Analyticity domains of critical points of polynomials. A proof of Sendov's conjecture

TL;DR

This paper proves that the zeros and critical points of certain polynomials depend analytically on their roots, providing a new proof of Sendov's conjecture regarding zeros and derivatives of polynomials within the unit disk.

Contribution

It establishes the analyticity of zeros and critical points of polynomials with multiple roots, leading to a novel proof of Sendov's conjecture.

Findings

Zeros and critical points are analytic functions of polynomial roots.

Provides a new proof of Sendov's conjecture for polynomials with roots in the unit disk.

Demonstrates continuity of these functions on boundary conditions.

Abstract

Let $\P_{n}^c(\bar{\mu},\bar{\nu})$ be the set of all complex polynomials $p(z)=\prod_{i=1}^{m}(z-z_i)^{\mu_i}$, $\sum_{i=1}^m\mu_i=n$, with derivatives of the form $$ p'(z)=n\prod_{i=1}^{m}(z-z_i)^{\mu_i-1}\prod_{j=1}^{k}(z-\xi_j)^{\nu_j}, ~\sum_{j=1}^k\nu_j=m-1. $$ In this note we prove the following:\par\medskip {\it \noindent For a fixed ordering $\alpha=(1,2,\ldots,m)$, the distinct zeros $\{z_i\}_{i=1}^m$ and the distinct critical points of the second kind $\{\xi_j\}_{j=1}^k$ of polynomials from $\P_{n}^c(\bar{\mu},\bar{\nu})$ are analytic functions $\{z_i^{\alpha\beta}\}_{i=1}^m$ and $\{\xi_j^{\alpha\beta}\}_{j=1}^k$, resp., $\beta=(i_1,i_2,\ldots,i_{k+1})$, of any of the variables $(z_{i_1},z_{i_2},\ldots,z_{i_{k+1}})$ in the domain $$ \{(z_{i_1},z_{i_2},\ldots,z_{i_{k+1}})\in \C^{k+1}~\vert~p\in \P_{n}^c(\bar{\mu},\bar{\nu}) \}, $$ being also continuous on its boundary.}\par\medskip\noindent This statement gives an immediate proof to the well-known conjecture of Bl. Sendov \cite{sen}: \par\medskip {\it \noindent If $n\ge 2$ and $p(z)=\prod_{i=1}^n (z-z_i)$ is a polynomial of degree $n$ such that $z_i\in \C$, $\vert z_i\vert\le 1$, $i=1,2,\ldots,n$, then for every $i=1,2,\ldots,n$, the disk $\{z\in \C\,|\,\vert z_i-z\vert \le 1\}$ contains at least one zero of $p'(z)$.}