Borcea's variance conjectures on the critical points of polynomials

TL;DR

This paper explores Borcea's variance conjectures related to the critical points of polynomials, proposing modifications, relaxations, and connections to matrix theory, statistics, and potential theory, with new bounds and open problems.

Contribution

It introduces new conjectures and relaxations inspired by Borcea's ideas, linking polynomial critical points to matrix spectra and potential theory, and provides illustrative bounds.

Findings

Links between polynomial critical points and matrix spectra established

Proposes new conjectures and relaxations inspired by Borcea's ideas

Provides examples with sharp bounds for the conjectures

Abstract

Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures.