Characterization of polynomials whose large powers have all positive coefficients

Tan, Colin; To, Wing-Keung

doi:10.1090/proc/13709

Characterization of polynomials whose large powers have all positive coefficients
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by Colin Tan and Wing-Keung To PDF

Proc. Amer. Math. Soc. 146 (2018), 589-600 Request permission

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 all positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bivariate 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 spectral radius functions of the defining matrix functions of Markov chains.

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