A unifying framework for generalizations of the Enestrom-Kakeya theorem

TL;DR

This paper introduces a unifying framework that generalizes the Enestrom-Kakeya theorem, providing new zero inclusion and exclusion regions for polynomials with positive coefficients, simplifying analysis through minimal algebraic manipulation.

Contribution

It develops a broad framework that encompasses existing theorems and produces new results on polynomial zero regions using generalized Cauchy observations and polynomial multipliers.

Findings

Unified framework for polynomial zero bounds

New zero inclusion/exclusion regions as disks

Simplified algebraic approach

Abstract

The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients. We establish a unifying framework that incorporates this theorem and several similar ones as special cases, while generating new theorems of a similar type. These establish zero inclusion and exclusion regions consisting of a single disk or the union of several disks in the complex plane. Our framework is built on two basic tools, namely a generalization of an observation by Cauchy, and a family of polynomial multipliers. Its approach is transparent and reduces algebraic manipulations to a minimum.