Maximal and inextensible polynomials and the geometry of the spectra of normal operators

TL;DR

This paper investigates the maximal zero-distance polynomials within a specific class, challenges recent proofs of Sendov's conjecture, and introduces an operator-theoretic approach linking polynomial critical points to spectra of normal operators.

Contribution

It characterizes 0-maximal polynomials, demonstrates limitations of current proof methods for Sendov's conjecture, and proposes a novel spectral approach using normal operators.

Findings

Identified all 0-maximal polynomials in the class S(n,0)

Showed some inextensible polynomials are not locally maximal for Sendov's conjecture

Linked polynomial critical points to spectra of normal matrices via an operator approach

Abstract

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are inextensible, they are not necessarily locally maximal for Sendov's conjecture. This invalidates the recently claimed proofs of the conjectures of Sendov and Smale and shows that the method used in these proofs can only lead to (already known) partial results. In the second part of the paper we obtain a characterization of the critical points of a complex polynomial by means of multivariate majorization relations. We also propose an operator theoretical approach to Sendov's conjecture, which we formulate in terms of the spectral variation of a normal operator and its compression to the orthogonal complement of a trace vector. Using a theorem of Gauss-Lucas type for normal operators, we relate the problem of locating the critical points of complex polynomials to the more general problem of describing the relationships between the spectra of normal matrices and the spectra of their principal submatrices.