A result related to the Sendov conjecture

Abstract

The Sendov conjecture asserts that if $p(z)={\prod }_{j=1}^N(z-z_j)$ is a polynomial with zeros $|z_j|\le 1$, then each disk $|z-z_j|\le 1$ contains a zero of $p^{\prime }$. Our purpose is the following: Given a zero $z_j$ of order $n\ge 2$, determine whether there exists $\zeta \ne z_j$ such that $p^{\prime }(\zeta )=0$ and $|z_j-\zeta |\le 1$. In this paper we present some partial results on the problem.

Annales Polonici Mathematici, 2023