Cauchy-like and Pellet-like results for polynomials

TL;DR

This paper introduces new Cauchy-like and Pellet-like theorems for complex polynomial zeros using similarity transformations of the companion matrix and polynomial eigenvalue reformulations.

Contribution

It presents novel zero localization results for polynomials based on matrix similarity and eigenvalue problem reformulations, expanding classical bounds.

Findings

New Cauchy-like bounds for polynomial zeros

Pellet-like results for zero distribution

Eigenvalue-based polynomial zero localization

Abstract

We obtain several Cauchy-like and Pellet-like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and the reformulation of the zeros of a scalar polynomial as the eigenvalues of a polynomial eigenvalue problem.