Characterization of polynomials whose large powers have fully positive coefficients

TL;DR

This paper provides a criterion to determine when large powers of a multivariate Laurent polynomial with a smooth Newton polytope have all positive coefficients, extending previous results in special cases.

Contribution

It generalizes earlier results by characterizing multivariate Laurent polynomials with positive coefficients in large powers, based on their Newton polytope properties.

Findings

Provides a new criterion for positivity in large powers of Laurent polynomials.

Generalizes previous univariate and simplex cases.

Applies to polynomial spectral radius functions of Markov chain matrices.

Abstract

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of the authors, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.