Sendov conjecture for high degree polynomials

TL;DR

This paper proves the Sendov conjecture for polynomials of sufficiently high degree, providing estimates for the degree threshold based on the zero's position, by analyzing the geometry of zeros and critical points.

Contribution

It establishes the conjecture for high-degree polynomials and offers bounds on the degree needed based on the zero's location.

Findings

Sendov conjecture holds for polynomials with degree above a certain threshold

Derived estimates for the degree N in terms of |a|

Analyzed zero and critical point geometry to support the proof

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.