Sendov conjecture for high degree polynomials

Abstract

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture.