Title: Characterization of polynomials whose large powers have all positive coefficients,
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 Pólya, 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.