A result related to the Sendov conjecture

TL;DR

This paper investigates a specific aspect of the Sendov conjecture, focusing on the existence of zeros of the derivative near a multiple zero of a polynomial with zeros inside the unit disk, providing partial results.

Contribution

It offers new partial results on the problem of locating zeros of the derivative near multiple zeros of polynomials related to the Sendov conjecture.

Findings

Partial results on the existence of derivative zeros near multiple zeros

Conditions under which zeros of the derivative are within distance 1

Insights into the structure of polynomials with zeros in the unit disk

Abstract

The Sendov conjecture asserts that if $p(z) = \prod_{j=1}^{N}(z-z_j)$ is a polynomial with zeros $|z_j| \leq 1$, then each disk $|z-z_j| \leq 1$ contains a zero of $p'$. Our purpose is the following: Given a zero $z_j$ of order $n \geq 2$, determine whether there exists $\zeta \not= z_j$ such that $p'(\zeta) = 0$ and $|z_j - \zeta| \leq 1$. In this paper we present some partial results on the problem.