-4607	258	9	)	)	PUNCT
ap-4607	258	10	,	,	PUNCT
ap-4607	258	11	g2(tx0	g2(tx0	PROPN
ap-4607	258	12	:	:	PUNCT
ap-4607	258	13	x1	x1	NUM
ap-4607	258	14	:	:	PUNCT
ap-4607	258	15	tx2	tx2	NOUN
ap-4607	258	16	)	)	PUNCT
ap-4607	258	17	)	)	PUNCT
ap-4607	258	18	.	.	PUNCT
ap-4607	259	1	(	(	PUNCT
ap-4607	259	2	29	29	NUM
ap-4607	259	3	)	)	PUNCT
ap-4607	259	4	according	accord	VERB
ap-4607	259	5	to	to	ADP
ap-4607	259	6	theorem	theorem	ADJ
ap-4607	259	7	1a	1a	NOUN
ap-4607	259	8	,	,	PUNCT
ap-4607	259	9	see	see	VERB
ap-4607	259	10	appendix	appendix	NOUN
ap-4607	259	11	,	,	PUNCT
ap-4607	259	12	we	we	PRON
ap-4607	259	13	have	have	VERB
ap-4607	259	14	r1	r1	NOUN
ap-4607	259	15	(	(	PUNCT
ap-4607	259	16	g1(tx0	g1(tx0	X
ap-4607	259	17	:	:	PUNCT
ap-4607	260	1	x1	x1	NUM
ap-4607	260	2	:	:	PUNCT
ap-4607	260	3	tx2	tx2	NOUN
ap-4607	260	4	)	)	PUNCT
ap-4607	260	5	,	,	PUNCT
ap-4607	260	6	g2(tx0	g2(tx0	PROPN
ap-4607	260	7	:	:	PUNCT
ap-4607	260	8	x1	x1	NUM
ap-4607	260	9	:	:	PUNCT
ap-4607	260	10	tx2	tx2	NOUN
ap-4607	260	11	)	)	PUNCT
ap-4607	260	12	)	)	PUNCT
ap-4607	261	1	=	=	PUNCT
ap-4607	261	2	det	det	PROPN
ap-4607	261	3	sul	sul	PROPN
ap-4607	261	4	(	(	PUNCT
ap-4607	261	5	g1(tx0	g1(tx0	PROPN
ap-4607	261	6	:	:	PUNCT
ap-4607	261	7	x1	x1	NUM
ap-4607	261	8	:	:	PUNCT
ap-4607	261	9	tx2	tx2	NOUN
ap-4607	261	10	)	)	PUNCT
ap-4607	261	11	,	,	PUNCT
ap-4607	261	12	g2(tx0	g2(tx0	PROPN
ap-4607	261	13	:	:	PUNCT
ap-4607	261	14	x1	x1	NUM
ap-4607	261	15	:	:	PUNCT
ap-4607	261	16	tx2	tx2	NOUN
ap-4607	261	17	)	)	PUNCT
ap-4607	261	18	)	)	PUNCT
ap-4607	261	19	.	.	PUNCT
ap-4607	262	1	(	(	PUNCT
ap-4607	262	2	30	30	NUM
ap-4607	262	3	)	)	PUNCT
ap-4607	262	4	where	where	SCONJ
ap-4607	262	5	the	the	DET
ap-4607	262	6	right	right	ADJ
ap-4607	262	7	-	-	PUNCT
ap-4607	262	8	hand	hand	NOUN
ap-4607	262	9	side	side	NOUN
ap-4607	262	10	of	of	ADP
ap-4607	262	11	expression	expression	NOUN
ap-4607	262	12	(	(	PUNCT
ap-4607	262	13	30	30	NUM
ap-4607	262	14	)	)	PUNCT
ap-4607	262	15	is	be	AUX
ap-4607	262	16	the	the	DET
ap-4607	262	17	determinant	determinant	ADJ
ap-4607	262	18	of	of	ADP
ap-4607	262	19	the	the	DET
ap-4607	262	20	sylvester	sylvest	ADJ
ap-4607	262	21	matrix	matrix	NOUN
ap-4607	262	22	.	.	PUNCT
ap-4607	263	1	the	the	DET
ap-4607	263	2	order	order	NOUN
ap-4607	263	3	of	of	ADP
ap-4607	263	4	this	this	DET
ap-4607	263	5	determinant	determinant	ADJ
ap-4607	263	6	is	be	AUX
ap-4607	263	7	n+m	n+m	NOUN
ap-4607	263	8	.	.	PUNCT
ap-4607	264	1	elements	element	NOUN
ap-4607	264	2	of	of	ADP
ap-4607	264	3	the	the	DET
ap-4607	264	4	sylvester	sylvest	ADJ
ap-4607	264	5	matrix	matrix	NOUN
ap-4607	264	6	are	be	AUX
ap-4607	264	7	coeficies	coeficie	NOUN
ap-4607	264	8	from	from	ADP
ap-4607	264	9	the	the	DET
ap-4607	264	10	lexicographic	lexicographic	ADJ
ap-4607	264	11	representation	representation	NOUN
ap-4607	264	12	of	of	ADP
ap-4607	264	13	homogeneous	homogeneous	ADJ
ap-4607	264	14	polynomials	polynomial	NOUN
ap-4607	264	15	g1	g1	NOUN
ap-4607	264	16	and	and	CCONJ
ap-4607	264	17	g2	g2	PROPN
ap-4607	264	18	.	.	PUNCT
ap-4607	265	1	we	we	PRON
ap-4607	265	2	have	have	VERB
ap-4607	265	3	that	that	DET
ap-4607	265	4	g1	g1	PROPN
ap-4607	265	5	=	=	PUNCT
ap-4607	265	6	n∑	n∑	PROPN
ap-4607	265	7	i=0	i=0	PROPN
ap-4607	265	8	aix	aix	PROPN
ap-4607	265	9	n−i	n−i	PROPN
ap-4607	265	10	1	1	NUM
ap-4607	265	11	and	and	CCONJ
ap-4607	265	12	g2	g2	PROPN
ap-4607	265	13	=	=	PUNCT
ap-4607	265	14	m∑	m∑	CCONJ
ap-4607	265	15	j=0	j=0	VERB
ap-4607	265	16	bjx	bjx	VERB
ap-4607	265	17	m−j	m−j	PROPN
ap-4607	265	18	1	1	NUM
ap-4607	265	19	.	.	PUNCT
ap-4607	266	1	the	the	DET
ap-4607	266	2	coefficients	coefficient	NOUN
ap-4607	266	3	ai	ai	VERB
ap-4607	266	4	=	=	NOUN
ap-4607	266	5	ai(x0	ai(x0	NOUN
ap-4607	266	6	:	:	PUNCT
ap-4607	266	7	x2	x2	X
ap-4607	266	8	)	)	PUNCT
ap-4607	266	9	and	and	CCONJ
ap-4607	266	10	bj	bj	VERB
ap-4607	266	11	=	=	VERB
ap-4607	266	12	bj(x0	bj(x0	NOUN
ap-4607	266	13	:	:	PUNCT
ap-4607	267	1	x2	x2	X
ap-4607	267	2	)	)	PUNCT
ap-4607	267	3	are	be	AUX
ap-4607	267	4	homogeneous	homogeneous	ADJ
ap-4607	267	5	polynomials	polynomial	NOUN
ap-4607	267	6	.	.	PUNCT
ap-4607	268	1	the	the	DET
ap-4607	268	2	degree	degree	NOUN
ap-4607	268	3	is	be	AUX
ap-4607	268	4	deg	deg	ADJ
ap-4607	268	5	ai(x0	ai(x0	NOUN
ap-4607	268	6	:	:	PUNCT
ap-4607	268	7	x2	x2	X
ap-4607	268	8	)	)	PUNCT
ap-4607	269	1	=	=	SYM
ap-4607	269	2	i	i	PRON
ap-4607	269	3	and	and	CCONJ
ap-4607	269	4	deg	deg	VERB
ap-4607	269	5	bj(x0	bj(x0	ADP
ap-4607	269	6	:	:	PUNCT
ap-4607	270	1	x2	x2	X
ap-4607	270	2	)	)	PUNCT
ap-4607	270	3	=	=	SYM
ap-4607	270	4	j	j	PROPN
ap-4607	270	5	,	,	PUNCT
ap-4607	270	6	that	that	ADV
ap-4607	270	7	is	is	ADV
ap-4607	270	8	,	,	PUNCT
ap-4607	270	9	ai(tx0	ai(tx0	PRON
ap-4607	270	10	:	:	PUNCT
ap-4607	270	11	tx2	tx2	X
ap-4607	270	12	)	)	PUNCT
ap-4607	271	1	=	=	NOUN
ap-4607	271	2	tiai(x0	tiai(x0	NOUN
ap-4607	271	3	:	:	PUNCT
ap-4607	271	4	x2	x2	X
ap-4607	271	5	)	)	PUNCT
ap-4607	271	6	and	and	CCONJ
ap-4607	271	7	bj(tx0	bj(tx0	NUM
ap-4607	271	8	:	:	PUNCT
ap-4607	272	1	tx2	tx2	X
ap-4607	272	2	)	)	PUNCT
ap-4607	272	3	=	=	NOUN
ap-4607	272	4	tjbj(x0	tjbj(x0	NOUN
ap-4607	272	5	:	:	PUNCT
ap-4607	272	6	x2	x2	X
ap-4607	272	7	)	)	PUNCT
ap-4607	272	8	.	.	PUNCT
ap-4607	273	1	we	we	PRON
ap-4607	273	2	multiply	multiply	VERB
ap-4607	273	3	every	every	DET
ap-4607	273	4	row	row	NOUN
ap-4607	273	5	of	of	ADP
ap-4607	273	6	the	the	DET
ap-4607	273	7	determinant	determinant	NOUN
ap-4607	273	8	of	of	ADP
ap-4607	273	9	the	the	DET
ap-4607	273	10	sylvester	sylvest	ADJ
ap-4607	273	11	matrix	matrix	NOUN
ap-4607	273	12	by	by	ADP
ap-4607	273	13	the	the	DET
ap-4607	273	14	parameter	parameter	NOUN
ap-4607	273	15	t	t	PROPN
ap-4607	273	16	in	in	ADP
ap-4607	273	17	some	some	DET
ap-4607	273	18	degree	degree	NOUN
ap-4607	273	19	.	.	PUNCT
ap-4607	274	1	we	we	PRON
ap-4607	274	2	choose	choose	VERB
ap-4607	274	3	a	a	DET
ap-4607	274	4	degree	degree	NOUN
ap-4607	274	5	so	so	SCONJ
ap-4607	274	6	that	that	SCONJ
ap-4607	274	7	all	all	DET
ap-4607	274	8	elements	element	NOUN
ap-4607	274	9	of	of	ADP
ap-4607	274	10	the	the	DET
ap-4607	274	11	column	column	NOUN
ap-4607	274	12	are	be	AUX
ap-4607	274	13	of	of	ADP
ap-4607	274	14	the	the	DET
ap-4607	274	15	same	same	ADJ
ap-4607	274	16	degree	degree	NOUN
ap-4607	274	17	with	with	ADP
ap-4607	274	18	respect	respect	NOUN
ap-4607	274	19	to	to	ADP
ap-4607	274	20	t.	t.	NOUN
ap-4607	274	21	we	we	PRON
ap-4607	274	22	multiply	multiply	VERB
ap-4607	274	23	the	the	DET
ap-4607	274	24	i	i	PROPN
ap-4607	274	25	-	-	PUNCT
ap-4607	274	26	th	th	X
ap-4607	274	27	row	row	NOUN
ap-4607	274	28	of	of	ADP
ap-4607	274	29	the	the	DET
ap-4607	274	30	determinant	determinant	NOUN
ap-4607	274	31	of	of	ADP
ap-4607	274	32	the	the	DET
ap-4607	274	33	sylvester	sylvest	ADJ
ap-4607	274	34	matrix	matrix	NOUN
ap-4607	274	35	by	by	ADP
ap-4607	274	36	ti	ti	PROPN
ap-4607	274	37	where	where	SCONJ
ap-4607	274	38	i	i	PRON
ap-4607	274	39	=	=	NOUN
ap-4607	274	40	1	1	NUM
ap-4607	274	41	,	,	PUNCT
ap-4607	274	42	2	2	NUM
ap-4607	274	43	,	,	PUNCT
ap-4607	274	44	.	.	PUNCT
ap-4607	274	45	.	.	PUNCT
ap-4607	274	46	.	.	PUNCT
ap-4607	275	1	,	,	PUNCT
ap-4607	275	2	m	m	PROPN
ap-4607	275	3	,	,	PUNCT
ap-4607	275	4	and	and	CCONJ
ap-4607	275	5	j	j	X
ap-4607	275	6	-	-	PUNCT
ap-4607	275	7	th	th	VERB
ap-4607	275	8	row	row	NOUN
ap-4607	275	9	by	by	ADP
ap-4607	275	10	tj	tj	PROPN
ap-4607	275	11	where	where	SCONJ
ap-4607	275	12	j	j	PROPN
ap-4607	276	1	=	=	NOUN
ap-4607	276	2	m	m	PROPN
ap-4607	276	3	+	+	NOUN
ap-4607	276	4	1,m	1,m	X
ap-4607	276	5	+	+	NUM
ap-4607	276	6	2	2	NUM
ap-4607	276	7	,	,	PUNCT
ap-4607	276	8	.	.	PUNCT
ap-4607	276	9	.	.	PUNCT
ap-4607	276	10	.	.	PUNCT
ap-4607	277	1	,	,	PUNCT
ap-4607	277	2	m	m	VERB
ap-4607	277	3	+	+	X
ap-4607	277	4	n.	n.	NOUN
ap-4607	277	5	we	we	PRON
ap-4607	277	6	take	take	VERB
ap-4607	277	7	the	the	DET
ap-4607	277	8	factor	factor	NOUN
ap-4607	277	9	ts	ts	ADP
ap-4607	277	10	from	from	ADP
ap-4607	277	11	the	the	DET
ap-4607	277	12	s	s	PROPN
ap-4607	277	13	-	-	PUNCT
ap-4607	277	14	th	th	VERB
ap-4607	277	15	column	column	NOUN
ap-4607	277	16	where	where	SCONJ
ap-4607	277	17	s	s	VERB
ap-4607	277	18	=	=	SYM
ap-4607	277	19	1	1	NUM
ap-4607	277	20	,	,	PUNCT
ap-4607	277	21	2	2	NUM
ap-4607	277	22	,	,	PUNCT
ap-4607	277	23	.	.	PUNCT
ap-4607	277	24	.	.	PUNCT
ap-4607	278	1	.	.	PUNCT
ap-4607	279	1	,	,	PUNCT
ap-4607	279	2	m	m	VERB
ap-4607	279	3	+	+	CCONJ
ap-4607	279	4	n.	n.	NOUN
ap-4607	279	5	we	we	PRON
ap-4607	279	6	denote	denote	VERB
ap-4607	279	7	the	the	DET
ap-4607	279	8	total	total	ADJ
ap-4607	279	9	power	power	NOUN
ap-4607	279	10	of	of	ADP
ap-4607	279	11	t	t	PROPN
ap-4607	279	12	by	by	ADP
ap-4607	279	13	s.	s.	PROPN
ap-4607	279	14	we	we	PRON
ap-4607	279	15	have	have	VERB
ap-4607	279	16	s	s	NOUN
ap-4607	279	17	=	=	PUNCT
ap-4607	279	18	m+n∑	m+n∑	PROPN
ap-4607	279	19	i=1	i=1	PROPN
ap-4607	280	1	i−	i−	PROPN
ap-4607	280	2	n∑	n∑	PROPN
ap-4607	280	3	i=1	i=1	PROPN
ap-4607	280	4	i−	i−	PROPN
ap-4607	280	5	m∑	m∑	NOUN
ap-4607	280	6	i=1	i=1	PROPN
ap-4607	281	1	i	i	PRON
ap-4607	281	2	=	=	PUNCT
ap-4607	281	3	(	(	PUNCT
ap-4607	281	4	n+m)(n+m+	n+m)(n+m+	NOUN
ap-4607	281	5	1	1	NUM
ap-4607	281	6	)	)	PUNCT
ap-4607	281	7	2	2	NUM
ap-4607	281	8	−	−	NOUN
ap-4607	281	9	n(n+	n(n+	NUM
ap-4607	281	10	1	1	NUM
ap-4607	281	11	)	)	SYM
ap-4607	281	12	2	2	NUM
ap-4607	281	13	−	−	NOUN
ap-4607	281	14	m(m+	m(m+	VERB
ap-4607	281	15	1	1	NUM
ap-4607	281	16	)	)	PUNCT
ap-4607	281	17	2	2	NUM
ap-4607	281	18	=	=	SYM
ap-4607	281	19	nm	nm	NOUN
ap-4607	281	20	.	.	PUNCT
ap-4607	282	1	(	(	PUNCT
ap-4607	282	2	31	31	NUM
ap-4607	282	3	)	)	PUNCT
ap-4607	282	4	in	in	ADP
ap-4607	282	5	this	this	DET
ap-4607	282	6	way	way	NOUN
ap-4607	282	7	,	,	PUNCT
ap-4607	282	8	det	det	PROPN
ap-4607	282	9	sul	sul	PROPN
ap-4607	282	10	(	(	PUNCT
ap-4607	282	11	g1(tx0	g1(tx0	PROPN
ap-4607	282	12	:	:	PUNCT
ap-4607	282	13	x1	x1	NUM
ap-4607	282	14	:	:	PUNCT
ap-4607	282	15	tx2	tx2	NOUN
ap-4607	282	16	)	)	PUNCT
ap-4607	282	17	,	,	PUNCT
ap-4607	282	18	g2(tx0	g2(tx0	PROPN
ap-4607	282	19	:	:	PUNCT
ap-4607	282	20	x1	x1	NUM
ap-4607	282	21	:	:	PUNCT
ap-4607	282	22	tx2	tx2	NOUN
ap-4607	282	23	)	)	PUNCT
ap-4607	282	24	)	)	PUNCT
ap-4607	283	1	=	=	SYM
ap-4607	283	2	tnm	tnm	PROPN
ap-4607	283	3	det	det	PROPN
ap-4607	283	4	sul	sul	PROPN
ap-4607	283	5	(	(	PUNCT
ap-4607	283	6	g1(x0	g1(x0	NOUN
ap-4607	283	7	:	:	PUNCT
ap-4607	283	8	x1	x1	NUM
ap-4607	283	9	:	:	PUNCT
ap-4607	283	10	x2	x2	ADJ
ap-4607	283	11	)	)	PUNCT
ap-4607	283	12	,	,	PUNCT
ap-4607	283	13	g2(x0	g2(x0	NOUN
ap-4607	283	14	:	:	PUNCT
ap-4607	283	15	x1	x1	NUM
ap-4607	283	16	:	:	PUNCT
ap-4607	283	17	x2	x2	NUM
ap-4607	283	18	)	)	PUNCT
ap-4607	283	19	)	)	PUNCT
ap-4607	283	20	.	.	PUNCT
ap-4607	284	1	(	(	PUNCT
ap-4607	284	2	32	32	NUM
ap-4607	284	3	)	)	PUNCT
ap-4607	284	4	the	the	DET
ap-4607	284	5	determinant	determinant	NOUN
ap-4607	284	6	of	of	ADP
ap-4607	284	7	the	the	DET
ap-4607	284	8	sylvester	sylvest	ADJ
ap-4607	284	9	matrix	matrix	NOUN
ap-4607	284	10	does	do	AUX
ap-4607	284	11	not	not	PART
ap-4607	284	12	depend	depend	VERB
ap-4607	284	13	on	on	ADP
ap-4607	284	14	the	the	DET
ap-4607	284	15	parameter	parameter	NOUN
ap-4607	284	16	t.	t.	PROPN
ap-4607	284	17	substituting	substituting	NOUN
ap-4607	284	18	(	(	PUNCT
ap-4607	284	19	32	32	NUM
ap-4607	284	20	)	)	PUNCT
ap-4607	284	21	in	in	ADP
ap-4607	284	22	(	(	PUNCT
ap-4607	284	23	30	30	NUM
ap-4607	284	24	)	)	PUNCT
ap-4607	284	25	and	and	CCONJ
ap-4607	284	26	further	far	ADV
ap-4607	284	27	in	in	ADP
ap-4607	284	28	(	(	PUNCT
ap-4607	284	29	29	29	NUM
ap-4607	284	30	)	)	PUNCT
ap-4607	284	31	we	we	PRON
ap-4607	284	32	have	have	VERB
ap-4607	284	33	f	f	PROPN
ap-4607	284	34	(	(	PUNCT
ap-4607	284	35	tx0	tx0	ADJ
ap-4607	284	36	,	,	PUNCT
ap-4607	284	37	tx2	tx2	NOUN
ap-4607	284	38	)	)	PUNCT
ap-4607	284	39	=	=	SYM
ap-4607	284	40	tnmf	tnmf	NOUN
ap-4607	284	41	(	(	PUNCT
ap-4607	284	42	x0	x0	PROPN
ap-4607	284	43	,	,	PUNCT
ap-4607	284	44	x2	x2	PROPN
ap-4607	284	45	)	)	PUNCT
ap-4607	284	46	.	.	PUNCT
ap-4607	285	1	(	(	PUNCT
ap-4607	285	2	33	33	NUM
ap-4607	285	3	)	)	PUNCT
ap-4607	285	4	consequently	consequently	ADV
ap-4607	285	5	,	,	PUNCT
ap-4607	285	6	the	the	DET
ap-4607	285	7	resultant	resultant	NOUN
ap-4607	285	8	is	be	AUX
ap-4607	285	9	a	a	DET
ap-4607	285	10	homogeneous	homogeneous	ADJ
ap-4607	285	11	polynomial	polynomial	NOUN
ap-4607	285	12	of	of	ADP
ap-4607	285	13	degree	degree	NOUN
ap-4607	285	14	nm	nm	PROPN
ap-4607	285	15	.	.	PUNCT
ap-4607	286	1	theorem	theorem	ADJ
ap-4607	286	2	1	1	NUM
ap-4607	286	3	admits	admit	VERB
ap-4607	286	4	a	a	DET
ap-4607	286	5	generalization	generalization	NOUN
ap-4607	286	6	.	.	PUNCT
ap-4607	287	1	we	we	PRON
ap-4607	287	2	have	have	AUX
ap-4607	287	3	proved	prove	VERB
ap-4607	287	4	an	an	DET
ap-4607	287	5	analogous	analogous	ADJ
ap-4607	287	6	assertion	assertion	NOUN
ap-4607	287	7	for	for	ADP
ap-4607	287	8	systems	system	NOUN
ap-4607	287	9	of	of	ADP
ap-4607	287	10	equations	equation	NOUN
ap-4607	287	11	of	of	ADP
ap-4607	287	12	several	several	ADJ
ap-4607	287	13	variables	variable	NOUN
ap-4607	287	14	,	,	PUNCT
ap-4607	287	15	see	see	VERB
ap-4607	287	16	[	[	X
ap-4607	287	17	7	7	NUM
ap-4607	287	18	,	,	PUNCT
ap-4607	287	19	16	16	NUM
ap-4607	287	20	]	]	PUNCT
ap-4607	287	21	.	.	PUNCT
ap-4607	288	1	we	we	PRON
ap-4607	288	2	transform	transform	VERB
ap-4607	288	3	system	system	NOUN
ap-4607	288	4	(	(	PUNCT
ap-4607	288	5	3	3	NUM
ap-4607	288	6	)	)	PUNCT
ap-4607	288	7	and	and	CCONJ
ap-4607	288	8	apply	apply	VERB
ap-4607	288	9	bezout	bezout	PROPN
ap-4607	288	10	’s	’s	PART
ap-4607	288	11	theorem	theorem	NOUN
ap-4607	288	12	to	to	ADP
ap-4607	288	13	its	its	PRON
ap-4607	288	14	study	study	NOUN
ap-4607	288	15	.	.	PUNCT
ap-4607	289	1	in	in	ADP
ap-4607	289	2	the	the	DET
ap-4607	289	3	equations	equation	NOUN
ap-4607	289	4	of	of	ADP
ap-4607	289	5	system	system	NOUN
ap-4607	289	6	(	(	PUNCT
ap-4607	289	7	3	3	X
ap-4607	289	8	)	)	PUNCT
ap-4607	289	9	we	we	PRON
ap-4607	289	10	proceed	proceed	VERB
ap-4607	289	11	to	to	ADP
ap-4607	289	12	homogeneous	homogeneous	ADJ
ap-4607	289	13	coordinates	coordinate	NOUN
ap-4607	289	14	.	.	PUNCT
ap-4607	290	1	let	let	VERB
ap-4607	290	2	{	{	PUNCT
ap-4607	290	3	x1	x1	PROPN
ap-4607	290	4	=	=	SYM
ap-4607	290	5	x1	x1	PROPN
ap-4607	290	6	/	/	SYM
ap-4607	290	7	x0	x0	PROPN
ap-4607	290	8	,	,	PUNCT
ap-4607	291	1	x2	x2	NOUN
ap-4607	291	2	=	=	SYM
ap-4607	291	3	x2	x2	PROPN
ap-4607	291	4	/	/	SYM
ap-4607	291	5	x0	x0	PROPN
ap-4607	291	6	.	.	PUNCT
ap-4607	292	1	(	(	PUNCT
ap-4607	292	2	34	34	NUM
ap-4607	292	3	)	)	PUNCT
ap-4607	292	4	after	after	ADP
ap-4607	292	5	reducing	reduce	VERB
ap-4607	292	6	the	the	DET
ap-4607	292	7	equations	equation	NOUN
ap-4607	292	8	of	of	ADP
ap-4607	292	9	the	the	DET
ap-4607	292	10	system	system	NOUN
ap-4607	292	11	to	to	ADP
ap-4607	292	12	a	a	DET
ap-4607	292	13	polynomial	polynomial	ADJ
ap-4607	292	14	form	form	NOUN
ap-4607	292	15	,	,	PUNCT
ap-4607	292	16	we	we	PRON
ap-4607	292	17	have	have	NUM
ap-4607	292	18	x2n+1	x2n+1	PROPN
ap-4607	292	19	0	0	NUM
ap-4607	292	20	f1	f1	PROPN
ap-4607	292	21	(	(	PUNCT
ap-4607	292	22	x1	x1	NOUN
ap-4607	292	23	x0	x0	PROPN
ap-4607	292	24	,	,	PUNCT
ap-4607	292	25	x2	x2	PRON
ap-4607	292	26	x0	x0	PROPN
ap-4607	292	27	,	,	PUNCT
ap-4607	292	28	y1	y1	INTJ
ap-4607	292	29	)	)	PUNCT
ap-4607	292	30	=	=	PUNCT
ap-4607	293	1	φ1(x0	φ1(x0	NOUN
ap-4607	293	2	:	:	PUNCT
ap-4607	293	3	x1	x1	NUM
ap-4607	293	4	:	:	PUNCT
ap-4607	293	5	x2	x2	ADJ
ap-4607	293	6	)	)	PUNCT
ap-4607	293	7	=	=	SYM
ap-4607	293	8	0	0	NUM
ap-4607	293	9	,	,	PUNCT
ap-4607	293	10	x2n+1	x2n+1	PROPN
ap-4607	293	11	0	0	PUNCT
ap-4607	294	1	f2	f2	PRON
ap-4607	294	2	(	(	PUNCT
ap-4607	294	3	x1	x1	NOUN
ap-4607	294	4	x0	x0	PROPN
ap-4607	294	5	,	,	PUNCT
ap-4607	294	6	x2	x2	PRON
ap-4607	294	7	x0	x0	PROPN
ap-4607	294	8	,	,	PUNCT
ap-4607	294	9	y2	y2	INTJ
ap-4607	294	10	)	)	PUNCT
ap-4607	295	1	=	=	PUNCT
ap-4607	295	2	φ2(x0	φ2(x0	NOUN
ap-4607	295	3	:	:	PUNCT
ap-4607	295	4	x1	x1	NUM
ap-4607	295	5	:	:	PUNCT
ap-4607	295	6	x2	x2	ADJ
ap-4607	295	7	)	)	PUNCT
ap-4607	295	8	=	=	SYM
ap-4607	295	9	0	0	X
ap-4607	295	10	.	.	PUNCT
ap-4607	296	1	(	(	PUNCT
ap-4607	296	2	35	35	NUM
ap-4607	296	3	)	)	PUNCT
ap-4607	296	4	the	the	DET
ap-4607	296	5	coordinatesx0	coordinatesx0	NOUN
ap-4607	296	6	,	,	PUNCT
ap-4607	296	7	x1	x1	PROPN
ap-4607	296	8	,	,	PUNCT
ap-4607	296	9	x2	x2	PRON
ap-4607	296	10	are	be	AUX
ap-4607	296	11	obviously	obviously	ADV
ap-4607	296	12	projective	projective	ADJ
ap-4607	296	13	coordinates	coordinate	NOUN
ap-4607	296	14	.	.	PUNCT
ap-4607	297	1	the	the	DET
ap-4607	297	2	system	system	NOUN
ap-4607	297	3	(	(	PUNCT
ap-4607	297	4	34	34	NUM
ap-4607	297	5	)	)	PUNCT
ap-4607	297	6	defines	define	VERB
ap-4607	297	7	surjective	surjective	ADJ
ap-4607	297	8	mapping	mapping	NOUN
ap-4607	297	9	,	,	PUNCT
ap-4607	297	10	=	=	PUNCT
ap-4607	297	11	:	:	PUNCT
ap-4607	297	12	c2	c2	PROPN
ap-4607	297	13	→	→	SYM
ap-4607	297	14	cp	cp	PROPN
ap-4607	297	15	2	2	NUM
ap-4607	297	16	.	.	PUNCT
ap-4607	298	1	the	the	DET
ap-4607	298	2	triple	triple	NOUN
ap-4607	298	3	of	of	ADP
ap-4607	298	4	complex	complex	ADJ
ap-4607	298	5	numbers	number	NOUN
ap-4607	298	6	(	(	PUNCT
ap-4607	298	7	x0	x0	PROPN
ap-4607	298	8	:	:	PUNCT
ap-4607	298	9	x1	x1	NUM
ap-4607	298	10	:	:	PUNCT
ap-4607	298	11	x2	x2	X
ap-4607	298	12	)	)	PUNCT
ap-4607	298	13	are	be	AUX
ap-4607	298	14	the	the	DET
ap-4607	298	15	coordinates	coordinate	NOUN
ap-4607	298	16	of	of	ADP
ap-4607	298	17	the	the	DET
ap-4607	298	18	point	point	NOUN
ap-4607	298	19	and	and	CCONJ
ap-4607	298	20	defines	define	VERB
ap-4607	298	21	the	the	DET
ap-4607	298	22	point	point	NOUN
ap-4607	298	23	p	p	X
ap-4607	298	24	∈	∈	PROPN
ap-4607	299	1	cp	cp	INTJ
ap-4607	299	2	2	2	NUM
ap-4607	299	3	in	in	ADP
ap-4607	299	4	the	the	DET
ap-4607	299	5	projective	projective	ADJ
ap-4607	299	6	plane	plane	NOUN
ap-4607	299	7	cp	cp	INTJ
ap-4607	299	8	2	2	NUM
ap-4607	299	9	.	.	PUNCT
ap-4607	300	1	the	the	DET
ap-4607	300	2	triple	triple	ADJ
ap-4607	300	3	(	(	PUNCT
ap-4607	300	4	λx0	λx0	NOUN
ap-4607	300	5	:	:	PUNCT
ap-4607	300	6	λx1	λx1	X
ap-4607	300	7	:	:	PUNCT
ap-4607	300	8	λx2	λx2	NOUN
ap-4607	300	9	)	)	PUNCT
ap-4607	300	10	specifies	specify	VERB
ap-4607	300	11	the	the	DET
ap-4607	300	12	same	same	ADJ
ap-4607	300	13	point	point	NOUN
ap-4607	300	14	if	if	SCONJ
ap-4607	300	15	λ	λ	PROPN
ap-4607	300	16	6=	6=	PRON
ap-4607	300	17	0	0	NUM
ap-4607	300	18	.	.	PUNCT
ap-4607	301	1	therefore	therefore	ADV
ap-4607	301	2	we	we	PRON
ap-4607	301	3	have	have	VERB
ap-4607	301	4	the	the	DET
ap-4607	301	5	following	follow	VERB
ap-4607	301	6	result	result	NOUN
ap-4607	301	7	.	.	PUNCT
ap-4607	302	1	theorem	theorem	NOUN
ap-4607	302	2	2	2	NUM
ap-4607	302	3	.	.	PUNCT
ap-4607	303	1	the	the	DET
ap-4607	303	2	system	system	NOUN
ap-4607	303	3	of	of	ADP
ap-4607	303	4	polynomial	polynomial	ADJ
ap-4607	303	5	equations	equation	NOUN
ap-4607	303	6	{	{	PUNCT
ap-4607	303	7	φ1(x0	φ1(x0	NOUN
ap-4607	303	8	:	:	PUNCT
ap-4607	303	9	x1	x1	NUM
ap-4607	303	10	:	:	PUNCT
ap-4607	303	11	x2	x2	ADJ
ap-4607	303	12	)	)	PUNCT
ap-4607	303	13	=	=	SYM
ap-4607	303	14	0	0	NUM
ap-4607	303	15	,	,	PUNCT
ap-4607	303	16	φ2(x0	φ2(x0	NOUN
ap-4607	303	17	:	:	PUNCT
ap-4607	303	18	x1	x1	NUM
ap-4607	303	19	:	:	PUNCT
ap-4607	303	20	x2	x2	ADJ
ap-4607	303	21	)	)	PUNCT
ap-4607	303	22	=	=	SYM
ap-4607	303	23	0	0	NUM
ap-4607	303	24	(	(	PUNCT
ap-4607	303	25	36	36	NUM
ap-4607	303	26	)	)	PUNCT
ap-4607	303	27	has	have	AUX
ap-4607	303	28	in	in	ADP
ap-4607	303	29	the	the	DET
ap-4607	303	30	projective	projective	ADJ
ap-4607	303	31	plane	plane	NOUN
ap-4607	303	32	cp	cp	PROPN
ap-4607	303	33	2	2	NUM
ap-4607	303	34	,	,	PUNCT
ap-4607	303	35	counting	count	VERB
ap-4607	303	36	multiplicity	multiplicity	NOUN
ap-4607	303	37	,	,	PUNCT
ap-4607	303	38	exactly	exactly	ADV
ap-4607	303	39	m	m	PROPN
ap-4607	303	40	·	·	PUNCT
ap-4607	303	41	n	n	PRON
ap-4607	303	42	solutions	solution	NOUN
ap-4607	303	43	,	,	PUNCT
ap-4607	303	44	where	where	SCONJ
ap-4607	303	45	,	,	PUNCT
ap-4607	303	46	m	m	VERB
ap-4607	303	47	=	=	ADJ
ap-4607	303	48	deg	deg	PROPN
ap-4607	303	49	φ1	φ1	PROPN
ap-4607	303	50	,	,	PUNCT
ap-4607	303	51	and	and	CCONJ
ap-4607	303	52	n	n	CCONJ
ap-4607	303	53	=	=	NUM
ap-4607	303	54	deg	deg	PROPN
ap-4607	303	55	φ2	φ2	PROPN
ap-4607	303	56	,	,	PUNCT
ap-4607	303	57	if	if	SCONJ
ap-4607	303	58	gcd(φ1,φ2	gcd(φ1,φ2	PROPN
ap-4607	303	59	)	)	PUNCT
ap-4607	303	60	belongs	belong	VERB
ap-4607	303	61	to	to	ADP
ap-4607	303	62	the	the	DET
ap-4607	303	63	coefficient	coefficient	NOUN
ap-4607	303	64	field	field	NOUN
ap-4607	303	65	c.	c.	PROPN
ap-4607	303	66	functions	function	NOUN
ap-4607	303	67	φ1	φ1	NOUN
ap-4607	303	68	=	=	PUNCT
ap-4607	304	1	φ1(x0	φ1(x0	NUM
ap-4607	304	2	:	:	PUNCT
ap-4607	304	3	x1	x1	NUM
ap-4607	304	4	:	:	PUNCT
ap-4607	304	5	x2	x2	ADJ
ap-4607	304	6	)	)	PUNCT
ap-4607	304	7	and	and	CCONJ
ap-4607	304	8	φ2	φ2	PROPN
ap-4607	304	9	=	=	PUNCT
ap-4607	305	1	φ2(x0	φ2(x0	NOUN
ap-4607	305	2	:	:	PUNCT
ap-4607	305	3	x1	x1	NUM
ap-4607	305	4	:	:	PUNCT
ap-4607	305	5	x2	x2	ADJ
ap-4607	305	6	)	)	PUNCT
ap-4607	305	7	are	be	AUX
ap-4607	305	8	homogeneous	homogeneous	ADJ
ap-4607	305	9	functions	function	NOUN
ap-4607	305	10	of	of	ADP
ap-4607	305	11	degree	degree	NOUN
ap-4607	305	12	2n	2n	NUM
ap-4607	306	1	+	+	CCONJ
ap-4607	306	2	1	1	X
ap-4607	306	3	.	.	X
ap-4607	307	1	if	if	SCONJ
ap-4607	307	2	,	,	PUNCT
ap-4607	307	3	at	at	ADV
ap-4607	307	4	least	least	ADJ
ap-4607	307	5	one	one	NUM
ap-4607	307	6	of	of	ADP
ap-4607	307	7	the	the	DET
ap-4607	307	8	coordinates	coordinate	NOUN
ap-4607	307	9	of	of	ADP
ap-4607	307	10	the	the	DET
ap-4607	307	11	point	point	NOUN
ap-4607	307	12	p	p	NOUN
ap-4607	307	13	is	be	AUX
ap-4607	307	14	equal	equal	ADJ
ap-4607	307	15	to	to	ADP
ap-4607	307	16	zero	zero	NUM
ap-4607	307	17	,	,	PUNCT
ap-4607	307	18	say	say	VERB
ap-4607	307	19	that	that	SCONJ
ap-4607	307	20	this	this	DET
ap-4607	307	21	point	point	NOUN
ap-4607	307	22	is	be	AUX
ap-4607	307	23	irregular	irregular	ADJ
ap-4607	307	24	.	.	PUNCT
ap-4607	308	1	otherwise	otherwise	ADV
ap-4607	308	2	,	,	PUNCT
ap-4607	308	3	the	the	DET
ap-4607	308	4	point	point	NOUN
ap-4607	308	5	is	be	AUX
ap-4607	308	6	called	call	VERB
ap-4607	308	7	regular	regular	ADJ
ap-4607	308	8	.	.	PUNCT
ap-4607	309	1	a	a	DET
ap-4607	309	2	straight	straight	ADJ
ap-4607	309	3	line	line	NOUN
ap-4607	309	4	that	that	PRON
ap-4607	309	5	consists	consist	VERB
ap-4607	309	6	of	of	ADP
ap-4607	309	7	irregular	irregular	ADJ
ap-4607	309	8	points	point	NOUN
ap-4607	309	9	is	be	AUX
ap-4607	309	10	called	call	VERB
ap-4607	309	11	an	an	DET
ap-4607	309	12	irregular	irregular	ADJ
ap-4607	309	13	line	line	NOUN
ap-4607	309	14	.	.	PUNCT
ap-4607	310	1	the	the	DET
ap-4607	310	2	projective	projective	ADJ
ap-4607	310	3	plane	plane	NOUN
ap-4607	310	4	cp	cp	PROPN
ap-4607	310	5	2	2	NUM
ap-4607	310	6	has	have	VERB
ap-4607	310	7	three	three	NUM
ap-4607	310	8	irregular	irregular	ADJ
ap-4607	310	9	straight	straight	ADJ
ap-4607	310	10	lines	line	NOUN
ap-4607	310	11	,	,	PUNCT
ap-4607	310	12	which	which	PRON
ap-4607	310	13	are	be	AUX
ap-4607	310	14	given	give	VERB
ap-4607	310	15	by	by	ADP
ap-4607	310	16	the	the	DET
ap-4607	310	17	equations	equation	NOUN
ap-4607	310	18	x0	x0	PROPN
ap-4607	311	1	=	=	PUNCT
ap-4607	311	2	0	0	PROPN
ap-4607	311	3	,	,	PUNCT
ap-4607	311	4	x1	x1	PROPN
ap-4607	311	5	=	=	SYM
ap-4607	311	6	0	0	PROPN
ap-4607	311	7	,	,	PUNCT
ap-4607	311	8	x2	x2	NOUN
ap-4607	311	9	=	=	NOUN
ap-4607	311	10	0	0	PROPN
ap-4607	311	11	.	.	PUNCT
ap-4607	312	1	(	(	PUNCT
ap-4607	312	2	37	37	NUM
ap-4607	312	3	)	)	PUNCT
ap-4607	312	4	the	the	DET
ap-4607	312	5	set	set	NOUN
ap-4607	312	6	of	of	ADP
ap-4607	312	7	points	point	NOUN
ap-4607	312	8	cp	cp	INTJ
ap-4607	312	9	2	2	NUM
ap-4607	312	10	one	one	NUM
ap-4607	312	11	of	of	ADP
ap-4607	312	12	the	the	DET
ap-4607	312	13	coordinates	coordinate	NOUN
ap-4607	312	14	,	,	PUNCT
ap-4607	312	15	which	which	PRON
ap-4607	312	16	is	be	AUX
ap-4607	312	17	equal	equal	ADJ
ap-4607	312	18	to	to	ADP
ap-4607	312	19	the	the	DET
ap-4607	312	20	number	number	NOUN
ap-4607	312	21	h	h	NOUN
ap-4607	312	22	6=	6=	PROPN
ap-4607	312	23	0	0	NUM
ap-4607	312	24	,	,	PUNCT
ap-4607	312	25	is	be	AUX
ap-4607	312	26	called	call	VERB
ap-4607	312	27	affine	affine	NOUN
ap-4607	312	28	map	map	NOUN
ap-4607	312	29	on	on	ADP
ap-4607	312	30	cp	cp	PROPN
ap-4607	312	31	2	2	NUM
ap-4607	312	32	and	and	CCONJ
ap-4607	312	33	denoted	denote	VERB
ap-4607	312	34	by	by	ADP
ap-4607	312	35	a2(h	a2(h	PROPN
ap-4607	312	36	)	)	PUNCT
ap-4607	312	37	.	.	PUNCT
ap-4607	313	1	the	the	DET
ap-4607	313	2	complement	complement	NOUN
ap-4607	313	3	of	of	ADP
ap-4607	313	4	a	a	DET
ap-4607	313	5	cp	cp	PROPN
ap-4607	313	6	2\a2(h	2\a2(h	NUM
ap-4607	313	7	)	)	PUNCT
ap-4607	313	8	consists	consist	VERB
ap-4607	313	9	of	of	ADP
ap-4607	313	10	a	a	DET
ap-4607	313	11	one	one	NUM
ap-4607	313	12	-	-	PUNCT
ap-4607	313	13	dimensional	dimensional	ADJ
ap-4607	313	14	complex	complex	ADJ
ap-4607	313	15	projective	projective	ADJ
ap-4607	313	16	subspace	subspace	NOUN
ap-4607	313	17	,	,	PUNCT
ap-4607	313	18	which	which	PRON
ap-4607	313	19	is	be	AUX
ap-4607	313	20	called	call	VERB
ap-4607	313	21	an	an	DET
ap-4607	313	22	infinitely	infinitely	ADV
ap-4607	313	23	distant	distant	ADJ
ap-4607	313	24	line	line	NOUN
ap-4607	313	25	of	of	ADP
ap-4607	313	26	the	the	DET
ap-4607	313	27	affine	affine	NOUN
ap-4607	313	28	map	map	NOUN
ap-4607	313	29	,	,	PUNCT
ap-4607	313	30	see	see	VERB
ap-4607	313	31	for	for	ADP
ap-4607	313	32	example	example	NOUN
ap-4607	313	33	[	[	X
ap-4607	313	34	11	11	NUM
ap-4607	313	35	,	,	PUNCT
ap-4607	313	36	15	15	NUM
ap-4607	313	37	]	]	PUNCT
ap-4607	313	38	.	.	PUNCT
ap-4607	314	1	the	the	DET
ap-4607	314	2	infinitely	infinitely	ADV
ap-4607	314	3	distant	distant	ADJ
ap-4607	314	4	line	line	NOUN
ap-4607	314	5	of	of	ADP
ap-4607	314	6	any	any	DET
ap-4607	314	7	affine	affine	NOUN
ap-4607	314	8	map	map	NOUN
ap-4607	314	9	a2(h	a2(h	NOUN
ap-4607	314	10	)	)	PUNCT
ap-4607	314	11	is	be	AUX
ap-4607	314	12	evidently	evidently	ADV
ap-4607	314	13	irregular	irregular	ADJ
ap-4607	314	14	.	.	PUNCT
ap-4607	315	1	in	in	ADP
ap-4607	315	2	particular	particular	ADJ
ap-4607	315	3	,	,	PUNCT
ap-4607	315	4	if	if	SCONJ
ap-4607	315	5	we	we	PRON
ap-4607	315	6	put	put	VERB
ap-4607	315	7	x0	x0	PROPN
ap-4607	315	8	=	=	SYM
ap-4607	315	9	1	1	NUM
ap-4607	315	10	,	,	PUNCT
ap-4607	315	11	then	then	ADV
ap-4607	315	12	the	the	DET
ap-4607	315	13	set	set	NOUN
ap-4607	315	14	of	of	ADP
ap-4607	315	15	points	point	NOUN
ap-4607	315	16	cp	cp	INTJ
ap-4607	315	17	2	2	NUM
ap-4607	315	18	with	with	ADP
ap-4607	315	19	coordinates	coordinate	NOUN
ap-4607	315	20	(	(	PUNCT
ap-4607	315	21	1	1	NUM
ap-4607	315	22	:	:	PUNCT
ap-4607	316	1	x1	x1	NUM
ap-4607	316	2	:	:	PUNCT
ap-4607	316	3	x2	x2	X
ap-4607	316	4	)	)	PUNCT
ap-4607	316	5	will	will	AUX
ap-4607	316	6	be	be	AUX
ap-4607	316	7	408	408	NUM
ap-4607	316	8	vol	vol	NOUN
ap-4607	316	9	.	.	PUNCT
ap-4607	317	1	57	57	NUM
ap-4607	318	1	no	no	NOUN
ap-4607	318	2	.	.	PUNCT
ap-4607	319	1	6/2017	6/2017	X
ap-4607	319	2	the	the	DET
ap-4607	319	3	analysis	analysis	NOUN
ap-4607	319	4	of	of	ADP
ap-4607	319	5	images	image	NOUN
ap-4607	319	6	in	in	ADP
ap-4607	319	7	n	n	PRON
ap-4607	319	8	-point	-point	NOUN
ap-4607	319	9	gravitational	gravitational	ADJ
ap-4607	319	10	lens	lens	NOUN
ap-4607	319	11	affine	affine	NOUN
ap-4607	319	12	map	map	NOUN
ap-4607	319	13	of	of	ADP
ap-4607	319	14	a2(1	a2(1	NOUN
ap-4607	319	15	)	)	PUNCT
ap-4607	319	16	,	,	PUNCT
ap-4607	319	17	and	and	CCONJ
ap-4607	319	18	the	the	DET
ap-4607	319	19	infinity	infinity	NOUN
ap-4607	319	20	of	of	ADP
ap-4607	319	21	the	the	DET
ap-4607	319	22	straight	straight	ADJ
ap-4607	319	23	line	line	NOUN
ap-4607	319	24	of	of	ADP
ap-4607	319	25	this	this	DET
ap-4607	319	26	map	map	NOUN
ap-4607	319	27	will	will	AUX
ap-4607	319	28	be	be	AUX
ap-4607	319	29	given	give	VERB
ap-4607	319	30	by	by	ADP
ap-4607	319	31	equation	equation	NOUN
ap-4607	319	32	x0	x0	PROPN
ap-4607	320	1	=	=	PUNCT
ap-4607	320	2	0	0	X
ap-4607	320	3	.	.	PUNCT
ap-4607	320	4	consider	consider	VERB
ap-4607	320	5	the	the	DET
ap-4607	320	6	situation	situation	NOUN
ap-4607	320	7	of	of	ADP
ap-4607	320	8	general	general	ADJ
ap-4607	320	9	position	position	NOUN
ap-4607	320	10	,	,	PUNCT
ap-4607	320	11	i.e.	i.e.	X
ap-4607	320	12	the	the	DET
ap-4607	320	13	source	source	NOUN
ap-4607	320	14	is	be	AUX
ap-4607	320	15	not	not	PART
ap-4607	320	16	on	on	ADP
ap-4607	320	17	the	the	DET
ap-4607	320	18	caustic	caustic	NOUN
ap-4607	320	19	.	.	PUNCT
ap-4607	321	1	in	in	ADP
ap-4607	321	2	this	this	DET
ap-4607	321	3	case	case	NOUN
ap-4607	321	4	,	,	PUNCT
ap-4607	321	5	the	the	DET
ap-4607	321	6	jacobian	jacobian	NOUN
ap-4607	321	7	of	of	ADP
ap-4607	321	8	the	the	DET
ap-4607	321	9	system	system	NOUN
ap-4607	321	10	of	of	ADP
ap-4607	321	11	lens	lens	NOUN
ap-4607	321	12	equations	equation	NOUN
ap-4607	321	13	is	be	AUX
ap-4607	321	14	not	not	PART
ap-4607	321	15	equal	equal	ADJ
ap-4607	321	16	to	to	ADP
ap-4607	321	17	zero	zero	NUM
ap-4607	321	18	.	.	PUNCT
ap-4607	322	1	theorem	theorem	NOUN
ap-4607	322	2	3	3	NUM
ap-4607	322	3	.	.	PUNCT
ap-4607	323	1	in	in	ADP
ap-4607	323	2	a	a	DET
ap-4607	323	3	situation	situation	NOUN
ap-4607	323	4	of	of	ADP
ap-4607	323	5	general	general	ADJ
ap-4607	323	6	position	position	NOUN
ap-4607	323	7	(	(	PUNCT
ap-4607	323	8	the	the	DET
ap-4607	323	9	jacobian	jacobian	NOUN
ap-4607	323	10	of	of	ADP
ap-4607	323	11	the	the	DET
ap-4607	323	12	system	system	NOUN
ap-4607	323	13	of	of	ADP
ap-4607	323	14	lens	lens	NOUN
ap-4607	323	15	equations	equation	NOUN
ap-4607	323	16	is	be	AUX
ap-4607	323	17	not	not	PART
ap-4607	323	18	equal	equal	ADJ
ap-4607	323	19	to	to	ADP
ap-4607	323	20	zero	zero	NUM
ap-4607	323	21	)	)	PUNCT
ap-4607	323	22	,	,	PUNCT
ap-4607	323	23	the	the	DET
ap-4607	323	24	number	number	NOUN
ap-4607	323	25	of	of	ADP
ap-4607	323	26	point	point	NOUN
ap-4607	323	27	images	image	NOUN
ap-4607	323	28	in	in	ADP
ap-4607	323	29	an	an	DET
ap-4607	323	30	n	n	CCONJ
ap-4607	323	31	-	-	PUNCT
ap-4607	323	32	point	point	NOUN
ap-4607	323	33	gravitational	gravitational	ADJ
ap-4607	323	34	lens	lens	NOUN
ap-4607	323	35	has	have	AUX
ap-4607	323	36	parity	parity	NOUN
ap-4607	323	37	opposite	opposite	ADJ
ap-4607	323	38	to	to	ADP
ap-4607	323	39	the	the	DET
ap-4607	323	40	parity	parity	NOUN
ap-4607	323	41	of	of	ADP
ap-4607	323	42	the	the	DET
ap-4607	323	43	number	number	NOUN
ap-4607	323	44	n	n	NOUN
ap-4607	323	45	.	.	PUNCT
ap-4607	324	1	in	in	ADP
ap-4607	324	2	the	the	DET
ap-4607	324	3	proof	proof	NOUN
ap-4607	324	4	of	of	ADP
ap-4607	324	5	theorem	theorem	NOUN
ap-4607	324	6	3	3	NUM
ap-4607	324	7	we	we	PRON
ap-4607	324	8	use	use	VERB
ap-4607	324	9	the	the	DET
ap-4607	324	10	following	follow	VERB
ap-4607	324	11	lemma	lemma	PROPN
ap-4607	324	12	.	.	PUNCT
ap-4607	325	1	lemma	lemma	PROPN
ap-4607	325	2	1	1	NUM
ap-4607	325	3	.	.	PUNCT
ap-4607	326	1	the	the	DET
ap-4607	326	2	number	number	NOUN
ap-4607	326	3	of	of	ADP
ap-4607	326	4	irregular	irregular	ADJ
ap-4607	326	5	solutions	solution	NOUN
ap-4607	326	6	of	of	ADP
ap-4607	326	7	system	system	NOUN
ap-4607	326	8	(	(	PUNCT
ap-4607	326	9	37	37	NUM
ap-4607	326	10	)	)	PUNCT
ap-4607	326	11	,	,	PUNCT
ap-4607	326	12	on	on	ADP
ap-4607	326	13	line	line	NOUN
ap-4607	326	14	x0	x0	PROPN
ap-4607	326	15	=	=	SYM
ap-4607	326	16	0	0	NUM
ap-4607	326	17	,	,	PUNCT
ap-4607	326	18	is	be	AUX
ap-4607	326	19	2n	2n	NUM
ap-4607	326	20	.	.	PUNCT
ap-4607	327	1	proof	proof	NOUN
ap-4607	327	2	.	.	PUNCT
ap-4607	328	1	using	use	VERB
ap-4607	328	2	(	(	PUNCT
ap-4607	328	3	37	37	NUM
ap-4607	328	4	)	)	PUNCT
ap-4607	328	5	,	,	PUNCT
ap-4607	328	6	we	we	PRON
ap-4607	328	7	reduce	reduce	VERB
ap-4607	328	8	the	the	DET
ap-4607	328	9	system	system	NOUN
ap-4607	328	10	to	to	ADP
ap-4607	328	11	the	the	DET
ap-4607	328	12	form	form	PROPN
ap-4607	328	13	(	(	PUNCT
ap-4607	328	14	x1	x1	PROPN
ap-4607	328	15	−x0y1	−x0y1	PROPN
ap-4607	328	16	)	)	PUNCT
ap-4607	328	17	n∏	n∏	PROPN
ap-4607	328	18	i=1	i=1	PROPN
ap-4607	329	1	hi	hi	INTJ
ap-4607	329	2	−x2	−x2	PROPN
ap-4607	329	3	0	0	PUNCT
ap-4607	330	1	n∑	n∑	NOUN
ap-4607	330	2	j=1	j=1	NOUN
ap-4607	330	3	mj(x1	mj(x1	PROPN
ap-4607	330	4	−x0aj	−x0aj	NUM
ap-4607	330	5	)	)	PUNCT
ap-4607	330	6	n∏	n∏	PROPN
ap-4607	330	7	i=1,i6	i=1,i6	NOUN
ap-4607	330	8	=	=	SYM
ap-4607	330	9	j	j	NOUN
ap-4607	330	10	hi	hi	INTJ
ap-4607	330	11	=	=	NOUN
ap-4607	330	12	0	0	PROPN
ap-4607	330	13	,	,	PUNCT
ap-4607	330	14	(	(	PUNCT
ap-4607	330	15	x2	x2	PROPN
ap-4607	330	16	−x0y2	−x0y2	PROPN
ap-4607	330	17	)	)	PUNCT
ap-4607	330	18	n∏	n∏	PROPN
ap-4607	330	19	i=1	i=1	PROPN
ap-4607	331	1	hi	hi	INTJ
ap-4607	331	2	−x2	−x2	PROPN
ap-4607	331	3	0	0	PUNCT
ap-4607	332	1	n∑	n∑	DET
ap-4607	332	2	j=1	j=1	PROPN
ap-4607	332	3	mj(x2	mj(x2	PROPN
ap-4607	332	4	−x0bj	−x0bj	NUM
ap-4607	332	5	)	)	PUNCT
ap-4607	332	6	n∏	n∏	PROPN
ap-4607	332	7	i=1,i6	i=1,i6	NOUN
ap-4607	332	8	=	=	SYM
ap-4607	332	9	j	j	NOUN
ap-4607	332	10	hi	hi	INTJ
ap-4607	332	11	=	=	NOUN
ap-4607	332	12	0	0	PROPN
ap-4607	332	13	,	,	PUNCT
ap-4607	332	14	(	(	PUNCT
ap-4607	332	15	38	38	NUM
ap-4607	332	16	)	)	PUNCT
ap-4607	332	17	where	where	SCONJ
ap-4607	332	18	hi	hi	ADV
ap-4607	332	19	=	=	SYM
ap-4607	332	20	(	(	PUNCT
ap-4607	332	21	x1	x1	PROPN
ap-4607	332	22	−x0ai)2	−x0ai)2	PROPN
ap-4607	333	1	+	+	CCONJ
ap-4607	333	2	(	(	PUNCT
ap-4607	333	3	x2	x2	NOUN
ap-4607	333	4	−x0bi)2	−x0bi)2	PROPN
ap-4607	333	5	.	.	PUNCT
ap-4607	334	1	let	let	VERB
ap-4607	334	2	x0	x0	PROPN
ap-4607	334	3	=	=	PUNCT
ap-4607	335	1	0	0	X
ap-4607	335	2	.	.	PUNCT
ap-4607	336	1	we	we	PRON
ap-4607	336	2	have	have	PROPN
ap-4607	336	3	x1	x1	PROPN
ap-4607	336	4	n∏	n∏	PROPN
ap-4607	337	1	i=1	i=1	PROPN
ap-4607	338	1	(	(	PUNCT
ap-4607	338	2	x2	x2	NOUN
ap-4607	338	3	1	1	NUM
ap-4607	338	4	+	+	NOUN
ap-4607	338	5	x2	x2	NOUN
ap-4607	338	6	2	2	X
ap-4607	338	7	)	)	PUNCT
ap-4607	338	8	=	=	SYM
ap-4607	338	9	0	0	NUM
ap-4607	338	10	,	,	PUNCT
ap-4607	338	11	x2	x2	PROPN
ap-4607	338	12	n∏	n∏	PROPN
ap-4607	338	13	i=1	i=1	PROPN
ap-4607	339	1	(	(	PUNCT
ap-4607	339	2	x2	x2	NOUN
ap-4607	339	3	1	1	NUM
ap-4607	339	4	+	+	NOUN
ap-4607	339	5	x2	x2	NOUN
ap-4607	339	6	2	2	X
ap-4607	339	7	)	)	PUNCT
ap-4607	339	8	=	=	SYM
ap-4607	339	9	0	0	NUM
ap-4607	339	10	⇒	⇒	NOUN
ap-4607	339	11	{	{	PUNCT
ap-4607	339	12	x1(x2	x1(x2	NOUN
ap-4607	339	13	1	1	NUM
ap-4607	339	14	+	+	NOUN
ap-4607	339	15	x2	x2	NOUN
ap-4607	339	16	2	2	NUM
ap-4607	339	17	)	)	PUNCT
ap-4607	339	18	n	n	NOUN
ap-4607	339	19	=	=	SYM
ap-4607	339	20	0	0	NUM
ap-4607	339	21	,	,	PUNCT
ap-4607	339	22	x2(x2	x2(x2	NOUN
ap-4607	339	23	1	1	NUM
ap-4607	339	24	+	+	NOUN
ap-4607	339	25	x2	x2	NOUN
ap-4607	339	26	2	2	NUM
ap-4607	339	27	)	)	PUNCT
ap-4607	339	28	n	n	NOUN
ap-4607	339	29	=	=	SYM
ap-4607	339	30	0	0	NUM
ap-4607	339	31	⇒	⇒	NOUN
ap-4607	339	32	(	(	PUNCT
ap-4607	339	33	x2	x2	NOUN
ap-4607	339	34	1	1	NUM
ap-4607	339	35	+	+	NOUN
ap-4607	339	36	x2	x2	NOUN
ap-4607	339	37	2	2	NUM
ap-4607	339	38	)	)	PUNCT
ap-4607	339	39	n	n	NOUN
ap-4607	339	40	=	=	SYM
ap-4607	339	41	0	0	NUM
ap-4607	339	42	⇒	⇒	NOUN
ap-4607	339	43	x1	x1	X
ap-4607	340	1	=	=	SYM
ap-4607	341	1	±ix2	±ix2	ADJ
ap-4607	341	2	⇒	⇒	NOUN
ap-4607	341	3	{	{	PUNCT
ap-4607	341	4	x1	x1	PROPN
ap-4607	341	5	=	=	SYM
ap-4607	341	6	c	c	X
ap-4607	341	7	,	,	PUNCT
ap-4607	341	8	x2	x2	NOUN
ap-4607	341	9	=	=	PUNCT
ap-4607	341	10	±ic	±ic	PROPN
ap-4607	341	11	.	.	PUNCT
ap-4607	342	1	(	(	PUNCT
ap-4607	342	2	39	39	NUM
ap-4607	342	3	)	)	PUNCT
ap-4607	342	4	finally	finally	ADV
ap-4607	342	5	we	we	PRON
ap-4607	342	6	have	have	VERB
ap-4607	342	7	two	two	NUM
ap-4607	342	8	n	n	CCONJ
ap-4607	342	9	-fold	-fold	ADJ
ap-4607	342	10	solutions	solution	NOUN
ap-4607	342	11	:	:	PUNCT
ap-4607	342	12	p1	p1	NOUN
ap-4607	342	13	=	=	SYM
ap-4607	342	14	(	(	PUNCT
ap-4607	342	15	0	0	NUM
ap-4607	342	16	:	:	PUNCT
ap-4607	342	17	a	a	DET
ap-4607	342	18	:	:	PUNCT
ap-4607	342	19	ic	ic	NUM
ap-4607	342	20	)	)	PUNCT
ap-4607	342	21	and	and	CCONJ
ap-4607	342	22	p2	p2	PROPN
ap-4607	342	23	=	=	SYM
ap-4607	343	1	(	(	PUNCT
ap-4607	343	2	0	0	NUM
ap-4607	343	3	:	:	PUNCT
ap-4607	343	4	a	a	DET
ap-4607	343	5	:	:	PUNCT
ap-4607	343	6	−ic	−ic	PROPN
ap-4607	343	7	)	)	PUNCT
ap-4607	343	8	.	.	PUNCT
ap-4607	344	1	proof	proof	NOUN
ap-4607	344	2	of	of	ADP
ap-4607	344	3	theorem	theorem	NOUN
ap-4607	344	4	2	2	NUM
ap-4607	344	5	.	.	X
ap-4607	344	6	for	for	ADP
ap-4607	344	7	the	the	DET
ap-4607	344	8	degrees	degree	NOUN
ap-4607	344	9	of	of	ADP
ap-4607	344	10	the	the	DET
ap-4607	344	11	polynomials	polynomial	NOUN
ap-4607	344	12	of	of	ADP
ap-4607	344	13	systems	system	NOUN
ap-4607	344	14	(	(	PUNCT
ap-4607	344	15	3	3	NUM
ap-4607	344	16	)	)	PUNCT
ap-4607	344	17	and	and	CCONJ
ap-4607	344	18	(	(	PUNCT
ap-4607	344	19	5	5	X
ap-4607	344	20	)	)	PUNCT
ap-4607	344	21	we	we	PRON
ap-4607	344	22	have	have	VERB
ap-4607	344	23	degf1	degf1	NOUN
ap-4607	345	1	=	=	X
ap-4607	345	2	degf2	degf2	NOUN
ap-4607	345	3	=	=	PUNCT
ap-4607	345	4	deg	deg	PROPN
ap-4607	345	5	φ1	φ1	PROPN
ap-4607	345	6	=	=	SYM
ap-4607	345	7	deg	deg	NOUN
ap-4607	345	8	φ2	φ2	PROPN
ap-4607	345	9	=	=	PUNCT
ap-4607	345	10	2n	2n	NUM
ap-4607	346	1	+	+	CCONJ
ap-4607	346	2	1	1	X
ap-4607	346	3	.	.	PUNCT
ap-4607	346	4	by	by	ADP
ap-4607	346	5	bezout	bezout	PROPN
ap-4607	346	6	’s	’s	PART
ap-4607	346	7	theorem	theorem	PROPN
ap-4607	346	8	,	,	PUNCT
ap-4607	346	9	the	the	DET
ap-4607	346	10	system	system	NOUN
ap-4607	346	11	of	of	ADP
ap-4607	346	12	equations	equation	NOUN
ap-4607	346	13	(	(	PUNCT
ap-4607	346	14	36	36	NUM
ap-4607	346	15	)	)	PUNCT
ap-4607	346	16	has	have	VERB
ap-4607	346	17	(	(	PUNCT
ap-4607	346	18	2n	2n	NUM
ap-4607	346	19	+	+	CCONJ
ap-4607	346	20	1)2	1)2	NUM
ap-4607	346	21	solutions	solution	NOUN
ap-4607	346	22	,	,	PUNCT
ap-4607	346	23	which	which	PRON
ap-4607	346	24	include	include	VERB
ap-4607	346	25	an	an	DET
ap-4607	346	26	even	even	ADJ
ap-4607	346	27	number	number	NOUN
ap-4607	346	28	of	of	ADP
ap-4607	346	29	2q	2q	NUM
ap-4607	346	30	complex	complex	ADJ
ap-4607	346	31	conjugate	conjugate	ADJ
ap-4607	346	32	solutions	solution	NOUN
ap-4607	346	33	and	and	CCONJ
ap-4607	346	34	p	p	NOUN
ap-4607	346	35	=	=	X
ap-4607	346	36	2n	2n	NUM
ap-4607	346	37	irregular	irregular	ADJ
ap-4607	346	38	solutions	solution	NOUN
ap-4607	346	39	.	.	PUNCT
ap-4607	347	1	therefore	therefore	ADV
ap-4607	347	2	,	,	PUNCT
ap-4607	347	3	the	the	DET
ap-4607	347	4	number	number	NOUN
ap-4607	347	5	of	of	ADP
ap-4607	347	6	real	real	ADJ
ap-4607	347	7	solutions	solution	NOUN
ap-4607	347	8	of	of	ADP
ap-4607	347	9	system	system	NOUN
ap-4607	347	10	(	(	PUNCT
ap-4607	347	11	36	36	NUM
ap-4607	347	12	)	)	PUNCT
ap-4607	347	13	,	,	PUNCT
ap-4607	347	14	card	card	NOUN
ap-4607	347	15	(	(	PUNCT
ap-4607	347	16	realv	realv	PROPN
ap-4607	347	17	0(f1	0(f1	PROPN
ap-4607	347	18	,	,	PUNCT
ap-4607	347	19	f2	f2	PROPN
ap-4607	347	20	)	)	PUNCT
ap-4607	347	21	)	)	PUNCT
ap-4607	348	1	=	=	PUNCT
ap-4607	348	2	(	(	PUNCT
ap-4607	348	3	2n	2n	NUM
ap-4607	348	4	+	+	CCONJ
ap-4607	348	5	1)2	1)2	NUM
ap-4607	348	6	−	−	NOUN
ap-4607	348	7	2q	2q	NOUN
ap-4607	349	1	−	−	PROPN
ap-4607	349	2	p	p	X
ap-4607	349	3	=	=	X
ap-4607	349	4	(	(	PUNCT
ap-4607	349	5	2n	2n	NUM
ap-4607	349	6	+	+	CCONJ
ap-4607	349	7	1)2	1)2	NUM
ap-4607	349	8	−	−	NOUN
ap-4607	349	9	2q	2q	NUM
ap-4607	349	10	−	−	PROPN
ap-4607	349	11	2n	2n	NUM
ap-4607	349	12	=	=	SYM
ap-4607	349	13	4n2	4n2	PROPN
ap-4607	350	1	+	+	NUM
ap-4607	350	2	2n	2n	NUM
ap-4607	350	3	+	+	CCONJ
ap-4607	350	4	1−	1−	NUM
ap-4607	350	5	2q	2q	NUM
ap-4607	350	6	.	.	PUNCT
ap-4607	351	1	(	(	PUNCT
ap-4607	351	2	40	40	NUM
ap-4607	351	3	)	)	PUNCT
ap-4607	351	4	from	from	ADP
ap-4607	351	5	the	the	DET
ap-4607	351	6	fact	fact	NOUN
ap-4607	351	7	that	that	SCONJ
ap-4607	351	8	the	the	DET
ap-4607	351	9	restriction	restriction	NOUN
ap-4607	351	10	of	of	ADP
ap-4607	351	11	the	the	DET
ap-4607	351	12	inverse	inverse	NOUN
ap-4607	351	13	mapping	mapping	NOUN
ap-4607	351	14	=	=	NOUN
ap-4607	351	15	−1	−1	NOUN
ap-4607	351	16	:	:	PUNCT
ap-4607	351	17	cp	cp	X
ap-4607	351	18	2	2	NUM
ap-4607	351	19	→	→	SYM
ap-4607	351	20	c2	c2	PROPN
ap-4607	351	21	to	to	ADP
ap-4607	351	22	the	the	DET
ap-4607	351	23	affine	affine	NOUN
ap-4607	351	24	map	map	NOUN
ap-4607	351	25	a2(1	a2(1	PROPN
ap-4607	351	26	)	)	PUNCT
ap-4607	351	27	is	be	AUX
ap-4607	351	28	a	a	DET
ap-4607	351	29	bijection	bijection	NOUN
ap-4607	351	30	that	that	PRON
ap-4607	351	31	is	be	AUX
ap-4607	351	32	given	give	VERB
ap-4607	351	33	by	by	ADP
ap-4607	351	34	the	the	DET
ap-4607	351	35	equations	equation	NOUN
ap-4607	351	36	x0	x0	PROPN
ap-4607	352	1	=	=	PUNCT
ap-4607	352	2	1	1	NUM
ap-4607	352	3	,	,	PUNCT
ap-4607	352	4	x1	x1	PROPN
ap-4607	353	1	=	=	SYM
ap-4607	353	2	x1	x1	PROPN
ap-4607	353	3	,	,	PUNCT
ap-4607	353	4	x2	x2	PROPN
ap-4607	353	5	=	=	SYM
ap-4607	353	6	x2	x2	PROPN
ap-4607	353	7	,	,	PUNCT
ap-4607	353	8	we	we	PRON
ap-4607	353	9	have	have	VERB
ap-4607	353	10	card	card	NOUN
ap-4607	353	11	(	(	PUNCT
ap-4607	353	12	m0(f1	m0(f1	NOUN
ap-4607	353	13	,	,	PUNCT
ap-4607	353	14	f2	f2	PROPN
ap-4607	353	15	)	)	PUNCT
ap-4607	353	16	)	)	PUNCT
ap-4607	354	1	=	=	SYM
ap-4607	354	2	card	card	NOUN
ap-4607	354	3	(	(	PUNCT
ap-4607	354	4	realv	realv	NOUN
ap-4607	354	5	0(f1	0(f1	PROPN
ap-4607	354	6	,	,	PUNCT
ap-4607	354	7	f2	f2	PROPN
ap-4607	354	8	)	)	PUNCT
ap-4607	354	9	)	)	PUNCT
ap-4607	355	1	−n	−n	ADV
ap-4607	355	2	=	=	SYM
ap-4607	355	3	4n2	4n2	PROPN
ap-4607	356	1	+	+	NOUN
ap-4607	356	2	n	n	NOUN
ap-4607	356	3	+	+	CCONJ
ap-4607	356	4	1−	1−	NUM
ap-4607	356	5	2q	2q	NUM
ap-4607	356	6	.	.	PUNCT
ap-4607	357	1	(	(	PUNCT
ap-4607	357	2	41	41	NUM
ap-4607	357	3	)	)	PUNCT
ap-4607	357	4	in	in	ADP
ap-4607	357	5	a	a	DET
ap-4607	357	6	situation	situation	NOUN
ap-4607	357	7	of	of	ADP
ap-4607	357	8	general	general	ADJ
ap-4607	357	9	position	position	NOUN
ap-4607	357	10	,	,	PUNCT
ap-4607	357	11	the	the	DET
ap-4607	357	12	point	point	NOUN
ap-4607	357	13	source	source	NOUN
ap-4607	357	14	is	be	AUX
ap-4607	357	15	not	not	PART
ap-4607	357	16	on	on	ADP
ap-4607	357	17	the	the	DET
ap-4607	357	18	caustic	caustic	ADJ
ap-4607	357	19	,	,	PUNCT
ap-4607	357	20	therefore	therefore	ADV
ap-4607	357	21	,	,	PUNCT
ap-4607	357	22	all	all	DET
ap-4607	357	23	elements	element	NOUN
ap-4607	357	24	of	of	ADP
ap-4607	357	25	the	the	DET
ap-4607	357	26	set	set	NOUN
ap-4607	357	27	realv	realv	NOUN
ap-4607	357	28	(	(	PUNCT
ap-4607	357	29	f1	f1	NOUN
ap-4607	357	30	,	,	PUNCT
ap-4607	357	31	f2	f2	PROPN
ap-4607	357	32	)	)	PUNCT
ap-4607	357	33	are	be	AUX
ap-4607	357	34	different	different	ADJ
ap-4607	357	35	.	.	PUNCT
ap-4607	358	1	in	in	ADP
ap-4607	358	2	this	this	DET
ap-4607	358	3	case	case	NOUN
ap-4607	358	4	,	,	PUNCT
ap-4607	358	5	each	each	DET
ap-4607	358	6	point	point	NOUN
ap-4607	358	7	of	of	ADP
ap-4607	358	8	the	the	DET
ap-4607	358	9	set	set	NOUN
ap-4607	358	10	realv	realv	NOUN
ap-4607	358	11	(	(	PUNCT
ap-4607	358	12	f1	f1	NOUN
ap-4607	358	13	,	,	PUNCT
ap-4607	358	14	f2	f2	PROPN
ap-4607	358	15	)	)	PUNCT
ap-4607	358	16	is	be	AUX
ap-4607	358	17	,	,	PUNCT
ap-4607	358	18	by	by	ADP
ap-4607	358	19	definition	definition	NOUN
ap-4607	358	20	,	,	PUNCT
ap-4607	358	21	an	an	DET
ap-4607	358	22	image	image	NOUN
ap-4607	358	23	.	.	PUNCT
ap-4607	359	1	it	it	PRON
ap-4607	359	2	follows	follow	VERB
ap-4607	359	3	from	from	ADP
ap-4607	359	4	(	(	PUNCT
ap-4607	359	5	9	9	NUM
ap-4607	359	6	)	)	PUNCT
ap-4607	359	7	that	that	SCONJ
ap-4607	359	8	the	the	DET
ap-4607	359	9	parity	parity	NOUN
ap-4607	359	10	of	of	ADP
ap-4607	359	11	the	the	DET
ap-4607	359	12	number	number	NOUN
ap-4607	359	13	of	of	ADP
ap-4607	359	14	images	image	NOUN
ap-4607	359	15	is	be	AUX
ap-4607	359	16	opposite	opposite	ADJ
ap-4607	359	17	to	to	ADP
ap-4607	359	18	the	the	DET
ap-4607	359	19	parity	parity	NOUN
ap-4607	359	20	of	of	ADP
ap-4607	359	21	the	the	DET
ap-4607	359	22	number	number	NOUN
ap-4607	359	23	n	n	X
ap-4607	359	24	.	.	PUNCT
ap-4607	360	1	theorem	theorem	NOUN
ap-4607	360	2	3	3	NUM
ap-4607	360	3	does	do	AUX
ap-4607	360	4	not	not	PART
ap-4607	360	5	contradict	contradict	VERB
ap-4607	360	6	the	the	DET
ap-4607	360	7	theorem	theorem	NOUN
ap-4607	360	8	on	on	ADP
ap-4607	360	9	the	the	DET
ap-4607	360	10	oddness	oddness	NOUN
ap-4607	360	11	of	of	ADP
ap-4607	360	12	the	the	DET
ap-4607	360	13	number	number	NOUN
ap-4607	360	14	of	of	ADP
ap-4607	360	15	images	image	NOUN
ap-4607	360	16	in	in	ADP
ap-4607	360	17	transparent	transparent	ADJ
ap-4607	360	18	lenses	lense	NOUN
ap-4607	360	19	[	[	X
ap-4607	360	20	9	9	NUM
ap-4607	360	21	,	,	PUNCT
ap-4607	360	22	10	10	NUM
ap-4607	360	23	]	]	PUNCT
ap-4607	360	24	.	.	PUNCT
ap-4607	361	1	example	example	NOUN
ap-4607	362	1	1	1	NUM
ap-4607	362	2	.	.	X
ap-4607	362	3	for	for	ADP
ap-4607	362	4	a	a	DET
ap-4607	362	5	1	1	NUM
ap-4607	362	6	-	-	PUNCT
ap-4607	362	7	point	point	NOUN
ap-4607	362	8	lens	len	NOUN
ap-4607	362	9	,	,	PUNCT
ap-4607	362	10	the	the	DET
ap-4607	362	11	number	number	NOUN
ap-4607	362	12	of	of	ADP
ap-4607	362	13	images	image	NOUN
ap-4607	362	14	is	be	AUX
ap-4607	362	15	2	2	NUM
ap-4607	362	16	,	,	PUNCT
ap-4607	362	17	see	see	VERB
ap-4607	362	18	[	[	X
ap-4607	362	19	9	9	NUM
ap-4607	362	20	,	,	PUNCT
ap-4607	362	21	10	10	NUM
ap-4607	362	22	,	,	PUNCT
ap-4607	362	23	14	14	NUM
ap-4607	362	24	]	]	PUNCT
ap-4607	362	25	.	.	PUNCT
ap-4607	362	26	example	example	NOUN
ap-4607	363	1	2	2	NUM
ap-4607	363	2	.	.	X
ap-4607	363	3	for	for	ADP
ap-4607	363	4	a	a	DET
ap-4607	363	5	2	2	NUM
ap-4607	363	6	-	-	PUNCT
ap-4607	363	7	point	point	NOUN
ap-4607	363	8	lens	len	NOUN
ap-4607	363	9	,	,	PUNCT
ap-4607	363	10	the	the	DET
ap-4607	363	11	number	number	NOUN
ap-4607	363	12	of	of	ADP
ap-4607	363	13	images	image	NOUN
ap-4607	363	14	is	be	AUX
ap-4607	363	15	3	3	NUM
ap-4607	363	16	or	or	CCONJ
ap-4607	363	17	5	5	NUM
ap-4607	363	18	see	see	VERB
ap-4607	363	19	[	[	X
ap-4607	363	20	17	17	NUM
ap-4607	363	21	]	]	SYM
ap-4607	363	22	.	.	PUNCT
ap-4607	364	1	6	6	X
ap-4607	364	2	.	.	PUNCT
ap-4607	364	3	conclusions	conclusion	NOUN
ap-4607	364	4	applying	apply	VERB
ap-4607	364	5	methods	method	NOUN
ap-4607	364	6	of	of	ADP
ap-4607	364	7	algebraic	algebraic	ADJ
ap-4607	364	8	geometry	geometry	NOUN
ap-4607	364	9	,	,	PUNCT
ap-4607	364	10	we	we	PRON
ap-4607	364	11	constructed	construct	VERB
ap-4607	364	12	an	an	DET
ap-4607	364	13	algorithm	algorithm	NOUN
ap-4607	364	14	that	that	PRON
ap-4607	364	15	separates	separate	VERB
ap-4607	364	16	images	image	NOUN
ap-4607	364	17	of	of	ADP
ap-4607	364	18	dimensions	dimension	NOUN
ap-4607	364	19	1	1	NUM
ap-4607	364	20	and	and	CCONJ
ap-4607	364	21	0	0	NUM
ap-4607	364	22	.	.	PUNCT
ap-4607	365	1	in	in	ADP
ap-4607	365	2	the	the	DET
ap-4607	365	3	present	present	ADJ
ap-4607	365	4	paper	paper	NOUN
ap-4607	365	5	,	,	PUNCT
ap-4607	365	6	for	for	ADP
ap-4607	365	7	an	an	DET
ap-4607	365	8	image	image	NOUN
ap-4607	365	9	of	of	ADP
ap-4607	365	10	dimension	dimension	NOUN
ap-4607	365	11	1	1	NUM
ap-4607	365	12	,	,	PUNCT
ap-4607	365	13	it	it	PRON
ap-4607	365	14	is	be	AUX
ap-4607	365	15	proved	prove	VERB
ap-4607	365	16	that	that	SCONJ
ap-4607	365	17	for	for	ADP
ap-4607	365	18	single	single	ADJ
ap-4607	365	19	-	-	PUNCT
ap-4607	365	20	point	point	NOUN
ap-4607	365	21	sources	source	NOUN
ap-4607	365	22	there	there	PRON
ap-4607	365	23	exists	exist	VERB
ap-4607	365	24	a	a	DET
ap-4607	365	25	unique	unique	ADJ
ap-4607	365	26	image	image	NOUN
ap-4607	365	27	of	of	ADP
ap-4607	365	28	dimension	dimension	NOUN
ap-4607	365	29	1	1	NUM
ap-4607	365	30	-	-	NUM
ap-4607	365	31	the	the	DET
ap-4607	365	32	einstein	einstein	PROPN
ap-4607	365	33	ring	ring	NOUN
ap-4607	365	34	;	;	PUNCT
ap-4607	365	35	einstein	einstein	PROPN
ap-4607	365	36	’s	’s	PART
ap-4607	365	37	ring	ring	NOUN
ap-4607	365	38	is	be	AUX
ap-4607	365	39	only	only	ADV
ap-4607	365	40	in	in	ADP
ap-4607	365	41	a	a	DET
ap-4607	365	42	single	single	ADJ
ap-4607	365	43	-	-	PUNCT
ap-4607	365	44	point	point	NOUN
ap-4607	365	45	lens	len	NOUN
ap-4607	365	46	;	;	PUNCT
ap-4607	365	47	the	the	DET
ap-4607	365	48	point	point	NOUN
ap-4607	365	49	source	source	NOUN
ap-4607	365	50	in	in	ADP
ap-4607	365	51	other	other	ADJ
ap-4607	365	52	lenses	lense	NOUN
ap-4607	365	53	does	do	AUX
ap-4607	365	54	not	not	PART
ap-4607	365	55	have	have	VERB
ap-4607	365	56	images	image	NOUN
ap-4607	365	57	of	of	ADP
ap-4607	365	58	dimension	dimension	NOUN
ap-4607	365	59	1	1	NUM
ap-4607	365	60	for	for	ADP
ap-4607	365	61	n	n	X
ap-4607	365	62	>	>	X
ap-4607	365	63	1	1	X
ap-4607	365	64	.	.	PUNCT
ap-4607	366	1	for	for	ADP
ap-4607	366	2	an	an	DET
ap-4607	366	3	image	image	NOUN
ap-4607	366	4	of	of	ADP
ap-4607	366	5	dimension	dimension	NOUN
ap-4607	366	6	0	0	NUM
ap-4607	366	7	,	,	PUNCT
ap-4607	366	8	it	it	PRON
ap-4607	366	9	is	be	AUX
ap-4607	366	10	proved	prove	VERB
ap-4607	366	11	that	that	SCONJ
ap-4607	366	12	in	in	ADP
ap-4607	366	13	any	any	DET
ap-4607	366	14	n	n	CCONJ
ap-4607	366	15	-point	-point	NOUN
ap-4607	366	16	gravitational	gravitational	ADJ
ap-4607	366	17	lens	len	NOUN
ap-4607	366	18	:	:	PUNCT
ap-4607	366	19	there	there	PRON
ap-4607	366	20	are	be	VERB
ap-4607	366	21	a	a	DET
ap-4607	366	22	finite	finite	ADJ
ap-4607	366	23	number	number	NOUN
ap-4607	366	24	of	of	ADP
ap-4607	366	25	images	image	NOUN
ap-4607	366	26	;	;	PUNCT
ap-4607	366	27	the	the	DET
ap-4607	366	28	parity	parity	NOUN
ap-4607	366	29	of	of	ADP
ap-4607	366	30	the	the	DET
ap-4607	366	31	number	number	NOUN
ap-4607	366	32	of	of	ADP
ap-4607	366	33	images	image	NOUN
ap-4607	366	34	is	be	AUX
ap-4607	366	35	always	always	ADV
ap-4607	366	36	the	the	DET
ap-4607	366	37	opposite	opposite	NOUN
ap-4607	366	38	of	of	ADP
ap-4607	366	39	the	the	DET
ap-4607	366	40	parity	parity	NOUN
ap-4607	366	41	of	of	ADP
ap-4607	366	42	the	the	DET
ap-4607	366	43	number	number	NOUN
ap-4607	366	44	n	n	NOUN
ap-4607	366	45	.	.	PUNCT
ap-4607	367	1	the	the	DET
ap-4607	367	2	assertion	assertion	NOUN
ap-4607	367	3	for	for	ADP
ap-4607	367	4	the	the	DET
ap-4607	367	5	number	number	NOUN
ap-4607	367	6	of	of	ADP
ap-4607	367	7	images	image	NOUN
ap-4607	367	8	of	of	ADP
ap-4607	367	9	dimension	dimension	NOUN
ap-4607	367	10	0	0	NUM
ap-4607	367	11	was	be	AUX
ap-4607	367	12	proved	prove	VERB
ap-4607	367	13	by	by	ADP
ap-4607	367	14	us	we	PRON
ap-4607	367	15	earlier	early	ADV
ap-4607	367	16	,	,	PUNCT
ap-4607	367	17	see	see	VERB
ap-4607	367	18	[	[	X
ap-4607	367	19	18	18	NUM
ap-4607	367	20	]	]	PUNCT
ap-4607	367	21	.	.	PUNCT
ap-4607	368	1	in	in	ADP
ap-4607	368	2	[	[	X
ap-4607	368	3	18	18	NUM
ap-4607	368	4	]	]	PUNCT
ap-4607	368	5	,	,	PUNCT
ap-4607	368	6	we	we	PRON
ap-4607	368	7	used	use	VERB
ap-4607	368	8	the	the	DET
ap-4607	368	9	geometric	geometric	ADJ
ap-4607	368	10	method	method	NOUN
ap-4607	368	11	of	of	ADP
ap-4607	368	12	algebraic	algebraic	ADJ
ap-4607	368	13	geometry	geometry	NOUN
ap-4607	368	14	-	-	PUNCT
ap-4607	368	15	the	the	DET
ap-4607	368	16	newton	newton	PROPN
ap-4607	368	17	diagram	diagram	PROPN
ap-4607	368	18	.	.	PUNCT
ap-4607	369	1	in	in	ADP
ap-4607	369	2	the	the	DET
ap-4607	369	3	present	present	ADJ
ap-4607	369	4	paper	paper	NOUN
ap-4607	369	5	all	all	DET
ap-4607	369	6	the	the	DET
ap-4607	369	7	assertions	assertion	NOUN
ap-4607	369	8	are	be	AUX
ap-4607	369	9	proved	prove	VERB
ap-4607	369	10	algebraically	algebraically	ADV
ap-4607	369	11	.	.	PUNCT
ap-4607	370	1	this	this	PRON
ap-4607	370	2	opens	open	VERB
ap-4607	370	3	the	the	DET
ap-4607	370	4	possibility	possibility	NOUN
ap-4607	370	5	,	,	PUNCT
ap-4607	370	6	to	to	PART
ap-4607	370	7	use	use	VERB
ap-4607	370	8	n	n	NUM
ap-4607	370	9	-point	-point	NOUN
ap-4607	370	10	gravitational	gravitational	ADJ
ap-4607	370	11	lenses	lense	NOUN
ap-4607	370	12	,	,	PUNCT
ap-4607	370	13	not	not	PART
ap-4607	370	14	only	only	ADV
ap-4607	370	15	approximate	approximate	ADJ
ap-4607	370	16	or	or	CCONJ
ap-4607	370	17	numerical	numerical	ADJ
ap-4607	370	18	methods	method	NOUN
ap-4607	370	19	,	,	PUNCT
ap-4607	370	20	but	but	CCONJ
ap-4607	370	21	also	also	ADV
ap-4607	370	22	computer	computer	NOUN
ap-4607	370	23	algebra	algebra	NOUN
ap-4607	370	24	systems	system	NOUN
ap-4607	370	25	.	.	PUNCT
ap-4607	371	1	a.	a.	NOUN
ap-4607	371	2	appendix	appendix	PROPN
ap-4607	371	3	let	let	VERB
ap-4607	371	4	f(x	f(x	PROPN
ap-4607	371	5	,	,	PUNCT
ap-4607	371	6	y	y	PROPN
ap-4607	371	7	)	)	PUNCT
ap-4607	371	8	be	be	VERB
ap-4607	371	9	a	a	DET
ap-4607	371	10	function	function	NOUN
ap-4607	371	11	of	of	ADP
ap-4607	371	12	two	two	NUM
ap-4607	371	13	variables	variable	NOUN
ap-4607	371	14	,	,	PUNCT
ap-4607	371	15	and	and	CCONJ
ap-4607	371	16	f(x	f(x	PROPN
ap-4607	371	17	,	,	PUNCT
ap-4607	371	18	y	y	PROPN
ap-4607	371	19	)	)	PUNCT
ap-4607	371	20	,	,	PUNCT
ap-4607	371	21	at	at	ADP
ap-4607	371	22	the	the	DET
ap-4607	371	23	point	point	NOUN
ap-4607	371	24	(	(	PUNCT
ap-4607	371	25	x0	x0	PROPN
ap-4607	371	26	,	,	PUNCT
ap-4607	371	27	y0	y0	PROPN
ap-4607	371	28	)	)	PUNCT
ap-4607	371	29	,	,	PUNCT
ap-4607	371	30	n	n	CCONJ
ap-4607	371	31	-	-	PUNCT
ap-4607	371	32	times	time	NOUN
ap-4607	371	33	continuous	continuous	ADJ
ap-4607	371	34	,	,	PUNCT
ap-4607	371	35	differentiable	differentiable	ADJ
ap-4607	371	36	function	function	NOUN
ap-4607	371	37	.	.	PUNCT
ap-4607	372	1	then	then	ADV
ap-4607	372	2	taylor	taylor	PROPN
ap-4607	372	3	’s	’s	PART
ap-4607	372	4	formula	formula	NOUN
ap-4607	372	5	holds	hold	VERB
ap-4607	372	6	:	:	PUNCT
ap-4607	372	7	f(x	f(x	PROPN
ap-4607	372	8	,	,	PUNCT
ap-4607	372	9	y	y	NOUN
ap-4607	372	10	)	)	PUNCT
ap-4607	372	11	=	=	SYM
ap-4607	372	12	f(x0	f(x0	PROPN
ap-4607	372	13	,	,	PUNCT
ap-4607	372	14	y0	y0	PROPN
ap-4607	372	15	)	)	PUNCT
ap-4607	373	1	+	+	CCONJ
ap-4607	374	1	n∑	n∑	INTJ
ap-4607	374	2	k=1	k=1	X
ap-4607	374	3	f	f	PROPN
ap-4607	374	4	(	(	PUNCT
ap-4607	374	5	k)(x−	k)(x−	PROPN
ap-4607	374	6	x0	x0	PROPN
ap-4607	374	7	,	,	PUNCT
ap-4607	374	8	y	y	PROPN
ap-4607	374	9	−	−	PROPN
ap-4607	374	10	y0	y0	PROPN
ap-4607	374	11	)	)	PUNCT
ap-4607	374	12	+	+	CCONJ
ap-4607	374	13	rn(x	rn(x	X
ap-4607	374	14	,	,	PUNCT
ap-4607	374	15	y	y	PROPN
ap-4607	374	16	)	)	PUNCT
ap-4607	374	17	,	,	PUNCT
ap-4607	374	18	(	(	PUNCT
ap-4607	374	19	42	42	NUM
ap-4607	374	20	)	)	PUNCT
ap-4607	374	21	409	409	NUM
ap-4607	374	22	a.	a.	NOUN
ap-4607	374	23	t.	t.	PROPN
ap-4607	374	24	kotvytskiy	kotvytskiy	PROPN
ap-4607	374	25	,	,	PUNCT
ap-4607	374	26	s.	s.	PROPN
ap-4607	374	27	d.	d.	PROPN
ap-4607	374	28	bronza	bronza	PROPN
ap-4607	374	29	,	,	PUNCT
ap-4607	374	30	v.	v.	PROPN
ap-4607	374	31	yu	yu	PROPN
ap-4607	374	32	.	.	PUNCT
ap-4607	375	1	shablenko	shablenko	PROPN
ap-4607	375	2	acta	acta	PROPN
ap-4607	375	3	polytechnica	polytechnica	PROPN
ap-4607	375	4	where	where	SCONJ
ap-4607	375	5	f	f	PROPN
ap-4607	375	6	(	(	PUNCT
ap-4607	375	7	k)(x−	k)(x−	PROPN
ap-4607	375	8	x0	x0	PROPN
ap-4607	375	9	,	,	PUNCT
ap-4607	375	10	y	y	PROPN
ap-4607	375	11	−	−	PROPN
ap-4607	375	12	y0	y0	PROPN
ap-4607	375	13	)	)	PUNCT
ap-4607	375	14	=	=	SYM
ap-4607	376	1	n∑	n∑	NOUN
ap-4607	376	2	k=1	k=1	PROPN
ap-4607	376	3	ci	ci	PROPN
ap-4607	377	1	k	k	PROPN
ap-4607	377	2	∂kf(x0	∂kf(x0	PROPN
ap-4607	377	3	,	,	PUNCT
ap-4607	377	4	y0	y0	PROPN
ap-4607	377	5	)	)	PUNCT
ap-4607	377	6	∂xk−i∂yi	∂xk−i∂yi	NUM
ap-4607	377	7	(	(	PUNCT
ap-4607	377	8	x−	x−	PROPN
ap-4607	377	9	x0)k−i(y	x0)k−i(y	PUNCT
ap-4607	378	1	−	−	PROPN
ap-4607	378	2	y0)i	y0)i	PROPN
ap-4607	378	3	,	,	PUNCT
ap-4607	378	4	(	(	PUNCT
ap-4607	378	5	43	43	NUM
ap-4607	378	6	)	)	PUNCT
ap-4607	378	7	and	and	CCONJ
ap-4607	378	8	f(x	f(x	PROPN
ap-4607	378	9	,	,	PUNCT
ap-4607	378	10	y	y	PROPN
ap-4607	378	11	)	)	PUNCT
ap-4607	378	12	is	be	AUX
ap-4607	378	13	the	the	DET
ap-4607	378	14	remainder	remainder	ADJ
ap-4607	378	15	term	term	NOUN
ap-4607	378	16	.	.	PUNCT
ap-4607	379	1	if	if	SCONJ
ap-4607	379	2	f(x	f(x	PROPN
ap-4607	379	3	,	,	PUNCT
ap-4607	379	4	y	y	PROPN
ap-4607	379	5	)	)	PUNCT
ap-4607	379	6	is	be	AUX
ap-4607	379	7	a	a	DET
ap-4607	379	8	polynomial	polynomial	ADJ
ap-4607	379	9	,	,	PUNCT
ap-4607	379	10	andm	andm	PROPN
ap-4607	379	11	=	=	SYM
ap-4607	379	12	deg	deg	PROPN
ap-4607	379	13	f(x	f(x	PROPN
ap-4607	379	14	,	,	PUNCT
ap-4607	379	15	y	y	PROPN
ap-4607	379	16	)	)	PUNCT
ap-4607	379	17	,	,	PUNCT
ap-4607	379	18	then	then	ADV
ap-4607	379	19	rn(x	rn(x	X
ap-4607	379	20	,	,	PUNCT
ap-4607	379	21	y	y	NOUN
ap-4607	379	22	)	)	PUNCT
ap-4607	379	23	=	=	SYM
ap-4607	379	24	0	0	NUM
ap-4607	379	25	for	for	ADP
ap-4607	379	26	all	all	DET
ap-4607	379	27	n	n	PRON
ap-4607	379	28	≥	≥	NOUN
ap-4607	379	29	m.	m.	NOUN
ap-4607	379	30	definition	definition	NOUN
ap-4607	379	31	1a	1a	NOUN
ap-4607	379	32	.	.	PUNCT
ap-4607	380	1	we	we	PRON
ap-4607	380	2	say	say	VERB
ap-4607	380	3	that	that	SCONJ
ap-4607	380	4	a	a	DET
ap-4607	380	5	pair	pair	NOUN
ap-4607	380	6	of	of	ADP
ap-4607	380	7	numbers	number	NOUN
ap-4607	380	8	x0	x0	PROPN
ap-4607	380	9	,	,	PUNCT
ap-4607	380	10	y0	y0	PROPN
ap-4607	380	11	is	be	AUX
ap-4607	380	12	a	a	DET
ap-4607	380	13	s	s	ADJ
ap-4607	380	14	-	-	PUNCT
ap-4607	380	15	multiple	multiple	ADJ
ap-4607	380	16	solution	solution	NOUN
ap-4607	380	17	of	of	ADP
ap-4607	380	18	equation	equation	NOUN
ap-4607	380	19	f(x	f(x	PROPN
ap-4607	380	20	,	,	PUNCT
ap-4607	380	21	y	y	NOUN
ap-4607	380	22	)	)	PUNCT
ap-4607	380	23	=	=	SYM
ap-4607	380	24	0	0	PUNCT
ap-4607	381	1	if	if	SCONJ
ap-4607	381	2	:	:	PUNCT
ap-4607	381	3	•	•	NUM
ap-4607	381	4	f(x0	f(x0	NOUN
ap-4607	381	5	,	,	PUNCT
ap-4607	381	6	y0	y0	PROPN
ap-4607	381	7	)	)	PUNCT
ap-4607	381	8	=	=	SYM
ap-4607	381	9	0	0	NUM
ap-4607	381	10	;	;	PUNCT
ap-4607	381	11	•	•	NUM
ap-4607	381	12	f	f	PROPN
ap-4607	381	13	(	(	PUNCT
ap-4607	381	14	i)(x−	i)(x−	PROPN
ap-4607	381	15	x0	x0	PROPN
ap-4607	381	16	,	,	PUNCT
ap-4607	381	17	y	y	PROPN
ap-4607	381	18	−	−	PROPN
ap-4607	381	19	y0	y0	PROPN
ap-4607	381	20	)	)	PUNCT
ap-4607	381	21	≡	≡	PROPN
ap-4607	381	22	0	0	NUM
ap-4607	381	23	,	,	PUNCT
ap-4607	381	24	i	i	PRON
ap-4607	381	25	=	=	NOUN
ap-4607	381	26	1	1	NUM
ap-4607	381	27	,	,	PUNCT
ap-4607	381	28	2	2	NUM
ap-4607	381	29	,	,	PUNCT
ap-4607	381	30	.	.	PUNCT
ap-4607	381	31	.	.	PUNCT
ap-4607	381	32	.	.	PUNCT
ap-4607	382	1	,	,	PUNCT
ap-4607	382	2	s−	s−	PROPN
ap-4607	382	3	1	1	NUM
ap-4607	382	4	;	;	PUNCT
ap-4607	382	5	•	•	NUM
ap-4607	382	6	f	f	PROPN
ap-4607	382	7	(	(	PUNCT
ap-4607	382	8	s)(x	s)(x	PROPN
ap-4607	382	9	−	−	PROPN
ap-4607	382	10	x0	x0	PROPN
ap-4607	382	11	,	,	PUNCT
ap-4607	382	12	y	y	PROPN
ap-4607	382	13	−	−	PROPN
ap-4607	382	14	y0	y0	PROPN
ap-4607	382	15	)	)	PUNCT
ap-4607	382	16	6=	6=	ADP
ap-4607	382	17	0	0	NUM
ap-4607	382	18	,	,	PUNCT
ap-4607	382	19	s	s	VERB
ap-4607	382	20	≤	≤	NOUN
ap-4607	382	21	n	n	NOUN
ap-4607	382	22	in	in	ADP
ap-4607	382	23	some	some	DET
ap-4607	382	24	neighborhood	neighborhood	NOUN
ap-4607	382	25	of	of	ADP
ap-4607	382	26	the	the	DET
ap-4607	382	27	point	point	NOUN
ap-4607	382	28	(	(	PUNCT
ap-4607	382	29	x0	x0	PROPN
ap-4607	382	30	,	,	PUNCT
ap-4607	382	31	y0	y0	PROPN
ap-4607	382	32	)	)	PUNCT
ap-4607	382	33	;	;	PUNCT
ap-4607	382	34	we	we	PRON
ap-4607	382	35	will	will	AUX
ap-4607	382	36	write	write	VERB
ap-4607	382	37	this	this	DET
ap-4607	382	38	fact	fact	NOUN
ap-4607	382	39	as	as	ADP
ap-4607	382	40	mult(f(x0	mult(f(x0	PROPN
ap-4607	382	41	,	,	PUNCT
ap-4607	382	42	y0	y0	NOUN
ap-4607	382	43	)	)	PUNCT
ap-4607	382	44	)	)	PUNCT
ap-4607	383	1	=	=	VERB
ap-4607	383	2	s.	s.	PROPN
ap-4607	383	3	for	for	ADP
ap-4607	383	4	example	example	NOUN
ap-4607	383	5	point	point	NOUN
ap-4607	383	6	(	(	PUNCT
ap-4607	383	7	0	0	NUM
ap-4607	383	8	,	,	PUNCT
ap-4607	383	9	0	0	NUM
ap-4607	383	10	)	)	PUNCT
ap-4607	383	11	is	be	AUX
ap-4607	383	12	s	s	NOUN
ap-4607	383	13	-	-	PUNCT
ap-4607	383	14	multiple	multiple	ADJ
ap-4607	383	15	solution	solution	NOUN
ap-4607	383	16	of	of	ADP
ap-4607	383	17	equation	equation	NOUN
ap-4607	383	18	f(x	f(x	PROPN
ap-4607	383	19	,	,	PUNCT
ap-4607	383	20	y	y	NOUN
ap-4607	383	21	)	)	PUNCT
ap-4607	384	1	=	=	SYM
ap-4607	384	2	0	0	PUNCT
ap-4607	385	1	if	if	SCONJ
ap-4607	385	2	:	:	PUNCT
ap-4607	385	3	•	•	NUM
ap-4607	385	4	f(0	f(0	NOUN
ap-4607	385	5	,	,	PUNCT
ap-4607	385	6	0	0	NUM
ap-4607	385	7	)	)	PUNCT
ap-4607	385	8	=	=	SYM
ap-4607	385	9	0	0	NUM
ap-4607	385	10	;	;	PUNCT
ap-4607	385	11	•	•	NUM
ap-4607	385	12	f	f	X
ap-4607	385	13	(	(	PUNCT
ap-4607	385	14	i)(x	i)(x	NOUN
ap-4607	385	15	,	,	PUNCT
ap-4607	385	16	y	y	NOUN
ap-4607	385	17	)	)	PUNCT
ap-4607	385	18	≡	≡	PROPN
ap-4607	385	19	0	0	NUM
ap-4607	385	20	,	,	PUNCT
ap-4607	385	21	i	i	PRON
ap-4607	385	22	=	=	NOUN
ap-4607	385	23	1	1	NUM
ap-4607	385	24	,	,	PUNCT
ap-4607	385	25	2	2	NUM
ap-4607	385	26	,	,	PUNCT
ap-4607	385	27	...	...	PUNCT
ap-4607	385	28	,	,	PUNCT
ap-4607	385	29	s−	s−	PROPN
ap-4607	385	30	1	1	NUM
ap-4607	385	31	;	;	PUNCT
ap-4607	385	32	•	•	NUM
ap-4607	385	33	f	f	PROPN
ap-4607	385	34	(	(	PUNCT
ap-4607	385	35	s)(x	s)(x	PROPN
ap-4607	385	36	,	,	PUNCT
ap-4607	385	37	y	y	PROPN
ap-4607	385	38	)	)	PUNCT
ap-4607	385	39	6=	6=	ADP
ap-4607	385	40	0	0	NUM
ap-4607	385	41	,	,	PUNCT
ap-4607	385	42	s	s	VERB
ap-4607	385	43	≤	≤	NOUN
ap-4607	385	44	n	n	NOUN
ap-4607	385	45	in	in	ADP
ap-4607	385	46	some	some	DET
ap-4607	385	47	neighborhood	neighborhood	NOUN
ap-4607	385	48	of	of	ADP
ap-4607	385	49	the	the	DET
ap-4607	385	50	point	point	NOUN
ap-4607	385	51	(	(	PUNCT
ap-4607	385	52	0	0	NUM
ap-4607	385	53	,	,	PUNCT
ap-4607	385	54	0	0	NUM
ap-4607	385	55	)	)	PUNCT
ap-4607	385	56	.	.	PUNCT
ap-4607	386	1	such	such	DET
ap-4607	386	2	a	a	DET
ap-4607	386	3	solution	solution	NOUN
ap-4607	386	4	is	be	AUX
ap-4607	386	5	called	call	VERB
ap-4607	386	6	a	a	DET
ap-4607	386	7	s	s	NOUN
ap-4607	386	8	-	-	ADJ
ap-4607	386	9	multiple	multiple	ADJ
ap-4607	386	10	zero	zero	NUM
ap-4607	386	11	solution	solution	NOUN
ap-4607	386	12	.	.	PUNCT
ap-4607	387	1	definition	definition	NOUN
ap-4607	387	2	2a	2a	NUM
ap-4607	387	3	.	.	PUNCT
ap-4607	388	1	let	let	VERB
ap-4607	388	2	the	the	DET
ap-4607	388	3	pair	pair	NOUN
ap-4607	388	4	of	of	ADP
ap-4607	388	5	numbers	number	NOUN
ap-4607	388	6	x0	x0	PROPN
ap-4607	388	7	,	,	PUNCT
ap-4607	388	8	y0	y0	PROPN
ap-4607	388	9	be	be	VERB
ap-4607	388	10	a	a	DET
ap-4607	388	11	solution	solution	NOUN
ap-4607	388	12	of	of	ADP
ap-4607	388	13	the	the	DET
ap-4607	388	14	system	system	NOUN
ap-4607	388	15	of	of	ADP
ap-4607	388	16	equations	equation	NOUN
ap-4607	388	17	{	{	PUNCT
ap-4607	388	18	f(x	f(x	PROPN
ap-4607	388	19	,	,	PUNCT
ap-4607	388	20	y	y	PROPN
ap-4607	388	21	)	)	PUNCT
ap-4607	388	22	=	=	SYM
ap-4607	388	23	0	0	NUM
ap-4607	388	24	,	,	PUNCT
ap-4607	388	25	g(x	g(x	NOUN
ap-4607	388	26	,	,	PUNCT
ap-4607	388	27	y	y	NOUN
ap-4607	388	28	)	)	PUNCT
ap-4607	388	29	=	=	SYM
ap-4607	388	30	0	0	NUM
ap-4607	389	1	(	(	PUNCT
ap-4607	389	2	44	44	NUM
ap-4607	389	3	)	)	PUNCT
ap-4607	389	4	and	and	CCONJ
ap-4607	389	5	q	q	NOUN
ap-4607	389	6	=	=	SYM
ap-4607	389	7	min(mult(f(x0	min(mult(f(x0	PROPN
ap-4607	389	8	,	,	PUNCT
ap-4607	389	9	y0)),mult(g(x0	y0)),mult(g(x0	NOUN
ap-4607	389	10	,	,	PUNCT
ap-4607	389	11	y0	y0	PROPN
ap-4607	389	12	)	)	PUNCT
ap-4607	389	13	)	)	PUNCT
ap-4607	389	14	)	)	PUNCT
ap-4607	389	15	.	.	PUNCT
ap-4607	390	1	the	the	DET
ap-4607	390	2	solution	solution	NOUN
ap-4607	390	3	x0	x0	PROPN
ap-4607	390	4	,	,	PUNCT
ap-4607	390	5	y0	y0	PROPN
ap-4607	390	6	will	will	AUX
ap-4607	390	7	be	be	AUX
ap-4607	390	8	called	call	VERB
ap-4607	390	9	q	q	NOUN
ap-4607	390	10	-	-	PUNCT
ap-4607	390	11	multiple	multiple	NOUN
ap-4607	390	12	of	of	ADP
ap-4607	390	13	the	the	DET
ap-4607	390	14	solution	solution	NOUN
ap-4607	390	15	of	of	ADP
ap-4607	390	16	the	the	DET
ap-4607	390	17	system	system	NOUN
ap-4607	390	18	of	of	ADP
ap-4607	390	19	equations	equation	NOUN
ap-4607	390	20	(	(	PUNCT
ap-4607	390	21	44	44	NUM
ap-4607	390	22	)	)	PUNCT
ap-4607	390	23	,	,	PUNCT
ap-4607	390	24	while	while	SCONJ
ap-4607	390	25	we	we	PRON
ap-4607	390	26	will	will	AUX
ap-4607	390	27	write	write	VERB
ap-4607	390	28	q	q	PROPN
ap-4607	390	29	=	=	NOUN
ap-4607	390	30	min(mult(f	min(mult(f	NOUN
ap-4607	390	31	,	,	PUNCT
ap-4607	390	32	g)(x0	g)(x0	PROPN
ap-4607	390	33	,	,	PUNCT
ap-4607	390	34	y0	y0	NOUN
ap-4607	390	35	)	)	PUNCT
ap-4607	390	36	)	)	PUNCT
ap-4607	390	37	.	.	PUNCT
ap-4607	391	1	the	the	DET
ap-4607	391	2	concept	concept	NOUN
ap-4607	391	3	of	of	ADP
ap-4607	391	4	a	a	DET
ap-4607	391	5	multiple	multiple	ADJ
ap-4607	391	6	solution	solution	NOUN
ap-4607	391	7	of	of	ADP
ap-4607	391	8	a	a	DET
ap-4607	391	9	system	system	NOUN
ap-4607	391	10	of	of	ADP
ap-4607	391	11	equations	equation	NOUN
ap-4607	391	12	can	can	AUX
ap-4607	391	13	obviously	obviously	ADV
ap-4607	391	14	be	be	AUX
ap-4607	391	15	extended	extend	VERB
ap-4607	391	16	to	to	ADP
ap-4607	391	17	systems	system	NOUN
ap-4607	391	18	with	with	ADP
ap-4607	391	19	an	an	DET
ap-4607	391	20	arbitrary	arbitrary	ADJ
ap-4607	391	21	number	number	NOUN
ap-4607	391	22	of	of	ADP
ap-4607	391	23	equations	equation	NOUN
ap-4607	391	24	from	from	ADP
ap-4607	391	25	several	several	ADJ
ap-4607	391	26	variables	variable	NOUN
ap-4607	391	27	.	.	PUNCT
ap-4607	392	1	the	the	DET
ap-4607	392	2	resultant	resultant	NOUN
ap-4607	392	3	of	of	ADP
ap-4607	392	4	polynomials	polynomial	NOUN
ap-4607	392	5	is	be	AUX
ap-4607	392	6	one	one	NUM
ap-4607	392	7	of	of	ADP
ap-4607	392	8	the	the	DET
ap-4607	392	9	basic	basic	ADJ
ap-4607	392	10	concepts	concept	NOUN
ap-4607	392	11	of	of	ADP
ap-4607	392	12	classical	classical	ADJ
ap-4607	392	13	algebraic	algebraic	ADJ
ap-4607	392	14	geometry	geometry	NOUN
ap-4607	392	15	.	.	PUNCT
ap-4607	393	1	in	in	ADP
ap-4607	393	2	the	the	DET
ap-4607	393	3	modern	modern	ADJ
ap-4607	393	4	literature	literature	NOUN
ap-4607	393	5	[	[	X
ap-4607	393	6	12	12	NUM
ap-4607	393	7	,	,	PUNCT
ap-4607	393	8	19	19	NUM
ap-4607	393	9	,	,	PUNCT
ap-4607	393	10	20	20	NUM
ap-4607	393	11	]	]	PUNCT
ap-4607	393	12	,	,	PUNCT
ap-4607	393	13	the	the	DET
ap-4607	393	14	resultant	resultant	NOUN
ap-4607	393	15	of	of	ADP
ap-4607	393	16	polynomials	polynomial	NOUN
ap-4607	393	17	is	be	AUX
ap-4607	393	18	usually	usually	ADV
ap-4607	393	19	defined	define	VERB
ap-4607	393	20	as	as	SCONJ
ap-4607	393	21	follows	follow	VERB
ap-4607	393	22	.	.	PUNCT
ap-4607	394	1	definition	definition	NOUN
ap-4607	394	2	3a	3a	NUM
ap-4607	394	3	.	.	PUNCT
ap-4607	395	1	let	let	VERB
ap-4607	395	2	k	k	ADJ
ap-4607	395	3	-	-	ADJ
ap-4607	395	4	arbitrary	arbitrary	ADJ
ap-4607	395	5	field	field	NOUN
ap-4607	395	6	,	,	PUNCT
ap-4607	395	7	f(x	f(x	PROPN
ap-4607	395	8	)	)	PUNCT
ap-4607	395	9	and	and	CCONJ
ap-4607	395	10	g(x	g(x	NOUN
ap-4607	395	11	)	)	PUNCT
ap-4607	395	12	polynomials	polynomial	NOUN
ap-4607	395	13	in	in	ADP
ap-4607	395	14	k[x	k[x	NOUN
ap-4607	395	15	]	]	PUNCT
ap-4607	395	16	.	.	PUNCT
ap-4607	396	1	the	the	DET
ap-4607	396	2	resultant	resultant	NOUN
ap-4607	396	3	r(f	r(f	PROPN
ap-4607	396	4	,	,	PUNCT
ap-4607	396	5	g	g	NOUN
ap-4607	396	6	)	)	PUNCT
ap-4607	396	7	of	of	ADP
ap-4607	396	8	polynomials	polynomial	VERB
ap-4607	396	9	f(x	f(x	PROPN
ap-4607	396	10	)	)	PUNCT
ap-4607	396	11	and	and	CCONJ
ap-4607	396	12	g(x	g(x	NOUN
ap-4607	396	13	)	)	PUNCT
ap-4607	396	14	is	be	AUX
ap-4607	396	15	called	call	VERB
ap-4607	396	16	an	an	DET
ap-4607	396	17	element	element	NOUN
ap-4607	396	18	in	in	ADP
ap-4607	396	19	the	the	DET
ap-4607	396	20	field	field	NOUN
ap-4607	396	21	k	k	NOUN
ap-4607	396	22	,	,	PUNCT
ap-4607	396	23	defined	define	VERB
ap-4607	396	24	by	by	ADP
ap-4607	396	25	the	the	DET
ap-4607	396	26	formula	formula	NOUN
ap-4607	396	27	r(f	r(f	PROPN
ap-4607	396	28	,	,	PUNCT
ap-4607	396	29	g	g	NOUN
ap-4607	396	30	)	)	PUNCT
ap-4607	396	31	=	=	PUNCT
ap-4607	397	1	an	an	PRON
ap-4607	397	2	0	0	NUM
ap-4607	397	3	b	b	NOUN
ap-4607	397	4	m	m	NOUN
ap-4607	397	5	0	0	PROPN
ap-4607	397	6	n∏	n∏	PROPN
ap-4607	397	7	i=0	i=0	PROPN
ap-4607	397	8	m∏	m∏	PROPN
ap-4607	397	9	j=0	j=0	PROPN
ap-4607	397	10	(	(	PUNCT
ap-4607	397	11	αi	αi	NOUN
ap-4607	397	12	−	−	PROPN
ap-4607	397	13	βj	βj	PRON
ap-4607	397	14	)	)	PUNCT
ap-4607	397	15	,	,	PUNCT
ap-4607	397	16	(	(	PUNCT
ap-4607	397	17	45	45	NUM
ap-4607	397	18	)	)	PUNCT
ap-4607	397	19	where	where	SCONJ
ap-4607	397	20	αi	αi	VERB
ap-4607	397	21	,	,	PUNCT
ap-4607	397	22	βi	βi	PRON
ap-4607	397	23	are	be	AUX
ap-4607	397	24	roots	root	NOUN
ap-4607	397	25	of	of	ADP
ap-4607	397	26	polynomials	polynomial	NOUN
ap-4607	397	27	f(x	f(x	PROPN
ap-4607	397	28	)	)	PUNCT
ap-4607	398	1	=	=	NOUN
ap-4607	398	2	∑n	∑n	NUM
ap-4607	398	3	i=0	i=0	ADJ
ap-4607	398	4	aix	aix	NOUN
ap-4607	398	5	n−i	n−i	PROPN
ap-4607	398	6	and	and	CCONJ
ap-4607	398	7	g(x	g(x	NOUN
ap-4607	398	8	)	)	PUNCT
ap-4607	399	1	=	=	SYM
ap-4607	399	2	∑m	∑m	PROPN
ap-4607	399	3	j=0	j=0	PROPN
ap-4607	399	4	bjx	bjx	VERB
ap-4607	399	5	m−j	m−j	PROPN
ap-4607	399	6	,	,	PUNCT
ap-4607	399	7	correspondingly	correspondingly	ADV
ap-4607	399	8	,	,	PUNCT
ap-4607	399	9	with	with	ADP
ap-4607	399	10	the	the	DET
ap-4607	399	11	highest	high	ADJ
ap-4607	399	12	coefficients	coefficient	NOUN
ap-4607	399	13	,	,	PUNCT
ap-4607	399	14	a0	a0	NOUN
ap-4607	399	15	,	,	PUNCT
ap-4607	399	16	b0	b0	VERB
ap-4607	399	17	such	such	ADJ
ap-4607	399	18	that	that	DET
ap-4607	399	19	a0	a0	PROPN
ap-4607	399	20	6=	6=	ADP
ap-4607	399	21	0	0	NUM
ap-4607	399	22	,	,	PUNCT
ap-4607	399	23	b0	b0	NOUN
ap-4607	399	24	6=	6=	ADP
ap-4607	399	25	0	0	NUM
ap-4607	399	26	.	.	PUNCT
ap-4607	400	1	let	let	VERB
ap-4607	400	2	the	the	DET
ap-4607	400	3	roots	root	NOUN
ap-4607	400	4	of	of	ADP
ap-4607	400	5	the	the	DET
ap-4607	400	6	polynomials	polynomial	NOUN
ap-4607	400	7	f(x	f(x	PROPN
ap-4607	400	8	)	)	PUNCT
ap-4607	400	9	and	and	CCONJ
ap-4607	400	10	g(x	g(x	NOUN
ap-4607	400	11	)	)	PUNCT
ap-4607	400	12	be	be	AUX
ap-4607	400	13	known	know	VERB
ap-4607	400	14	.	.	PUNCT
ap-4607	401	1	to	to	PART
ap-4607	401	2	calculate	calculate	VERB
ap-4607	401	3	their	their	PRON
ap-4607	401	4	resultant	resultant	NOUN
ap-4607	401	5	,	,	PUNCT
ap-4607	401	6	one	one	PRON
ap-4607	401	7	can	can	AUX
ap-4607	401	8	use	use	VERB
ap-4607	401	9	formula	formula	NOUN
ap-4607	401	10	(	(	PUNCT
ap-4607	401	11	45	45	NUM
ap-4607	401	12	)	)	PUNCT
ap-4607	401	13	.	.	PUNCT
ap-4607	402	1	if	if	SCONJ
ap-4607	402	2	we	we	PRON
ap-4607	402	3	know	know	VERB
ap-4607	402	4	only	only	ADV
ap-4607	402	5	the	the	DET
ap-4607	402	6	coefficients	coefficient	NOUN
ap-4607	402	7	of	of	ADP
ap-4607	402	8	these	these	DET
ap-4607	402	9	polynomials	polynomial	NOUN
ap-4607	402	10	,	,	PUNCT
ap-4607	402	11	then	then	ADV
ap-4607	402	12	we	we	PRON
ap-4607	402	13	can	can	AUX
ap-4607	402	14	use	use	VERB
ap-4607	402	15	the	the	DET
ap-4607	402	16	sylvester	sylvest	ADJ
ap-4607	402	17	matrix	matrix	NOUN
ap-4607	402	18	to	to	PART
ap-4607	402	19	calculate	calculate	VERB
ap-4607	402	20	their	their	PRON
ap-4607	402	21	resultant	resultant	NOUN
ap-4607	402	22	.	.	PUNCT
ap-4607	403	1	the	the	DET
ap-4607	403	2	sylvester	sylvest	ADJ
ap-4607	403	3	matrix	matrix	NOUN
ap-4607	403	4	is	be	AUX
ap-4607	403	5	a	a	DET
ap-4607	403	6	block	block	NOUN
ap-4607	403	7	matrix	matrix	NOUN
ap-4607	403	8	of	of	ADP
ap-4607	403	9	two	two	NUM
ap-4607	403	10	blocks	block	NOUN
ap-4607	403	11	.	.	PUNCT
ap-4607	404	1	each	each	DET
ap-4607	404	2	block	block	NOUN
ap-4607	404	3	has	have	VERB
ap-4607	404	4	one	one	NUM
ap-4607	404	5	ribbon	ribbon	NOUN
ap-4607	404	6	matrix	matrix	NOUN
ap-4607	404	7	.	.	PUNCT
ap-4607	405	1	we	we	PRON
ap-4607	405	2	have	have	VERB
ap-4607	405	3	a	a	DET
ap-4607	405	4	definition	definition	NOUN
ap-4607	405	5	of	of	ADP
ap-4607	405	6	the	the	DET
ap-4607	405	7	sylvester	sylvest	ADJ
ap-4607	405	8	matrix	matrix	NOUN
ap-4607	405	9	.	.	PUNCT
ap-4607	406	1	definition	definition	NOUN
ap-4607	406	2	4a	4a	NUM
ap-4607	406	3	.	.	PUNCT
ap-4607	407	1	matrix	matrix	NOUN
ap-4607	407	2	sylvester	sylvester	NOUN
ap-4607	407	3	for	for	ADP
ap-4607	407	4	polynomials	polynomial	NOUN
ap-4607	407	5	f(x	f(x	PROPN
ap-4607	407	6	)	)	PUNCT
ap-4607	408	1	=	=	PUNCT
ap-4607	409	1	∑n	∑n	PROPN
ap-4607	409	2	i=0	i=0	ADJ
ap-4607	409	3	aix	aix	NOUN
ap-4607	409	4	n−i	n−i	PROPN
ap-4607	409	5	and	and	CCONJ
ap-4607	409	6	g(x	g(x	NOUN
ap-4607	409	7	)	)	PUNCT
ap-4607	410	1	=	=	SYM
ap-4607	410	2	∑m	∑m	PROPN
ap-4607	410	3	j=0	j=0	PROPN
ap-4607	410	4	bjx	bjx	VERB
ap-4607	410	5	m−j	m−j	PROPN
ap-4607	410	6	,	,	PUNCT
ap-4607	410	7	we	we	PRON
ap-4607	410	8	call	call	VERB
ap-4607	410	9	a	a	DET
ap-4607	410	10	square	square	ADJ
ap-4607	410	11	matrix	matrix	NOUN
ap-4607	410	12	s	s	PART
ap-4607	410	13	=	=	SYM
ap-4607	410	14	s(f	s(f	PROPN
ap-4607	410	15	,	,	PUNCT
ap-4607	410	16	g	g	NOUN
ap-4607	410	17	)	)	PUNCT
ap-4607	410	18	of	of	ADP
ap-4607	410	19	order	order	NOUN
ap-4607	410	20	n+m	n+m	NUM
ap-4607	410	21	with	with	ADP
ap-4607	410	22	elements	element	NOUN
ap-4607	410	23	sij	sij	PROPN
ap-4607	410	24	defined	define	VERB
ap-4607	410	25	by	by	ADP
ap-4607	410	26	the	the	DET
ap-4607	410	27	formula	formula	NOUN
ap-4607	410	28	sij	sij	PROPN
ap-4607	410	29	=	=	SYM
ap-4607	410	30			NUM
ap-4607	410	31	aj−i	aj−i	PROPN
ap-4607	410	32	,	,	PUNCT
ap-4607	410	33	if	if	SCONJ
ap-4607	410	34	0	0	NUM
ap-4607	410	35	≤	≤	NUM
ap-4607	411	1	j	j	PROPN
ap-4607	412	1	−	−	NOUN
ap-4607	413	1	i	i	PRON
ap-4607	413	2	≤	≤	PROPN
ap-4607	413	3	n	n	CCONJ
ap-4607	413	4	,	,	PUNCT
ap-4607	413	5	i	i	PRON
ap-4607	413	6	=	=	NOUN
ap-4607	413	7	1	1	NUM
ap-4607	413	8	,	,	PUNCT
ap-4607	413	9	.	.	PUNCT
ap-4607	413	10	.	.	PUNCT
ap-4607	413	11	.	.	PUNCT
ap-4607	414	1	,	,	PUNCT
ap-4607	414	2	m	m	PROPN
ap-4607	414	3	,	,	PUNCT
ap-4607	414	4	j	j	PROPN
ap-4607	414	5	=	=	SYM
ap-4607	414	6	1	1	NUM
ap-4607	414	7	,	,	PUNCT
ap-4607	414	8	.	.	PUNCT
ap-4607	414	9	.	.	PUNCT
ap-4607	415	1	.	.	PUNCT
ap-4607	416	1	,	,	PUNCT
ap-4607	416	2	n+m	n+m	NUM
ap-4607	416	3	,	,	PUNCT
ap-4607	416	4	bj−i+m	bj−i+m	NOUN
ap-4607	416	5	,	,	PUNCT
ap-4607	417	1	if	if	SCONJ
ap-4607	417	2	0	0	NUM
ap-4607	417	3	≤	≤	NUM
ap-4607	417	4	j	j	NOUN
ap-4607	417	5	−	−	NOUN
ap-4607	417	6	i+m	i+m	PROPN
ap-4607	417	7	≤	≤	NOUN
ap-4607	417	8	n	n	CCONJ
ap-4607	417	9	,	,	PUNCT
ap-4607	417	10	i	i	PRON
ap-4607	417	11	=	=	PUNCT
ap-4607	417	12	m+	m+	NUM
ap-4607	417	13	1	1	NUM
ap-4607	417	14	,	,	PUNCT
ap-4607	417	15	.	.	PUNCT
ap-4607	417	16	.	.	PUNCT
ap-4607	417	17	.	.	PUNCT
ap-4607	418	1	,	,	PUNCT
ap-4607	418	2	m+	m+	NUM
ap-4607	418	3	n	n	CCONJ
ap-4607	418	4	,	,	PUNCT
ap-4607	418	5	j	j	PROPN
ap-4607	418	6	=	=	SYM
ap-4607	418	7	1	1	NUM
ap-4607	418	8	,	,	PUNCT
ap-4607	418	9	...	...	PUNCT
ap-4607	418	10	,	,	PUNCT
ap-4607	418	11	n+m	n+m	PROPN
ap-4607	418	12	,	,	PUNCT
ap-4607	418	13	0	0	NUM
ap-4607	418	14	,	,	PUNCT
ap-4607	418	15	for	for	ADP
ap-4607	418	16	other	other	ADJ
ap-4607	418	17	i	i	PROPN
ap-4607	418	18	,	,	PUNCT
ap-4607	418	19	j	j	PROPN
ap-4607	418	20	,	,	PUNCT
ap-4607	418	21	(	(	PUNCT
ap-4607	418	22	46	46	NUM
ap-4607	418	23	)	)	PUNCT
ap-4607	418	24	i.e.	i.e.	X
ap-4607	418	25	,	,	PUNCT
ap-4607	418	26	s(f	s(f	PROPN
ap-4607	418	27	,	,	PUNCT
ap-4607	418	28	g	g	NOUN
ap-4607	418	29	)	)	PUNCT
ap-4607	418	30	=	=	PUNCT
ap-4607	419	1	[	[	X
ap-4607	419	2	sij	sij	X
ap-4607	419	3	]	]	PUNCT
ap-4607	419	4	=	=	SYM
ap-4607	419	5	n	n	NUM
ap-4607	419	6	rows	row	VERB
ap-4607	419	7			PROPN
ap-4607	419	8	m	m	VERB
ap-4607	419	9	rows	row	VERB
ap-4607	419	10			PROPN
ap-4607	419	11			NOUN
ap-4607	419	12	a0	a0	NOUN
ap-4607	419	13	a1	a1	NOUN
ap-4607	419	14	·	·	PUNCT
ap-4607	419	15	·	·	PUNCT
ap-4607	419	16	·	·	PUNCT
ap-4607	420	1	an	an	DET
ap-4607	420	2	0	0	NUM
ap-4607	420	3	·	·	PUNCT
ap-4607	420	4	·	·	PUNCT
ap-4607	420	5	·	·	PUNCT
ap-4607	420	6	0	0	NUM
ap-4607	420	7	0	0	NUM
ap-4607	420	8	a0	a0	NOUN
ap-4607	420	9	a1	a1	NOUN
ap-4607	420	10	·	·	PUNCT
ap-4607	420	11	·	·	PUNCT
ap-4607	420	12	·	·	PUNCT
ap-4607	421	1	an	an	DET
ap-4607	421	2	0	0	NUM
ap-4607	421	3	0	0	NUM
ap-4607	421	4	.	.	PUNCT
ap-4607	421	5	.	.	PUNCT
ap-4607	421	6	.	.	PUNCT
ap-4607	421	7	.	.	PUNCT
ap-4607	421	8	.	.	PUNCT
ap-4607	421	9	.	.	PUNCT
ap-4607	421	10	.	.	PUNCT
ap-4607	421	11	.	.	PUNCT
ap-4607	421	12	.	.	PUNCT
ap-4607	421	13	0	0	PUNCT
ap-4607	421	14	·	·	PUNCT
ap-4607	421	15	·	·	PUNCT
ap-4607	421	16	·	·	PUNCT
ap-4607	421	17	0	0	NUM
ap-4607	421	18	a0	a0	NOUN
ap-4607	421	19	a1	a1	NOUN
ap-4607	421	20	·	·	PUNCT
ap-4607	421	21	·	·	PUNCT
ap-4607	421	22	·	·	PUNCT
ap-4607	421	23	an	an	DET
ap-4607	421	24	b0	b0	NOUN
ap-4607	421	25	b1	b1	NOUN
ap-4607	421	26	·	·	PUNCT
ap-4607	421	27	·	·	PUNCT
ap-4607	421	28	·	·	PUNCT
ap-4607	421	29	bm	bm	X
ap-4607	421	30	0	0	NUM
ap-4607	421	31	·	·	PUNCT
ap-4607	421	32	·	·	PUNCT
ap-4607	421	33	·	·	PUNCT
ap-4607	421	34	0	0	NUM
ap-4607	421	35	0	0	X
ap-4607	422	1	b0	b0	VERB
ap-4607	422	2	b1	b1	NOUN
ap-4607	422	3	·	·	PUNCT
ap-4607	422	4	·	·	PUNCT
ap-4607	422	5	·	·	PUNCT
ap-4607	422	6	bm	bm	X
ap-4607	422	7	0	0	NUM
ap-4607	422	8	0	0	NUM
ap-4607	422	9	.	.	PUNCT
ap-4607	422	10	.	.	PUNCT
ap-4607	422	11	.	.	PUNCT
ap-4607	422	12	.	.	PUNCT
ap-4607	422	13	.	.	PUNCT
ap-4607	422	14	.	.	PUNCT
ap-4607	422	15	.	.	PUNCT
ap-4607	422	16	.	.	PUNCT
ap-4607	423	1	.	.	PUNCT
ap-4607	424	1	0	0	PUNCT
ap-4607	425	1	·	·	PUNCT
ap-4607	425	2	·	·	PUNCT
ap-4607	425	3	·	·	PUNCT
ap-4607	425	4	0	0	PUNCT
ap-4607	425	5	b0	b0	VERB
ap-4607	425	6	b1	b1	NOUN
ap-4607	425	7	·	·	PUNCT
ap-4607	425	8	·	·	PUNCT
ap-4607	425	9	·	·	PUNCT
ap-4607	425	10	bm	bm	PROPN
ap-4607	425	11			NUM
ap-4607	425	12	.	.	PUNCT
ap-4607	426	1	(	(	PUNCT
ap-4607	426	2	47	47	NUM
ap-4607	426	3	)	)	PUNCT
ap-4607	426	4	the	the	DET
ap-4607	426	5	resultant	resultant	NOUN
ap-4607	426	6	polynomials	polynomial	VERB
ap-4607	426	7	r(f	r(f	PROPN
ap-4607	426	8	,	,	PUNCT
ap-4607	426	9	g	g	NOUN
ap-4607	426	10	)	)	PUNCT
ap-4607	426	11	and	and	CCONJ
ap-4607	426	12	the	the	DET
ap-4607	426	13	sylvester	sylvest	ADJ
ap-4607	426	14	matrix	matrix	NOUN
ap-4607	426	15	sul(f	sul(f	PROPN
ap-4607	426	16	,	,	PUNCT
ap-4607	426	17	g	g	NOUN
ap-4607	426	18	)	)	PUNCT
ap-4607	426	19	are	be	AUX
ap-4607	426	20	connected	connect	VERB
ap-4607	426	21	by	by	ADP
ap-4607	426	22	the	the	DET
ap-4607	426	23	following	follow	VERB
ap-4607	426	24	theorem	theorem	PROPN
ap-4607	426	25	.	.	PUNCT
ap-4607	427	1	theorem	theorem	PROPN
ap-4607	427	2	1a	1a	NOUN
ap-4607	427	3	.	.	PUNCT
ap-4607	428	1	the	the	DET
ap-4607	428	2	resultant	resultant	NOUN
ap-4607	428	3	r(f	r(f	PROPN
ap-4607	428	4	,	,	PUNCT
ap-4607	428	5	g	g	NOUN
ap-4607	428	6	)	)	PUNCT
ap-4607	428	7	of	of	ADP
ap-4607	428	8	the	the	DET
ap-4607	428	9	polynomials	polynomial	NOUN
ap-4607	428	10	f	f	PROPN
ap-4607	428	11	and	and	CCONJ
ap-4607	428	12	g	g	PROPN
ap-4607	428	13	is	be	AUX
ap-4607	428	14	equal	equal	ADJ
ap-4607	428	15	to	to	ADP
ap-4607	428	16	the	the	DET
ap-4607	428	17	determinant	determinant	NOUN
ap-4607	428	18	of	of	ADP
ap-4607	428	19	sylvester	sylvest	ADJ
ap-4607	428	20	matrix	matrix	NOUN
ap-4607	428	21	these	these	DET
ap-4607	428	22	polynomials	polynomial	NOUN
ap-4607	428	23	,	,	PUNCT
ap-4607	428	24	i.e.	i.e.	X
ap-4607	428	25	,	,	PUNCT
ap-4607	428	26	r(f	r(f	PROPN
ap-4607	428	27	,	,	PUNCT
ap-4607	428	28	g	g	NOUN
ap-4607	428	29	)	)	PUNCT
ap-4607	428	30	=	=	SYM
ap-4607	429	1	s(f	s(f	PROPN
ap-4607	429	2	,	,	PUNCT
ap-4607	429	3	g	g	NOUN
ap-4607	429	4	)	)	PUNCT
ap-4607	429	5	.	.	PUNCT
ap-4607	430	1	for	for	SCONJ
ap-4607	430	2	the	the	DET
ap-4607	430	3	proof	proof	NOUN
ap-4607	430	4	see	see	VERB
ap-4607	430	5	,	,	PUNCT
ap-4607	430	6	e.g.	e.g.	ADV
ap-4607	430	7	,	,	PUNCT
ap-4607	430	8	[	[	X
ap-4607	430	9	19	19	NUM
ap-4607	430	10	,	,	PUNCT
ap-4607	430	11	20	20	NUM
ap-4607	430	12	]	]	PUNCT
ap-4607	430	13	.	.	PUNCT
ap-4607	431	1	theorem	theorem	PROPN
ap-4607	431	2	2a	2a	NUM
ap-4607	431	3	.	.	PUNCT
ap-4607	432	1	polynomials	polynomial	NOUN
ap-4607	432	2	f	f	PROPN
ap-4607	432	3	and	and	CCONJ
ap-4607	432	4	g	g	PROPN
ap-4607	432	5	have	have	VERB
ap-4607	432	6	a	a	DET
ap-4607	432	7	common	common	ADJ
ap-4607	432	8	root	root	NOUN
ap-4607	432	9	if	if	SCONJ
ap-4607	433	1	and	and	CCONJ
ap-4607	433	2	only	only	ADV
ap-4607	433	3	if	if	SCONJ
ap-4607	433	4	r(f	r(f	PROPN
ap-4607	433	5	,	,	PUNCT
ap-4607	433	6	g	g	NOUN
ap-4607	433	7	)	)	PUNCT
ap-4607	433	8	=	=	SYM
ap-4607	434	1	0	0	X
ap-4607	434	2	.	.	PUNCT
ap-4607	435	1	(	(	PUNCT
ap-4607	435	2	48	48	NUM
ap-4607	435	3	)	)	PUNCT
ap-4607	435	4	for	for	ADP
ap-4607	435	5	the	the	DET
ap-4607	435	6	proof	proof	NOUN
ap-4607	435	7	see	see	VERB
ap-4607	435	8	,	,	PUNCT
ap-4607	435	9	e.g.	e.g.	ADV
ap-4607	435	10	,	,	PUNCT
ap-4607	435	11	[	[	X
ap-4607	435	12	13	13	NUM
ap-4607	435	13	]	]	PUNCT
ap-4607	435	14	.	.	PUNCT
ap-4607	436	1	theorem	theorem	ADJ
ap-4607	436	2	3a	3a	NUM
ap-4607	436	3	(	(	PUNCT
ap-4607	436	4	bezout	bezout	NOUN
ap-4607	436	5	)	)	PUNCT
ap-4607	436	6	.	.	PUNCT
ap-4607	437	1	the	the	DET
ap-4607	437	2	number	number	NOUN
ap-4607	437	3	of	of	ADP
ap-4607	437	4	intersection	intersection	NOUN
ap-4607	437	5	points	point	NOUN
ap-4607	437	6	of	of	ADP
ap-4607	437	7	plane	plane	NOUN
ap-4607	437	8	curves	curve	NOUN
ap-4607	437	9	φ1	φ1	NOUN
ap-4607	437	10	and	and	CCONJ
ap-4607	437	11	φ2	φ2	PROPN
ap-4607	437	12	(	(	PUNCT
ap-4607	437	13	counted	count	VERB
ap-4607	437	14	taking	take	VERB
ap-4607	437	15	into	into	ADP
ap-4607	437	16	account	account	NOUN
ap-4607	437	17	the	the	DET
ap-4607	437	18	multiplicity	multiplicity	NOUN
ap-4607	437	19	)	)	PUNCT
ap-4607	437	20	is	be	AUX
ap-4607	437	21	equal	equal	ADJ
ap-4607	437	22	to	to	ADP
ap-4607	437	23	nm	nm	NOUN
ap-4607	437	24	,	,	PUNCT
ap-4607	437	25	where	where	SCONJ
ap-4607	437	26	m	m	VERB
ap-4607	437	27	=	=	ADJ
ap-4607	437	28	deg	deg	PROPN
ap-4607	437	29	φ1	φ1	PROPN
ap-4607	437	30	,	,	PUNCT
ap-4607	437	31	and	and	CCONJ
ap-4607	437	32	n	n	CCONJ
ap-4607	437	33	=	=	NUM
ap-4607	437	34	deg	deg	PROPN
ap-4607	437	35	φ2	φ2	PROPN
ap-4607	437	36	,	,	PUNCT
ap-4607	437	37	if	if	SCONJ
ap-4607	437	38	the	the	DET
ap-4607	437	39	curves	curve	NOUN
ap-4607	437	40	:	:	PUNCT
ap-4607	437	41	•	•	X
ap-4607	437	42	do	do	AUX
ap-4607	437	43	not	not	PART
ap-4607	437	44	have	have	VERB
ap-4607	437	45	common	common	ADJ
ap-4607	437	46	components	component	NOUN
ap-4607	437	47	;	;	PUNCT
ap-4607	437	48	•	•	NUM
ap-4607	437	49	are	be	AUX
ap-4607	437	50	defined	define	VERB
ap-4607	437	51	over	over	ADP
ap-4607	437	52	an	an	DET
ap-4607	437	53	algebraically	algebraically	ADV
ap-4607	437	54	closed	close	VERB
ap-4607	437	55	field	field	NOUN
ap-4607	437	56	;	;	PUNCT
ap-4607	437	57	•	•	NUM
ap-4607	437	58	are	be	AUX
ap-4607	437	59	considered	consider	VERB
ap-4607	437	60	on	on	ADP
ap-4607	437	61	the	the	DET
ap-4607	437	62	projective	projective	ADJ
ap-4607	437	63	plane	plane	NOUN
ap-4607	437	64	.	.	PUNCT
ap-4607	438	1	for	for	SCONJ
ap-4607	438	2	the	the	DET
ap-4607	438	3	proof	proof	NOUN
ap-4607	438	4	see	see	VERB
ap-4607	438	5	,	,	PUNCT
ap-4607	438	6	e.g.	e.g.	ADV
ap-4607	438	7	,	,	PUNCT
ap-4607	438	8	[	[	X
ap-4607	438	9	12	12	NUM
ap-4607	438	10	]	]	PUNCT
ap-4607	438	11	.	.	PUNCT
ap-4607	439	1	410	410	NUM
ap-4607	439	2	vol	vol	NOUN
ap-4607	439	3	.	.	PUNCT
ap-4607	439	4	57	57	NUM
ap-4607	439	5	no	no	NOUN
ap-4607	439	6	.	.	PUNCT
ap-4607	440	1	6/2017	6/2017	X
ap-4607	440	2	the	the	DET
ap-4607	440	3	analysis	analysis	NOUN
ap-4607	440	4	of	of	ADP
ap-4607	440	5	images	image	NOUN
ap-4607	440	6	in	in	ADP
ap-4607	440	7	n	n	PRON
ap-4607	440	8	-point	-point	NOUN
ap-4607	440	9	gravitational	gravitational	ADJ
ap-4607	440	10	lens	lens	NOUN
ap-4607	440	11	theorem	theorem	VERB
ap-4607	440	12	4a	4a	NOUN
ap-4607	440	13	.	.	PUNCT
ap-4607	441	1	let	let	VERB
ap-4607	441	2	f(x1	f(x1	ADJ
ap-4607	441	3	,	,	PUNCT
ap-4607	441	4	x2	x2	PROPN
ap-4607	441	5	)	)	PUNCT
ap-4607	441	6	=	=	SYM
ap-4607	442	1	∑i+j≤n	∑i+j≤n	X
ap-4607	443	1	i	i	PRON
ap-4607	443	2	,	,	PUNCT
ap-4607	443	3	j=0	j=0	PROPN
ap-4607	443	4	fijx	fijx	PROPN
ap-4607	444	1	i	i	PRON
ap-4607	444	2	1x	1x	NUM
ap-4607	444	3	j	j	NOUN
ap-4607	444	4	2	2	NUM
ap-4607	444	5	and	and	CCONJ
ap-4607	444	6	g(x1	g(x1	NOUN
ap-4607	444	7	,	,	PUNCT
ap-4607	444	8	x2	x2	PROPN
ap-4607	444	9	)	)	PUNCT
ap-4607	444	10	=	=	SYM
ap-4607	444	11	∑i+j≤m	∑i+j≤m	PROPN
ap-4607	445	1	i	i	PROPN
ap-4607	445	2	,	,	PUNCT
ap-4607	445	3	j=0	j=0	PROPN
ap-4607	445	4	gijx	gijx	PROPN
ap-4607	446	1	i	i	PRON
ap-4607	446	2	1x	1x	PROPN
ap-4607	446	3	j	j	PROPN
ap-4607	446	4	2	2	NUM
ap-4607	446	5	be	be	AUX
ap-4607	446	6	polynomials	polynomial	NOUN
ap-4607	446	7	.	.	PUNCT
ap-4607	447	1	let	let	VERB
ap-4607	447	2	their	their	PRON
ap-4607	447	3	coefficients	coefficient	NOUN
ap-4607	447	4	be	be	AUX
ap-4607	447	5	such	such	ADJ
ap-4607	447	6	that	that	DET
ap-4607	447	7	fn0	fn0	NOUN
ap-4607	447	8	6=	6=	PRON
ap-4607	447	9	0	0	NUM
ap-4607	447	10	,	,	PUNCT
ap-4607	447	11	f0n	f0n	VERB
ap-4607	447	12	6=	6=	ADP
ap-4607	447	13	0	0	NUM
ap-4607	447	14	,	,	PUNCT
ap-4607	447	15	gm0	gm0	VERB
ap-4607	447	16	6=	6=	PRON
ap-4607	447	17	0	0	NUM
ap-4607	447	18	,	,	PUNCT
ap-4607	447	19	g0	g0	PROPN
ap-4607	447	20	m	m	PROPN
ap-4607	447	21	6=	6=	NUM
ap-4607	447	22	0	0	NUM
ap-4607	447	23	.	.	PUNCT
ap-4607	448	1	then	then	ADV
ap-4607	448	2	degr1	degr1	PROPN
ap-4607	448	3	(	(	PUNCT
ap-4607	448	4	f(x1	f(x1	ADJ
ap-4607	448	5	,	,	PUNCT
ap-4607	448	6	x2	x2	PROPN
ap-4607	448	7	)	)	PUNCT
ap-4607	448	8	,	,	PUNCT
ap-4607	448	9	g(x1	g(x1	NOUN
ap-4607	448	10	,	,	PUNCT
ap-4607	448	11	x2	x2	PROPN
ap-4607	448	12	)	)	PUNCT
ap-4607	448	13	)	)	PUNCT
ap-4607	449	1	=	=	SYM
ap-4607	449	2	deg	deg	PROPN
ap-4607	449	3	f(x1	f(x1	ADJ
ap-4607	449	4	,	,	PUNCT
ap-4607	449	5	x2	x2	PROPN
ap-4607	449	6	)	)	PUNCT
ap-4607	449	7	deg	deg	PROPN
ap-4607	449	8	g(x1	g(x1	PROPN
ap-4607	449	9	,	,	PUNCT
ap-4607	449	10	x2	x2	PROPN
ap-4607	449	11	)	)	PUNCT
ap-4607	449	12	=	=	PUNCT
ap-4607	449	13	nm	nm	PROPN
ap-4607	449	14	.	.	PUNCT
ap-4607	449	15	(	(	PUNCT
ap-4607	449	16	49	49	NUM
ap-4607	449	17	)	)	PUNCT
ap-4607	449	18	for	for	ADP
ap-4607	449	19	the	the	DET
ap-4607	449	20	proof	proof	NOUN
ap-4607	449	21	see	see	VERB
ap-4607	449	22	,	,	PUNCT
ap-4607	449	23	e.g.	e.g.	ADV
ap-4607	449	24	,	,	PUNCT
ap-4607	449	25	[	[	X
ap-4607	449	26	12	12	NUM
ap-4607	449	27	]	]	PUNCT
ap-4607	449	28	.	.	PUNCT
ap-4607	450	1	theorem	theorem	PROPN
ap-4607	450	2	5a	5a	NUM
ap-4607	450	3	.	.	PUNCT
ap-4607	451	1	the	the	DET
ap-4607	451	2	polynoms	polynom	NOUN
ap-4607	451	3	f1(x1	f1(x1	NOUN
ap-4607	451	4	,	,	PUNCT
ap-4607	451	5	x2	x2	PROPN
ap-4607	451	6	)	)	PUNCT
ap-4607	451	7	,	,	PUNCT
ap-4607	451	8	f2(x1	f2(x1	X
ap-4607	451	9	,	,	PUNCT
ap-4607	451	10	x2	x2	PROPN
ap-4607	451	11	)	)	PUNCT
ap-4607	451	12	have	have	VERB
ap-4607	451	13	a	a	DET
ap-4607	451	14	non	non	ADJ
ap-4607	451	15	-	-	ADJ
ap-4607	451	16	trivial	trivial	ADJ
ap-4607	451	17	common	common	ADJ
ap-4607	451	18	component	component	NOUN
ap-4607	451	19	if	if	SCONJ
ap-4607	451	20	and	and	CCONJ
ap-4607	451	21	only	only	ADV
ap-4607	451	22	if	if	SCONJ
ap-4607	451	23	r1(f1(x1	r1(f1(x1	NUM
ap-4607	451	24	,	,	PUNCT
ap-4607	451	25	x2	x2	PROPN
ap-4607	451	26	)	)	PUNCT
ap-4607	451	27	,	,	PUNCT
ap-4607	451	28	f1(x1	f1(x1	NOUN
ap-4607	451	29	,	,	PUNCT
ap-4607	451	30	x2	x2	PROPN
ap-4607	451	31	)	)	PUNCT
ap-4607	451	32	)	)	PUNCT
ap-4607	451	33	or	or	CCONJ
ap-4607	451	34	r2(f1(x1	r2(f1(x1	X
ap-4607	451	35	,	,	PUNCT
ap-4607	451	36	x2	x2	PROPN
ap-4607	451	37	)	)	PUNCT
ap-4607	451	38	,	,	PUNCT
ap-4607	451	39	f1(x1	f1(x1	NOUN
ap-4607	451	40	,	,	PUNCT
ap-4607	451	41	x2	x2	PROPN
ap-4607	451	42	)	)	PUNCT
ap-4607	451	43	)	)	PUNCT
ap-4607	451	44	.	.	PUNCT
ap-4607	452	1	for	for	SCONJ
ap-4607	452	2	the	the	DET
ap-4607	452	3	proof	proof	NOUN
ap-4607	452	4	see	see	VERB
ap-4607	452	5	,	,	PUNCT
ap-4607	452	6	e.g.	e.g.	ADV
ap-4607	452	7	,	,	PUNCT
ap-4607	452	8	[	[	X
ap-4607	452	9	12	12	NUM
ap-4607	452	10	]	]	PUNCT
ap-4607	452	11	.	.	PUNCT
ap-4607	453	1	definition	definition	NOUN
ap-4607	453	2	5a	5a	NUM
ap-4607	453	3	.	.	PUNCT
ap-4607	454	1	a	a	DET
ap-4607	454	2	formal	formal	ADJ
ap-4607	454	3	sum	sum	NOUN
ap-4607	454	4	g	g	PROPN
ap-4607	454	5	=	=	SYM
ap-4607	454	6	g(x1	g(x1	NOUN
ap-4607	454	7	,	,	PUNCT
ap-4607	454	8	x2	x2	PROPN
ap-4607	454	9	)	)	PUNCT
ap-4607	454	10	of	of	ADP
ap-4607	454	11	the	the	DET
ap-4607	454	12	form	form	NOUN
ap-4607	454	13	g	g	NOUN
ap-4607	454	14	=	=	PUNCT
ap-4607	454	15	∑i+j≤n	∑i+j≤n	X
ap-4607	454	16	i	i	PRON
ap-4607	454	17	,	,	PUNCT
ap-4607	454	18	j=0	j=0	PROPN
ap-4607	454	19	gijx	gijx	PROPN
ap-4607	455	1	i	i	PRON
ap-4607	455	2	1x	1x	PROPN
ap-4607	455	3	j	j	NOUN
ap-4607	455	4	2	2	NUM
ap-4607	455	5	is	be	AUX
ap-4607	455	6	called	call	VERB
ap-4607	455	7	a	a	DET
ap-4607	455	8	polynomial	polynomial	ADJ
ap-4607	455	9	nform	nform	NOUN
ap-4607	455	10	of	of	ADP
ap-4607	455	11	variables	variable	NOUN
ap-4607	455	12	x1	x1	PROPN
ap-4607	455	13	,	,	PUNCT
ap-4607	455	14	x2	x2	PROPN
ap-4607	455	15	over	over	ADP
ap-4607	455	16	the	the	DET
ap-4607	455	17	field	field	NOUN
ap-4607	455	18	k.	k.	PROPN
ap-4607	456	1	that	that	ADV
ap-4607	456	2	is	be	AUX
ap-4607	456	3	,	,	PUNCT
ap-4607	456	4	g	g	PROPN
ap-4607	456	5	is	be	AUX
ap-4607	456	6	a	a	DET
ap-4607	456	7	polynomial	polynomial	NOUN
ap-4607	456	8	of	of	ADP
ap-4607	456	9	degree	degree	NOUN
ap-4607	456	10	n	n	NOUN
ap-4607	456	11	in	in	ADP
ap-4607	456	12	variables	variable	NOUN
ap-4607	456	13	x1	x1	PROPN
ap-4607	456	14	,	,	PUNCT
ap-4607	456	15	x2	x2	PROPN
ap-4607	456	16	with	with	ADP
ap-4607	456	17	indefinite	indefinite	ADJ
ap-4607	456	18	coefficients	coefficient	NOUN
ap-4607	456	19	gij	gij	ADJ
ap-4607	456	20	in	in	ADP
ap-4607	456	21	the	the	DET
ap-4607	456	22	field	field	NOUN
ap-4607	456	23	k.	k.	PUNCT
ap-4607	457	1	the	the	DET
ap-4607	457	2	expression	expression	NOUN
ap-4607	457	3	“	"	PUNCT
ap-4607	457	4	the	the	DET
ap-4607	457	5	function	function	NOUN
ap-4607	457	6	will	will	AUX
ap-4607	457	7	be	be	AUX
ap-4607	457	8	sought	seek	VERB
ap-4607	457	9	in	in	ADP
ap-4607	457	10	the	the	DET
ap-4607	457	11	form	form	NOUN
ap-4607	457	12	of	of	ADP
ap-4607	457	13	a	a	DET
ap-4607	457	14	n	n	CCONJ
ap-4607	457	15	-	-	PUNCT
ap-4607	457	16	form	form	NOUN
ap-4607	457	17	”	"	PUNCT
ap-4607	457	18	is	be	AUX
ap-4607	457	19	usually	usually	ADV
ap-4607	457	20	understood	understand	VERB
ap-4607	457	21	as	as	ADP
ap-4607	457	22	a	a	DET
ap-4607	457	23	procedure	procedure	NOUN
ap-4607	457	24	for	for	ADP
ap-4607	457	25	determining	determine	VERB
ap-4607	457	26	the	the	DET
ap-4607	457	27	undetermined	undetermined	ADJ
ap-4607	457	28	coefficients	coefficient	NOUN
ap-4607	457	29	of	of	ADP
ap-4607	457	30	a	a	DET
ap-4607	457	31	given	give	VERB
ap-4607	457	32	n	n	CCONJ
ap-4607	457	33	-	-	PUNCT
ap-4607	457	34	form	form	NOUN
ap-4607	457	35	.	.	PUNCT
ap-4607	458	1	theorem	theorem	VERB
ap-4607	458	2	6a	6a	NOUN
ap-4607	458	3	(	(	PUNCT
ap-4607	458	4	criterion	criterion	NOUN
ap-4607	458	5	of	of	ADP
ap-4607	458	6	non	non	ADJ
ap-4607	458	7	-	-	NOUN
ap-4607	458	8	decomposability	decomposability	NOUN
ap-4607	458	9	)	)	PUNCT
ap-4607	458	10	.	.	PUNCT
ap-4607	459	1	let	let	VERB
ap-4607	459	2	f	f	PRON
ap-4607	459	3	be	be	AUX
ap-4607	459	4	a	a	DET
ap-4607	459	5	polynomial	polynomial	NOUN
ap-4607	459	6	in	in	ADP
ap-4607	459	7	the	the	DET
ap-4607	459	8	variables	variable	NOUN
ap-4607	459	9	over	over	ADP
ap-4607	459	10	the	the	DET
ap-4607	459	11	field	field	NOUN
ap-4607	459	12	of	of	ADP
ap-4607	459	13	complex	complex	ADJ
ap-4607	459	14	numbers	number	NOUN
ap-4607	459	15	and	and	CCONJ
ap-4607	459	16	.	.	PUNCT
ap-4607	460	1	let	let	VERB
ap-4607	460	2	,	,	PUNCT
ap-4607	460	3	and	and	CCONJ
ap-4607	460	4	let	let	VERB
ap-4607	460	5	be	be	AUX
ap-4607	460	6	the	the	DET
ap-4607	460	7	-form	-form	NOUN
ap-4607	460	8	of	of	ADP
ap-4607	460	9	the	the	DET
ap-4607	460	10	variables	variable	NOUN
ap-4607	460	11	x1	x1	PROPN
ap-4607	460	12	and	and	CCONJ
ap-4607	460	13	x2	x2	PROPN
ap-4607	460	14	over	over	ADP
ap-4607	460	15	the	the	DET
ap-4607	460	16	field	field	NOUN
ap-4607	460	17	of	of	ADP
ap-4607	460	18	complex	complex	ADJ
ap-4607	460	19	numbers	number	NOUN
ap-4607	460	20	.	.	PUNCT
ap-4607	461	1	let	let	VERB
ap-4607	461	2	and	and	CCONJ
ap-4607	461	3	be	be	AUX
ap-4607	461	4	the	the	DET
ap-4607	461	5	resultants	resultant	NOUN
ap-4607	461	6	with	with	ADP
ap-4607	461	7	respect	respect	NOUN
ap-4607	461	8	to	to	ADP
ap-4607	461	9	the	the	DET
ap-4607	461	10	variables	variable	NOUN
ap-4607	461	11	and	and	CCONJ
ap-4607	461	12	respectively	respectively	ADV
ap-4607	461	13	.	.	PUNCT
ap-4607	462	1	the	the	DET
ap-4607	462	2	polynomial	polynomial	ADJ
ap-4607	462	3	f	f	PROPN
ap-4607	462	4	is	be	AUX
ap-4607	462	5	not	not	PART
ap-4607	462	6	decomposable	decomposable	ADJ
ap-4607	462	7	if	if	SCONJ
ap-4607	462	8	and	and	CCONJ
ap-4607	462	9	only	only	ADV
ap-4607	462	10	if	if	SCONJ
ap-4607	462	11	the	the	DET
ap-4607	462	12	systems	system	NOUN
ap-4607	462	13	of	of	ADP
ap-4607	462	14	equations	equation	NOUN
ap-4607	462	15	∂i	∂i	PROPN
ap-4607	462	16	∂xi	∂xi	PROPN
ap-4607	462	17	1	1	NUM
ap-4607	462	18	r1	r1	PROPN
ap-4607	462	19	(	(	PUNCT
ap-4607	462	20	f	f	PROPN
ap-4607	462	21	(	(	PUNCT
ap-4607	462	22	x1	x1	PROPN
ap-4607	462	23	,	,	PUNCT
ap-4607	462	24	x2	x2	PROPN
ap-4607	462	25	)	)	PUNCT
ap-4607	462	26	,	,	PUNCT
ap-4607	462	27	g(x1	g(x1	NOUN
ap-4607	462	28	,	,	PUNCT
ap-4607	462	29	x2	x2	PROPN
ap-4607	462	30	)	)	PUNCT
ap-4607	462	31	)	)	PUNCT
ap-4607	463	1	=	=	SYM
ap-4607	463	2	0	0	NUM
ap-4607	463	3	,	,	PUNCT
ap-4607	463	4	∂i	∂i	PROPN
ap-4607	463	5	∂xi	∂xi	PROPN
ap-4607	463	6	2	2	NUM
ap-4607	463	7	r2	r2	NOUN
ap-4607	463	8	(	(	PUNCT
ap-4607	463	9	f	f	PROPN
ap-4607	463	10	(	(	PUNCT
ap-4607	463	11	x1	x1	PROPN
ap-4607	463	12	,	,	PUNCT
ap-4607	463	13	x2	x2	PROPN
ap-4607	463	14	)	)	PUNCT
ap-4607	463	15	,	,	PUNCT
ap-4607	463	16	g(x1	g(x1	NOUN
ap-4607	463	17	,	,	PUNCT
ap-4607	463	18	x2	x2	PROPN
ap-4607	463	19	)	)	PUNCT
ap-4607	463	20	)	)	PUNCT
ap-4607	464	1	=	=	PUNCT
ap-4607	464	2	0	0	NUM
ap-4607	464	3	,	,	PUNCT
ap-4607	464	4	(	(	PUNCT
ap-4607	464	5	50	50	NUM
ap-4607	464	6	)	)	PUNCT
ap-4607	464	7	where	where	SCONJ
ap-4607	464	8	i	i	PRON
ap-4607	464	9	=	=	NOUN
ap-4607	464	10	1	1	NUM
ap-4607	464	11	,	,	PUNCT
ap-4607	464	12	...	...	PUNCT
ap-4607	464	13	,	,	PUNCT
ap-4607	464	14	m	m	PROPN
ap-4607	464	15	,	,	PUNCT
ap-4607	464	16	m	m	VERB
ap-4607	464	17	=	=	SYM
ap-4607	464	18	degr1(f	degr1(f	PROPN
ap-4607	464	19	,	,	PUNCT
ap-4607	464	20	g	g	NOUN
ap-4607	464	21	)	)	PUNCT
ap-4607	464	22	,	,	PUNCT
ap-4607	464	23	have	have	VERB
ap-4607	464	24	only	only	ADV
ap-4607	464	25	zero	zero	NUM
ap-4607	464	26	solutions	solution	NOUN
ap-4607	464	27	.	.	PUNCT
ap-4607	465	1	the	the	DET
ap-4607	465	2	system	system	NOUN
ap-4607	465	3	(	(	PUNCT
ap-4607	465	4	50	50	NUM
ap-4607	465	5	)	)	PUNCT
ap-4607	465	6	is	be	AUX
ap-4607	465	7	considered	consider	VERB
ap-4607	465	8	with	with	ADP
ap-4607	465	9	respect	respect	NOUN
ap-4607	465	10	to	to	ADP
ap-4607	465	11	the	the	DET
ap-4607	465	12	undetermined	undetermined	ADJ
ap-4607	465	13	coefficients	coefficient	NOUN
ap-4607	465	14	gij	gij	ADJ
ap-4607	465	15	n	n	CCONJ
ap-4607	465	16	-	-	PUNCT
ap-4607	465	17	formg	formg	NOUN
ap-4607	465	18	,	,	PUNCT
ap-4607	465	19	as	as	ADP
ap-4607	465	20	with	with	ADP
ap-4607	465	21	respect	respect	NOUN
ap-4607	465	22	to	to	ADP
ap-4607	465	23	unknown	unknown	ADJ
ap-4607	465	24	variables	variable	NOUN
ap-4607	465	25	.	.	PUNCT
ap-4607	466	1	for	for	SCONJ
ap-4607	466	2	the	the	DET
ap-4607	466	3	proof	proof	NOUN
ap-4607	466	4	see	see	VERB
ap-4607	466	5	,	,	PUNCT
ap-4607	466	6	e.g.	e.g.	ADV
ap-4607	466	7	,	,	PUNCT
ap-4607	466	8	[	[	X
ap-4607	466	9	7	7	NUM
ap-4607	466	10	]	]	PUNCT
ap-4607	466	11	.	.	PUNCT
ap-4607	467	1	the	the	DET
ap-4607	467	2	theorem	theorem	NOUN
ap-4607	467	3	admits	admit	VERB
ap-4607	467	4	a	a	DET
ap-4607	467	5	generalization	generalization	NOUN
ap-4607	467	6	to	to	ADP
ap-4607	467	7	the	the	DET
ap-4607	467	8	case	case	NOUN
ap-4607	467	9	of	of	ADP
ap-4607	467	10	systems	system	NOUN
ap-4607	467	11	of	of	ADP
ap-4607	467	12	equations	equation	NOUN
ap-4607	467	13	of	of	ADP
ap-4607	467	14	several	several	ADJ
ap-4607	467	15	variables	variable	NOUN
ap-4607	467	16	see	see	VERB
ap-4607	467	17	[	[	X
ap-4607	467	18	7	7	NUM
ap-4607	467	19	,	,	PUNCT
ap-4607	467	20	16	16	NUM
ap-4607	467	21	]	]	PUNCT
ap-4607	467	22	.	.	PUNCT
ap-4607	468	1	the	the	DET
ap-4607	468	2	criterion	criterion	NOUN
ap-4607	468	3	was	be	AUX
ap-4607	468	4	formulated	formulate	VERB
ap-4607	468	5	and	and	CCONJ
ap-4607	468	6	proved	prove	VERB
ap-4607	468	7	by	by	ADP
ap-4607	468	8	the	the	DET
ap-4607	468	9	authors	author	NOUN
ap-4607	468	10	earlier	early	ADV
ap-4607	468	11	[	[	X
ap-4607	468	12	7	7	NUM
ap-4607	468	13	]	]	PUNCT
ap-4607	468	14	.	.	PUNCT
ap-4607	469	1	references	reference	NOUN
ap-4607	469	2	[	[	X
ap-4607	469	3	1	1	X
ap-4607	469	4	]	]	PUNCT
ap-4607	469	5	e.	e.	PROPN
ap-4607	469	6	y.	y.	PROPN
ap-4607	469	7	bannikova	bannikova	PROPN
ap-4607	469	8	,	,	PUNCT
ap-4607	469	9	a.	a.	NOUN
ap-4607	469	10	t.	t.	PROPN
ap-4607	469	11	kotvytskiy	kotvytskiy	PROPN
ap-4607	469	12	.	.	PUNCT
ap-4607	470	1	three	three	NUM
ap-4607	470	2	einstein	einstein	PROPN
ap-4607	470	3	rings	ring	NOUN
ap-4607	470	4	:	:	PUNCT
ap-4607	470	5	explicit	explicit	ADJ
ap-4607	470	6	solution	solution	NOUN
ap-4607	470	7	and	and	CCONJ
ap-4607	470	8	numerical	numerical	PROPN
ap-4607	470	9	simulation	simulation	PROPN
ap-4607	470	10	.	.	PUNCT
ap-4607	471	1	mnras	mnras	PROPN
ap-4607	471	2	445(4):4435–4442	445(4):4435–4442	PROPN
ap-4607	471	3	,	,	PUNCT
ap-4607	471	4	2014	2014	NUM
ap-4607	471	5	.	.	PUNCT
ap-4607	472	1	doi:10.1093	doi:10.1093	NOUN
ap-4607	472	2	/	/	SYM
ap-4607	472	3	mnras	mnras	PROPN
ap-4607	472	4	/	/	SYM
ap-4607	472	5	stu2068	stu2068	ADJ
ap-4607	472	6	.	.	PUNCT
ap-4607	473	1	[	[	X
ap-4607	473	2	2	2	NUM
ap-4607	473	3	]	]	PUNCT
ap-4607	473	4	a.	a.	NOUN
ap-4607	473	5	t.	t.	PROPN
ap-4607	473	6	kotvytskiy	kotvytskiy	PROPN
ap-4607	473	7	.	.	PUNCT
ap-4607	474	1	gravitational	gravitational	ADJ
ap-4607	474	2	lensing	lense	VERB
ap-4607	474	3	by	by	ADP
ap-4607	474	4	straight	straight	ADJ
ap-4607	474	5	cosmic	cosmic	ADJ
ap-4607	474	6	strings	string	NOUN
ap-4607	474	7	.	.	PUNCT
ap-4607	475	1	tmp	tmp	VERB
ap-4607	475	2	184(1):160–174	184(1):160–174	PRON
ap-4607	475	3	,	,	PUNCT
ap-4607	475	4	2015	2015	NUM
ap-4607	475	5	.	.	PUNCT
ap-4607	476	1	doi:10.1007	doi:10.1007	VERB
ap-4607	476	2	/	/	SYM
ap-4607	476	3	s11232	s11232	NOUN
ap-4607	476	4	-	-	PUNCT
ap-4607	476	5	015	015	NUM
ap-4607	476	6	-	-	PUNCT
ap-4607	476	7	0315	0315	NUM
ap-4607	476	8	-	-	PUNCT
ap-4607	476	9	x.	x.	NOUN
ap-4607	477	1	[	[	X
ap-4607	477	2	3	3	NUM
ap-4607	477	3	]	]	PUNCT
ap-4607	477	4	a.	a.	NOUN
ap-4607	477	5	cassan	cassan	PROPN
ap-4607	477	6	.	.	PUNCT
ap-4607	478	1	an	an	DET
ap-4607	478	2	alternative	alternative	ADJ
ap-4607	478	3	parameterisation	parameterisation	NOUN
ap-4607	478	4	for	for	ADP
ap-4607	478	5	binary	binary	ADJ
ap-4607	478	6	-	-	PUNCT
ap-4607	478	7	lens	lens	NOUN
ap-4607	478	8	caustic	caustic	ADJ
ap-4607	478	9	-	-	PUNCT
ap-4607	478	10	crossing	cross	VERB
ap-4607	478	11	events	event	NOUN
ap-4607	478	12	.	.	PUNCT
ap-4607	479	1	astronomy	astronomy	NOUN
ap-4607	479	2	and	and	CCONJ
ap-4607	479	3	astrophysics	astrophysic	NOUN
ap-4607	479	4	491(2):587–595	491(2):587–595	PROPN
ap-4607	479	5	,	,	PUNCT
ap-4607	479	6	2008	2008	NUM
ap-4607	479	7	.	.	PUNCT
ap-4607	480	1	doi:10.1051/0004	doi:10.1051/0004	VERB
ap-4607	480	2	-	-	PUNCT
ap-4607	480	3	6361:200809795	6361:200809795	NUM
ap-4607	480	4	.	.	PUNCT
ap-4607	481	1	[	[	X
ap-4607	481	2	4	4	NUM
ap-4607	481	3	]	]	X
ap-4607	481	4	p.	p.	NOUN
ap-4607	481	5	schneider	schneider	PROPN
ap-4607	481	6	,	,	PUNCT
ap-4607	481	7	a.	a.	NOUN
ap-4607	481	8	weiss	weiss	PROPN
ap-4607	481	9	.	.	PUNCT
ap-4607	482	1	the	the	DET
ap-4607	482	2	two	two	NUM
ap-4607	482	3	-	-	PUNCT
ap-4607	482	4	point	point	NOUN
ap-4607	482	5	-	-	PUNCT
ap-4607	482	6	mass	mass	NOUN
ap-4607	482	7	lens	len	NOUN
ap-4607	482	8	:	:	PUNCT
ap-4607	482	9	detailed	detailed	ADJ
ap-4607	482	10	investigation	investigation	NOUN
ap-4607	482	11	of	of	ADP
ap-4607	482	12	a	a	DET
ap-4607	482	13	special	special	ADJ
ap-4607	482	14	asymmetric	asymmetric	ADJ
ap-4607	482	15	gravitational	gravitational	ADJ
ap-4607	482	16	lens	len	NOUN
ap-4607	482	17	.	.	PUNCT
ap-4607	483	1	astronomy	astronomy	NOUN
ap-4607	483	2	and	and	CCONJ
ap-4607	483	3	astrophysics	astrophysic	NOUN
ap-4607	483	4	164(2):237–259	164(2):237–259	NUM
ap-4607	483	5	,	,	PUNCT
ap-4607	483	6	1986	1986	NUM
ap-4607	483	7	.	.	PUNCT
ap-4607	484	1	[	[	X
ap-4607	484	2	5	5	NUM
ap-4607	484	3	]	]	PUNCT
ap-4607	484	4	a.	a.	NOUN
ap-4607	484	5	t.	t.	PROPN
ap-4607	484	6	kotvytskiy	kotvytskiy	PROPN
ap-4607	484	7	,	,	PUNCT
ap-4607	484	8	s.	s.	PROPN
ap-4607	484	9	d.	d.	PROPN
ap-4607	484	10	bronza	bronza	PROPN
ap-4607	484	11	,	,	PUNCT
ap-4607	484	12	k.	k.	PROPN
ap-4607	484	13	d.	d.	PROPN
ap-4607	484	14	nerushenko	nerushenko	PROPN
ap-4607	484	15	,	,	PUNCT
ap-4607	484	16	v.	v.	PROPN
ap-4607	484	17	y.	y.	PROPN
ap-4607	484	18	shablenko	shablenko	PROPN
ap-4607	484	19	.	.	PUNCT
ap-4607	485	1	matematychnii	matematychnii	NOUN
ap-4607	485	2	zmist	zmist	VERB
ap-4607	485	3	kiltsia	kiltsia	PROPN
ap-4607	485	4	einshteina	einshteina	NOUN
ap-4607	485	5	ta	ta	ADP
ap-4607	485	6	umovy	umovy	PROPN
ap-4607	485	7	yogo	yogo	PROPN
ap-4607	485	8	vynyknennja	vynyknennja	X
ap-4607	485	9	.	.	PUNCT
ap-4607	486	1	doslidzhennya	doslidzhennya	VERB
ap-4607	486	2	uzahal’nenyh	uzahal’nenyh	PROPN
ap-4607	486	3	umov[in	umov[in	PROPN
ap-4607	486	4	ukrainian	ukrainian	PROPN
ap-4607	486	5	]	]	X
ap-4607	486	6	.	.	PUNCT
ap-4607	487	1	in	in	ADP
ap-4607	487	2	zbirnyk	zbirnyk	PROPN
ap-4607	487	3	naukovyx	naukovyx	PROPN
ap-4607	487	4	prac	prac	PROPN
ap-4607	487	5	’	'	PUNCT
ap-4607	487	6	vi	vi	PROPN
ap-4607	487	7	-	-	PUNCT
ap-4607	487	8	i	i	PRON
ap-4607	487	9	mizhrehional’noi	mizhrehional’noi	PROPN
ap-4607	487	10	naukovo	naukovo	PROPN
ap-4607	487	11	-	-	PUNCT
ap-4607	487	12	praktychnoi	praktychnoi	PROPN
ap-4607	487	13	konferencii	konferencii	NOUN
ap-4607	487	14	"	"	PUNCT
ap-4607	487	15	astronomiya	astronomiya	VERB
ap-4607	487	16	i	i	PRON
ap-4607	487	17	s’ogodennya	s’ogodennya	VERB
ap-4607	487	18	"	"	PUNCT
ap-4607	487	19	,	,	PUNCT
ap-4607	487	20	pp	pp	ADJ
ap-4607	487	21	.	.	PUNCT
ap-4607	488	1	198–213	198–213	NUM
ap-4607	488	2	.	.	PUNCT
ap-4607	488	3	vinnytsia	vinnytsia	NOUN
ap-4607	488	4	,	,	PUNCT
ap-4607	488	5	2017	2017	NUM
ap-4607	488	6	.	.	PUNCT
ap-4607	489	1	[	[	X
ap-4607	489	2	6	6	NUM
ap-4607	489	3	]	]	PUNCT
ap-4607	489	4	a.	a.	NOUN
ap-4607	489	5	t.	t.	PROPN
ap-4607	489	6	kotvytskiy	kotvytskiy	PROPN
ap-4607	489	7	,	,	PUNCT
ap-4607	489	8	s.	s.	PROPN
ap-4607	489	9	d.	d.	PROPN
ap-4607	489	10	bronza	bronza	PROPN
ap-4607	489	11	,	,	PUNCT
ap-4607	489	12	s.	s.	PROPN
ap-4607	489	13	r.	r.	PROPN
ap-4607	489	14	vovk	vovk	PROPN
ap-4607	489	15	.	.	PUNCT
ap-4607	490	1	estimating	estimate	VERB
ap-4607	490	2	the	the	DET
ap-4607	490	3	number	number	NOUN
ap-4607	490	4	of	of	ADP
ap-4607	490	5	images	image	NOUN
ap-4607	490	6	n	n	CCONJ
ap-4607	490	7	-	-	PUNCT
ap-4607	490	8	point	point	NOUN
ap-4607	490	9	gravitational	gravitational	ADJ
ap-4607	490	10	lenses	lense	NOUN
ap-4607	490	11	algebraic	algebraic	ADJ
ap-4607	490	12	geometry	geometry	NOUN
ap-4607	490	13	methods	method	NOUN
ap-4607	490	14	.	.	PUNCT
ap-4607	491	1	visnyk	visnyk	PROPN
ap-4607	491	2	khnu	khnu	PROPN
ap-4607	491	3	,	,	PUNCT
ap-4607	491	4	serija	serija	VERB
ap-4607	491	5	"	"	PUNCT
ap-4607	491	6	fizyka	fizyka	ADJ
ap-4607	491	7	"	"	PUNCT
ap-4607	491	8	24:55–59	24:55–59	NUM
ap-4607	491	9	,	,	PUNCT
ap-4607	491	10	2016	2016	NUM
ap-4607	491	11	.	.	PUNCT
ap-4607	492	1	[	[	X
ap-4607	492	2	7	7	X
ap-4607	492	3	]	]	PUNCT
ap-4607	492	4	s.	s.	PROPN
ap-4607	492	5	d.	d.	PROPN
ap-4607	492	6	bronza	bronza	PROPN
ap-4607	492	7	,	,	PUNCT
ap-4607	492	8	a.	a.	NOUN
ap-4607	492	9	t.	t.	PROPN
ap-4607	492	10	kotvytskiy	kotvytskiy	PROPN
ap-4607	492	11	.	.	PUNCT
ap-4607	493	1	mathematical	mathematical	ADJ
ap-4607	493	2	bases	basis	NOUN
ap-4607	493	3	of	of	ADP
ap-4607	493	4	the	the	DET
ap-4607	493	5	theory	theory	NOUN
ap-4607	493	6	of	of	ADP
ap-4607	493	7	n	n	CCONJ
ap-4607	493	8	-	-	PUNCT
ap-4607	493	9	point	point	NOUN
ap-4607	493	10	gravitational	gravitational	ADJ
ap-4607	493	11	lenses	lense	NOUN
ap-4607	493	12	.	.	PUNCT
ap-4607	494	1	part	part	NOUN
ap-4607	494	2	1	1	NUM
ap-4607	494	3	.	.	PUNCT
ap-4607	495	1	elements	element	NOUN
ap-4607	495	2	of	of	ADP
ap-4607	495	3	algebraic	algebraic	ADJ
ap-4607	495	4	geometry	geometry	NOUN
ap-4607	495	5	.	.	PUNCT
ap-4607	496	1	visnyk	visnyk	PROPN
ap-4607	496	2	khnu	khnu	PROPN
ap-4607	496	3	,	,	PUNCT
ap-4607	496	4	seriya	seriya	NOUN
ap-4607	496	5	"	"	PUNCT
ap-4607	496	6	fizyka	fizyka	ADJ
ap-4607	496	7	"	"	PUNCT
ap-4607	496	8	26:6–32	26:6–32	NUM
ap-4607	496	9	,	,	PUNCT
ap-4607	496	10	2017	2017	NUM
ap-4607	496	11	.	.	PUNCT
ap-4607	497	1	[	[	X
ap-4607	497	2	8	8	NUM
ap-4607	497	3	]	]	PUNCT
ap-4607	497	4	a.	a.	NOUN
ap-4607	497	5	t.	t.	PROPN
ap-4607	497	6	kotvytskiy	kotvytskiy	PROPN
ap-4607	497	7	,	,	PUNCT
ap-4607	497	8	s.	s.	PROPN
ap-4607	497	9	d.	d.	PROPN
ap-4607	497	10	bronza	bronza	PROPN
ap-4607	497	11	.	.	PUNCT
ap-4607	498	1	mathematical	mathematical	ADJ
ap-4607	498	2	bases	basis	NOUN
ap-4607	498	3	of	of	ADP
ap-4607	498	4	the	the	DET
ap-4607	498	5	theory	theory	NOUN
ap-4607	498	6	of	of	ADP
ap-4607	498	7	n	n	CCONJ
ap-4607	498	8	-	-	PUNCT
ap-4607	498	9	point	point	NOUN
ap-4607	498	10	lenses	lense	NOUN
ap-4607	498	11	.	.	PUNCT
ap-4607	499	1	odessa	odessa	ADJ
ap-4607	499	2	astronomical	astronomical	ADJ
ap-4607	499	3	publications	publication	NOUN
ap-4607	499	4	29:31–33	29:31–33	NUM
ap-4607	499	5	,	,	PUNCT
ap-4607	499	6	2016	2016	NUM
ap-4607	499	7	.	.	PUNCT
ap-4607	500	1	doi:0.18524/1810	doi:0.18524/1810	PROPN
ap-4607	500	2	-	-	PUNCT
ap-4607	500	3	4215.2016.29.84958	4215.2016.29.84958	PROPN
ap-4607	500	4	.	.	PUNCT
ap-4607	501	1	[	[	X
ap-4607	501	2	9	9	NUM
ap-4607	501	3	]	]	PUNCT
ap-4607	501	4	a.	a.	NOUN
ap-4607	501	5	f.	f.	PROPN
ap-4607	501	6	zakharov	zakharov	PROPN
ap-4607	501	7	.	.	PUNCT
ap-4607	502	1	gravitacionnye	gravitacionnye	NOUN
ap-4607	502	2	linzy	linzy	NOUN
ap-4607	503	1	i	i	PRON
ap-4607	503	2	mikrolinzy	mikrolinzy	ADV
ap-4607	504	1	[	[	X
ap-4607	504	2	in	in	ADP
ap-4607	504	3	russian	russian	NOUN
ap-4607	504	4	]	]	PUNCT
ap-4607	504	5	.	.	PUNCT
ap-4607	505	1	janus	janus	PROPN
ap-4607	505	2	-	-	PUNCT
ap-4607	505	3	k	k	PROPN
ap-4607	505	4	,	,	PUNCT
ap-4607	505	5	moscow	moscow	PROPN
ap-4607	505	6	,	,	PUNCT
ap-4607	505	7	1997	1997	NUM
ap-4607	505	8	.	.	PUNCT
ap-4607	506	1	[	[	X
ap-4607	506	2	10	10	NUM
ap-4607	506	3	]	]	X
ap-4607	506	4	p.	p.	NOUN
ap-4607	506	5	schneider	schneider	PROPN
ap-4607	506	6	,	,	PUNCT
ap-4607	506	7	j.	j.	PROPN
ap-4607	506	8	ehlers	ehlers	PROPN
ap-4607	506	9	,	,	PUNCT
ap-4607	506	10	e.	e.	PROPN
ap-4607	506	11	falco	falco	PROPN
ap-4607	506	12	.	.	PUNCT
ap-4607	507	1	gravitational	gravitational	ADJ
ap-4607	507	2	lenses	lense	NOUN
ap-4607	507	3	.	.	PUNCT
ap-4607	508	1	second	second	ADJ
ap-4607	508	2	printing	printing	NOUN
ap-4607	508	3	.	.	PUNCT
ap-4607	508	4	springer	springer	NOUN
ap-4607	508	5	-	-	PUNCT
ap-4607	508	6	verlag	verlag	PROPN
ap-4607	508	7	berlin	berlin	PROPN
ap-4607	508	8	heidelberg	heidelberg	PROPN
ap-4607	508	9	,	,	PUNCT
ap-4607	508	10	1999	1999	NUM
ap-4607	508	11	.	.	PUNCT
ap-4607	509	1	[	[	X
ap-4607	509	2	11	11	NUM
ap-4607	509	3	]	]	X
ap-4607	509	4	i.	i.	PROPN
ap-4607	509	5	v.	v.	PROPN
ap-4607	509	6	arzhantcev	arzhantcev	PROPN
ap-4607	509	7	.	.	PUNCT
ap-4607	510	1	bazisy	bazisy	ADJ
ap-4607	510	2	grebnera	grebnera	NOUN
ap-4607	511	1	i	i	PRON
ap-4607	511	2	sistemy	sistemy	NOUN
ap-4607	511	3	algebraicheskih	algebraicheskih	NOUN
ap-4607	511	4	uravnenij	uravnenij	VERB
ap-4607	512	1	[	[	X
ap-4607	512	2	in	in	ADP
ap-4607	512	3	russian	russian	PROPN
ap-4607	512	4	]	]	PUNCT
ap-4607	512	5	.	.	PUNCT
ap-4607	513	1	mcnmo	mcnmo	PROPN
ap-4607	513	2	,	,	PUNCT
ap-4607	513	3	moscow	moscow	PROPN
ap-4607	513	4	,	,	PUNCT
ap-4607	513	5	2003	2003	NUM
ap-4607	513	6	.	.	PUNCT
ap-4607	514	1	[	[	X
ap-4607	514	2	12	12	NUM
ap-4607	514	3	]	]	PUNCT
ap-4607	514	4	r.	r.	PROPN
ap-4607	514	5	j.	j.	PROPN
ap-4607	514	6	walker	walker	PROPN
ap-4607	514	7	.	.	PUNCT
ap-4607	515	1	algebraic	algebraic	ADJ
ap-4607	515	2	curves	curve	NOUN
ap-4607	515	3	.	.	PUNCT
ap-4607	516	1	springer	springer	NOUN
ap-4607	516	2	-	-	PUNCT
ap-4607	516	3	verlag	verlag	PROPN
ap-4607	516	4	new	new	PROPN
ap-4607	516	5	york	york	PROPN
ap-4607	516	6	,	,	PUNCT
ap-4607	516	7	1978	1978	NUM
ap-4607	516	8	.	.	PUNCT
ap-4607	517	1	[	[	X
ap-4607	517	2	13	13	NUM
ap-4607	517	3	]	]	X
ap-4607	517	4	e.	e.	PROPN
ap-4607	517	5	kalinina	kalinina	PROPN
ap-4607	517	6	,	,	PUNCT
ap-4607	517	7	a.	a.	PROPN
ap-4607	517	8	j.	j.	PROPN
ap-4607	517	9	uteshev	uteshev	PROPN
ap-4607	517	10	.	.	PUNCT
ap-4607	518	1	teoria	teoria	NOUN
ap-4607	518	2	iskluchenij	iskluchenij	NOUN
ap-4607	518	3	[	[	X
ap-4607	518	4	in	in	ADP
ap-4607	518	5	russian	russian	PROPN
ap-4607	518	6	]	]	PUNCT
ap-4607	518	7	.	.	PUNCT
ap-4607	519	1	nii	nii	PROPN
ap-4607	519	2	chimii	chimii	PROPN
ap-4607	519	3	spbgu	spbgu	PROPN
ap-4607	519	4	saint	saint	PROPN
ap-4607	519	5	petersburg	petersburg	PROPN
ap-4607	519	6	,	,	PUNCT
ap-4607	519	7	2002	2002	NUM
ap-4607	519	8	.	.	PUNCT
ap-4607	520	1	[	[	X
ap-4607	520	2	14	14	NUM
ap-4607	520	3	]	]	PUNCT
ap-4607	520	4	p.	p.	NOUN
ap-4607	520	5	v.	v.	CCONJ
ap-4607	521	1	bliokh	bliokh	ADJ
ap-4607	521	2	,	,	PUNCT
ap-4607	521	3	a.	a.	NOUN
ap-4607	521	4	a.	a.	NOUN
ap-4607	521	5	minakov	minakov	PROPN
ap-4607	521	6	.	.	PUNCT
ap-4607	522	1	gravitational	gravitational	ADJ
ap-4607	522	2	lenses	lense	NOUN
ap-4607	522	3	[	[	X
ap-4607	522	4	in	in	ADP
ap-4607	522	5	russian	russian	NOUN
ap-4607	522	6	]	]	PUNCT
ap-4607	522	7	.	.	PUNCT
ap-4607	523	1	naukova	naukova	PROPN
ap-4607	523	2	dumka	dumka	PROPN
ap-4607	523	3	,	,	PUNCT
ap-4607	523	4	kiev	kiev	PROPN
ap-4607	523	5	,	,	PUNCT
ap-4607	523	6	1989	1989	NUM
ap-4607	523	7	.	.	PUNCT
ap-4607	524	1	[	[	X
ap-4607	524	2	15	15	NUM
ap-4607	524	3	]	]	X
ap-4607	524	4	m.	m.	NOUN
ap-4607	524	5	reid	reid	PROPN
ap-4607	524	6	.	.	PUNCT
ap-4607	525	1	undergraduate	undergraduate	VERB
ap-4607	525	2	algebraic	algebraic	ADJ
ap-4607	525	3	geometry	geometry	NOUN
ap-4607	525	4	.	.	PUNCT
ap-4607	526	1	math	math	PROPN
ap-4607	526	2	inst	inst	PROPN
ap-4607	526	3	.	.	PROPN
ap-4607	526	4	,	,	PUNCT
ap-4607	526	5	university	university	PROPN
ap-4607	526	6	of	of	ADP
ap-4607	526	7	warwick	warwick	PROPN
ap-4607	526	8	,	,	PUNCT
ap-4607	526	9	2013	2013	NUM
ap-4607	526	10	.	.	PUNCT
ap-4607	527	1	[	[	X
ap-4607	527	2	16	16	NUM
ap-4607	527	3	]	]	PUNCT
ap-4607	527	4	s.	s.	PROPN
ap-4607	527	5	d.	d.	PROPN
ap-4607	527	6	bronza	bronza	PROPN
ap-4607	527	7	.	.	PUNCT
ap-4607	528	1	kriteriy	kriteriy	PROPN
ap-4607	528	2	neprivodimosti	neprivodimosti	NOUN
ap-4607	528	3	mnogochlenov	mnogochlenov	PROPN
ap-4607	528	4	ot	ot	NOUN
ap-4607	528	5	dvuh	dvuh	PROPN
ap-4607	528	6	peremennyih	peremennyih	NOUN
ap-4607	528	7	nad	nad	PROPN
ap-4607	528	8	polem	polem	NOUN
ap-4607	528	9	kompleksnih	kompleksnih	PROPN
ap-4607	528	10	chisel[in	chisel[in	PROPN
ap-4607	528	11	russian	russian	PROPN
ap-4607	528	12	]	]	X
ap-4607	528	13	.	.	PUNCT
ap-4607	529	1	in	in	ADP
ap-4607	529	2	zbirnyk	zbirnyk	PROPN
ap-4607	529	3	naukovyx	naukovyx	PROPN
ap-4607	529	4	prac	prac	PROPN
ap-4607	529	5	’	'	PUNCT
ap-4607	529	6	,	,	PUNCT
ap-4607	529	7	pp	pp	ADJ
ap-4607	529	8	.	.	PUNCT
ap-4607	530	1	114–115	114–115	NUM
ap-4607	530	2	.	.	PUNCT
ap-4607	531	1	ukrduzt	ukrduzt	NOUN
ap-4607	531	2	,	,	PUNCT
ap-4607	531	3	2016	2016	NUM
ap-4607	531	4	.	.	PUNCT
ap-4607	532	1	[	[	X
ap-4607	532	2	17	17	NUM
ap-4607	532	3	]	]	PUNCT
ap-4607	532	4	s.	s.	PROPN
ap-4607	532	5	h.	h.	PROPN
ap-4607	532	6	phie	phie	PROPN
ap-4607	532	7	.	.	PUNCT
ap-4607	533	1	infimum	infimum	ADJ
ap-4607	533	2	microlensing	microlense	VERB
ap-4607	533	3	amplification	amplification	NOUN
ap-4607	533	4	of	of	ADP
ap-4607	533	5	the	the	DET
ap-4607	533	6	maximum	maximum	ADJ
ap-4607	533	7	number	number	NOUN
ap-4607	533	8	of	of	ADP
ap-4607	533	9	images	image	NOUN
ap-4607	533	10	of	of	ADP
ap-4607	533	11	n	n	CCONJ
ap-4607	533	12	-	-	PUNCT
ap-4607	533	13	point	point	NOUN
ap-4607	533	14	lens	lens	NOUN
ap-4607	533	15	systems	system	NOUN
ap-4607	533	16	.	.	PUNCT
ap-4607	534	1	the	the	DET
ap-4607	534	2	astrophysical	astrophysical	ADJ
ap-4607	534	3	journal	journal	NOUN
ap-4607	534	4	484(1):63–69	484(1):63–69	NUM
ap-4607	534	5	,	,	PUNCT
ap-4607	534	6	1997	1997	NUM
ap-4607	534	7	.	.	PUNCT
ap-4607	535	1	doi:10.1086/304336	doi:10.1086/304336	NOUN
ap-4607	535	2	.	.	PUNCT
ap-4607	536	1	[	[	X
ap-4607	536	2	18	18	NUM
ap-4607	536	3	]	]	PUNCT
ap-4607	536	4	a.	a.	NOUN
ap-4607	536	5	t.	t.	PROPN
ap-4607	536	6	kotvytskiy	kotvytskiy	PROPN
ap-4607	536	7	,	,	PUNCT
ap-4607	536	8	s.	s.	PROPN
ap-4607	536	9	d.	d.	PROPN
ap-4607	536	10	bronza	bronza	PROPN
ap-4607	536	11	,	,	PUNCT
ap-4607	536	12	v.	v.	PROPN
ap-4607	536	13	y.	y.	PROPN
ap-4607	536	14	shablenko	shablenko	PROPN
ap-4607	536	15	.	.	PUNCT
ap-4607	537	1	correlation	correlation	NOUN
ap-4607	537	2	of	of	ADP
ap-4607	537	3	the	the	DET
ap-4607	537	4	number	number	NOUN
ap-4607	537	5	of	of	ADP
ap-4607	537	6	images	image	NOUN
ap-4607	537	7	of	of	ADP
ap-4607	537	8	an	an	DET
ap-4607	537	9	n	n	CCONJ
ap-4607	537	10	-	-	PUNCT
ap-4607	537	11	point	point	NOUN
ap-4607	537	12	gravitational	gravitational	ADJ
ap-4607	537	13	lens	lens	NOUN
ap-4607	537	14	and	and	CCONJ
ap-4607	537	15	the	the	DET
ap-4607	537	16	number	number	NOUN
ap-4607	537	17	of	of	ADP
ap-4607	537	18	solutions	solution	NOUN
ap-4607	537	19	of	of	ADP
ap-4607	537	20	its	its	PRON
ap-4607	537	21	system	system	NOUN
ap-4607	537	22	.	.	PUNCT
ap-4607	538	1	in	in	ADP
ap-4607	538	2	astronomy	astronomy	NOUN
ap-4607	538	3	and	and	CCONJ
ap-4607	538	4	beyond	beyond	ADP
ap-4607	538	5	:	:	PUNCT
ap-4607	538	6	astrophysics	astrophysic	NOUN
ap-4607	538	7	,	,	PUNCT
ap-4607	538	8	cosmology	cosmology	NOUN
ap-4607	538	9	,	,	PUNCT
ap-4607	538	10	cosmomicrophysics	cosmomicrophysic	NOUN
ap-4607	538	11	,	,	PUNCT
ap-4607	538	12	astroparticle	astroparticle	NOUN
ap-4607	538	13	physics	physics	PROPN
ap-4607	538	14	,	,	PUNCT
ap-4607	538	15	radioastronomy	radioastronomy	NOUN
ap-4607	538	16	and	and	CCONJ
ap-4607	538	17	astrobiology	astrobiology	NOUN
ap-4607	538	18	,	,	PUNCT
ap-4607	538	19	p.	p.	NOUN
ap-4607	538	20	12	12	NUM
ap-4607	538	21	.	.	PUNCT
ap-4607	539	1	odessa	odessa	ADJ
ap-4607	539	2	international	international	ADJ
ap-4607	539	3	astronomical	astronomical	ADJ
ap-4607	539	4	gamow	gamow	NOUN
ap-4607	539	5	conference	conference	NOUN
ap-4607	539	6	-	-	PUNCT
ap-4607	539	7	school	school	NOUN
ap-4607	539	8	,	,	PUNCT
ap-4607	539	9	2017	2017	NUM
ap-4607	539	10	.	.	PUNCT
ap-4607	540	1	[	[	X
ap-4607	540	2	19	19	NUM
ap-4607	540	3	]	]	X
ap-4607	540	4	b.	b.	PROPN
ap-4607	540	5	l.	l.	PROPN
ap-4607	540	6	van	van	PROPN
ap-4607	540	7	der	der	PROPN
ap-4607	540	8	waerden	waerden	ADJ
ap-4607	540	9	.	.	PUNCT
ap-4607	541	1	modern	modern	ADJ
ap-4607	541	2	algebra	algebra	PROPN
ap-4607	541	3	i	i	PRON
ap-4607	541	4	,	,	PUNCT
ap-4607	541	5	ii	ii	PROPN
ap-4607	541	6	.	.	PUNCT
ap-4607	542	1	new	new	PROPN
ap-4607	542	2	york	york	PROPN
ap-4607	542	3	:	:	PUNCT
ap-4607	542	4	frederic	frederic	PROPN
ap-4607	542	5	ungar	ungar	PROPN
ap-4607	542	6	publishing	publishing	PROPN
ap-4607	542	7	co.	co.	PROPN
ap-4607	542	8	,	,	PUNCT
ap-4607	542	9	1950	1950	NUM
ap-4607	542	10	.	.	PUNCT
ap-4607	543	1	[	[	X
ap-4607	543	2	20	20	NUM
ap-4607	543	3	]	]	PUNCT
ap-4607	543	4	s.	s.	PROPN
ap-4607	543	5	lang	lang	PROPN
ap-4607	543	6	.	.	PUNCT
ap-4607	544	1	algebra	algebra	PROPN
ap-4607	544	2	.	.	PUNCT
ap-4607	545	1	revised	revise	VERB
ap-4607	545	2	third	third	PROPN
ap-4607	545	3	edition	edition	PROPN
ap-4607	545	4	.	.	PUNCT
ap-4607	546	1	columbia	columbia	PROPN
ap-4607	546	2	university	university	PROPN
ap-4607	546	3	.	.	PUNCT
ap-4607	547	1	new	new	PROPN
ap-4607	547	2	york	york	PROPN
ap-4607	547	3	,	,	PUNCT
ap-4607	547	4	1965	1965	NUM
ap-4607	547	5	.	.	PUNCT
ap-4607	548	1	411	411	NUM
ap-4607	548	2	http://dx.doi.org/10.1093/mnras/stu2068	http://dx.doi.org/10.1093/mnras/stu2068	NOUN
ap-4607	548	3	http://dx.doi.org/10.1007/s11232-015-0315-x	http://dx.doi.org/10.1007/s11232-015-0315-x	PROPN
ap-4607	548	4	http://dx.doi.org/10.1051/0004-6361:200809795	http://dx.doi.org/10.1051/0004-6361:200809795	PROPN
ap-4607	548	5	http://dx.doi.org/0.18524/1810-4215.2016.29.84958	http://dx.doi.org/0.18524/1810-4215.2016.29.84958	PROPN
ap-4607	548	6	http://dx.doi.org/10.1086/304336	http://dx.doi.org/10.1086/304336	NOUN
ap-4607	548	7	acta	acta	PROPN
ap-4607	548	8	polytechnica	polytechnica	PROPN
ap-4607	549	1	57(6):404–411	57(6):404–411	PROPN
ap-4607	549	2	,	,	PUNCT
ap-4607	549	3	2017	2017	NUM
ap-4607	549	4	1	1	NUM
ap-4607	549	5	introduction	introduction	NOUN
ap-4607	549	6	2	2	NUM
ap-4607	549	7	the	the	DET
ap-4607	549	8	physical	physical	ADJ
ap-4607	549	9	formulation	formulation	NOUN
ap-4607	549	10	of	of	ADP
ap-4607	549	11	the	the	DET
ap-4607	549	12	problem	problem	NOUN
ap-4607	549	13	from	from	ADP
ap-4607	549	14	an	an	DET
ap-4607	549	15	algebraic	algebraic	ADJ
ap-4607	549	16	point	point	NOUN
ap-4607	549	17	of	of	ADP
ap-4607	549	18	view	view	NOUN
ap-4607	549	19	3	3	NUM
ap-4607	549	20	reduction	reduction	NOUN
ap-4607	549	21	of	of	ADP
ap-4607	549	22	the	the	DET
ap-4607	549	23	problem	problem	NOUN
ap-4607	549	24	to	to	ADP
ap-4607	549	25	the	the	DET
ap-4607	549	26	fundamental	fundamental	ADJ
ap-4607	549	27	problem	problem	NOUN
ap-4607	549	28	of	of	ADP
ap-4607	549	29	classical	classical	ADJ
ap-4607	549	30	algebraic	algebraic	ADJ
ap-4607	549	31	geometry	geometry	NOUN
ap-4607	549	32	4	4	NUM
ap-4607	549	33	study	study	NOUN
ap-4607	549	34	of	of	ADP
ap-4607	549	35	the	the	DET
ap-4607	549	36	set	set	NOUN
ap-4607	549	37	v1(f1,f2	v1(f1,f2	NOUN
ap-4607	549	38	)	)	PUNCT
ap-4607	549	39	(	(	PUNCT
ap-4607	549	40	extended	extend	VERB
ap-4607	549	41	solutions	solution	NOUN
ap-4607	549	42	)	)	PUNCT
ap-4607	549	43	4.1	4.1	NUM
ap-4607	549	44	1	1	NUM
ap-4607	549	45	-	-	PUNCT
ap-4607	549	46	point	point	NOUN
ap-4607	549	47	lens	lens	NOUN
ap-4607	549	48	(	(	PUNCT
ap-4607	549	49	schwarzschild	schwarzschild	NOUN
ap-4607	549	50	lens	lens	PROPN
ap-4607	549	51	)	)	PUNCT
ap-4607	549	52	4.2	4.2	NUM
ap-4607	549	53	2	2	NUM
ap-4607	549	54	-	-	PUNCT
ap-4607	549	55	point	point	NOUN
ap-4607	549	56	lens	lens	NOUN
ap-4607	549	57	5	5	NUM
ap-4607	549	58	the	the	DET
ap-4607	549	59	study	study	NOUN
ap-4607	549	60	of	of	ADP
ap-4607	549	61	the	the	DET
ap-4607	549	62	set	set	NOUN
ap-4607	549	63	v0(f1,f2	v0(f1,f2	NOUN
ap-4607	549	64	)	)	PUNCT
ap-4607	549	65	(	(	PUNCT
ap-4607	549	66	point	point	NOUN
ap-4607	549	67	solutions	solution	NOUN
ap-4607	549	68	)	)	PUNCT
ap-4607	549	69	6	6	NUM
ap-4607	549	70	conclusions	conclusion	NOUN
ap-4607	549	71	a	a	DET
ap-4607	549	72	appendix	appendix	ADJ
ap-4607	549	73	references	reference	NOUN