Geometric aspects of Pellet's and related theorems

TL;DR

This paper refines Pellet's theorem by replacing its zero separation region with a lemniscate, providing more precise zero location information, and introduces a linear algebra-based generalization that yields smaller lemniscate-based inclusion regions.

Contribution

It offers a geometric refinement of Pellet's theorem and a new linear algebra approach to generate tighter zero inclusion regions.

Findings

Refined zero separation region using lemniscates

Derived a sequence of smaller inclusion regions

Enhanced understanding of polynomial zero localization

Abstract

Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, according to their moduli. We refine one of those regions and replace it with the closed interior of a lemniscate that provides more precise information on the location of the zeros. Moreover, Pellet's theorem is considered the generalization of a zero inclusion region due to Cauchy. Using linear algebra tools, we derive a different generalization that leads to a sequence of smaller inclusion regions, which are also the closed interiors of lemniscates.