Characterization of Polynomials Whose Large Powers have Fully Positive Coefficients

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 coefficien...

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Bibliographic Details
Published inThe Journal of geometric analysis Vol. 32; no. 11
Main Author To, Wing-Keung
Format Journal Article
LanguageEnglish
Published New York Springer US 01.11.2022
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Coefficients
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Functions (mathematics)
Geometry
Global Analysis and Analysis on Manifolds
Markov chains
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Polynomials
Positive coefficients
Projective toric manifolds
14M25
26C05
32L15
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Summary: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 Colin Tan and the author, 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.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-022-00998-w