Matricial Proofs of Some Classical Results about Critical Point Location

TL;DR

This paper introduces a unified matrix analysis framework to prove classical theorems about polynomial critical points, offering new insights and variants in the geometric understanding of polynomial roots.

Contribution

It provides a novel, unified matrix analysis approach to classical polynomial theorems, including a variant of Siebeck's result.

Findings

Unified proof framework for Gauss--Lucas and Bôcher--Grace--Marden theorems

New matrix analysis techniques using the field of values and matrix differentiator

A variant of Siebeck's theorem related to polynomial critical points

Abstract

The Gauss--Lucas and B\^{o}cher--Grace--Marden theorems are classical results in the geometry of polynomials. Proofs of the these results are available in the literature, but the approaches are seemingly different. In this work, we show that these theorems can be proven in a unified theoretical framework utilizing matrix analysis (in particular, using the field of values and the differentiator of a matrix). In addition, we provide a useful variant of a well-known result due to Siebeck.