Spectrally Perron Polynomials and the Cauchy-Ostrovsky Theorem

TL;DR

This paper simplifies classical theorems by Cauchy and Ostrovsky using combinatorial and matrix theory, introduces spectrally Perron polynomials, and discusses necessary and sufficient conditions for these theorems.

Contribution

It provides simplified proofs of classical theorems, establishes necessity of conditions, and introduces the concept of spectrally Perron polynomials.

Findings

Simplified statements and proofs of Cauchy and Ostrovsky theorems.

Identification of necessary and sufficient conditions.

Introduction of spectrally Perron polynomials and related open problem.

Abstract

In this note, we simplify the statements of theorems attributed to Cauchy and Ostrovsky and give proofs of each theorem via combinatorial and nonnegative matrix theory. We also show that each simple sufficient condition in each statement is also necessary in its respective case. In addition, we introduce the notion of a spectrally Perron polynomial and pose a problem that appeals to a wide mathematical audience.