Sendov's Conjecture: A note on a paper of D\'{e}got

TL;DR

This paper builds on Dégot's partial proof of Sendov's conjecture by deriving explicit bounds and formulas for critical points of polynomials with zeros in the unit disk, advancing understanding of the conjecture.

Contribution

It provides an explicit formula for the bound N depending on a parameter a, and extends results uniformly over intervals within (0,1), addressing open questions from Dégot's work.

Findings

Derived explicit formula for N(a) for each a in (0,1)

Established uniform bounds for N over intervals in (0,1)

Partially addressed open questions in Sendov's conjecture research

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, D\'{e}got 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 $|z-a|\leq 1$ 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 D\'{e}got's paper.