Characterization of polynomials whose large powers have all positive coefficients

TL;DR

This paper provides a criterion to identify multivariate polynomials whose large powers and products with positive polynomials have all positive coefficients, extending classical results and applying to spectral radius functions of Markov chains.

Contribution

It generalizes De Angelis's and Pólya's classical results by characterizing polynomials with positive coefficients in large powers and their applications to Markov chain spectral functions.

Findings

Established a new criterion for positivity of polynomial powers

Extended classical results to multivariate polynomials

Applied results to spectral radius functions of Markov chains

Abstract

We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bi-variate polynomials, as well as a classical result of P\'olya, which corresponds to the case of a specific linear polynomial. As an application, we also give a characterization of certain polynomial beta functions, which are the spectral radius functions of the defining matrix functions of Markov chains.