Sendov conjecture for high degree polynomials

Abstract

Sendov’s conjecture says that if all zeros of a complex polynomial P P lie in the closed unit disk and a a denotes one of them, then the closed disk of center a a and radius 1 1 contains a critical point of P P (i.e. a zero of its derivative P ′ P’ ). The main result of this paper is to prove that, for each a a , there exists an integer N N such that the disk | ζ − a | ≤ 1 |\zeta - a|\leq 1 contains a critical point of P P when the degree of P P is larger than N N . We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov’s conjecture.