Sendov's Conjecture: A note on a paper of Dégot

Abstract

Sendov's conjecture states that if all the zeroes of a complex polynomial $P(z)$ of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of $P(z)$. In a paper that appeared in 2014, Dégot proved that, for each $a\in (0,1)$, there exists an integer $N$ such that for any polynomial $P(z)$ with degree greater than $N$, if $P(a)=0$ and all zeroes lie inside the unit disk, the disk $|z-a|\le 1$ contains a critical point of $P(z)$. Based on this result, we derive an explicit formula $\mathcal{N}(a)$ for each $a\in (0,1)$ and, consequently obtain a uniform bound $N$ for all $a\in [\alpha ,\beta ]$ where $0<\alpha <\beta <1$. This (partially) addresses the questions posed in Dégot's paper.

19 pages