D:\User\sbornik_38_pdf\16.DVI


Mathematical Problems of Computer Science 38, 37, 2012.

Dynamic Geometr y of Some P olynomials

G. V . A g h e kya n , K . P . S a h a kya n

Department of Applied Mathematics and Informatics
of Russian-Armenian (Slavonic) University

In 1 8 3 6 , Ga u s s s h o we d t h a t a ll t h e r o o t s o f P 0, d is t in c t fr o m t h e m u lt ip le r o o t s o f t h e
p o lyn o m ia l P it s e lf, s e r ve a s t h e p o in t s o f e qu ilib r iu m fo r t h e ¯ e ld o f fo r c e s c r e a t e d b y
id e n t ic a l p a r t ic le s p la c e d a t t h e r o o t s o f P ( p r o vid e d t h a t r p a r t ic le s a r e lo c a t e d a t t h e r o o t
o f m u lt ip lic it y r ) . Th e fo llo win g e qu a lit y t o z e r o s p r o vid e s a qu ic k p r o o f o f Ga u s s -L u c a s
t h e o r e m ( s e e fo r e xa m p le [1 ] o r [2 ]) . Th u s wa s a p p e a r e d t h e b r a n c h o f m a t h e m a t ic s , wh ic h
a ft e r t h e b o o k o f Mo r r is Ma r d e n [3 ], wa s c a lle d Ge o m e t r y o f P o lyn o m ia ls . Th e p o lyn o m ia l
c o n je c t u r e s o f S e n d o v a n d S m a le a r e t wo c h a lle n g in g p r o b le m s o f t h is b r a n c h [4 ,5 ,6 ].

On e o f t h e b e a u t ifu l t h e o r e m s o f m a t h e m a t ic s is Ma r d e n 's t h e o r e m [3 ,7 ]. It g ive s a g e o -
m e t r ic r e la t io n s h ip b e t we e n t h e z e r o s o f a t h ir d -d e g r e e p o lyn o m ia l wit h c o m p le x c o e ± c ie n t s
a n d t h e z e r o s o f it s d e r iva t ive . A m o r e g e n e r a l ve r s io n o f t h is t h e o r e m , d u e t o L in ¯ e ld [8 ].

Th is a r t ic le fo c u s e s o n t h e d yn a m ic b e h a vio r o f c r it ic a l p o in t s in t h e c a s e o f m o vin g o n e
o f t h e r o o t s o f c u b ic p o lyn o m ia l o n a g ive n t r a je c t o r y. Th e e qu a t io n s o f t h e c u r ve s , wh e r e
t h e c r it ic a l p o in t s m o ve s , a r e o b t a in e d . D is c o ve r e d n e w g e o m e t r ic p r o p e r t ie s o f p o s it io n s
o f t h e z e r o s a n d c r it ic a l p o in t s o f a c o m p le x p o lyn o m ia l o f d e g r e e t h r e e . Th e c a s e o f m u lt ip le
r o o t s o f t h e g ive n p o lyn o m ia l is c o n s id e r e d a s we ll.

R eferences

[1 ] P r a s o lo v, V .V , P o lyn o m ia ls , Springer, 2 0 0 0 .
[2 ] R a h m a n , Q.I a n d S c h m e is s e r , G., A n a lyt ic t h e o r y o f p o lyn o m ia ls , Oxford Univ. P ress,
2 0 0 5 .
[3 ] Ma r d e n M, Ge o m e t r y o f P o lin o m ia ls , A MS , 1 9 6 6 .
[4 ] S c h m e is s e r , G., Th e c o n je c t u r e s o f S e n d o v a n d S m a le , Approximation Theory(a volume
dedicated to B lagovest Sendov),So¯a D arba, 2 0 0 2 , p p 3 5 3 -3 6 9 .
[5 ] S m a le S ., Th e fu n d a m e n t a l t h e o r e m o f a lg e b r a a n d c o m p le xit y t h e o r y, B ulletin of AM S
4(1981), p p . 1 -3 6 .
[6 ] S e n d o v B l., Ge n e r a liz a t io n o f a c o n je c t u r e in t h e Ge o m e t r y o f P o lyn o m ia ls , S e r d ic a Ma t h .
J. 2 8 ( 2 0 0 2 ) , p p .2 8 3 -3 0 4 .
[7 ] K a lm a n , D ., A n E le m e n t a r y P r o o f o f Ma r d e n 's Th e o r e m , The American M athematical
M anthly, vol.115, no. 4, 2008, pp.330-338.
[8 ] L in ¯ e ld , B . Z., On t h e r e la t io n o f t h e r o o t s a n d p o le s o f a r a t io n a l fu n c t io n t o t h e r o o t s
o f it s d e r iva t ive , B ulletin of the AM S, 2 7 ( 1 9 2 0 ) , p p . 1 7 -2 1 .

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