Sendov conjecture for high degree polynomials

Abstract:

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