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Characterization of polynomials whose large powers have all positive coefficients


Authors: Colin Tan and Wing-Keung To
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 26C05, 12D99, 32T15
DOI: https://doi.org/10.1090/proc/13709
Published electronically: October 23, 2017
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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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Additional Information

Colin Tan
Affiliation: Department of Statistics & Applied Probability, National University of Singapore, Block S16, 6 Science Drive 2, Singapore 117546
Address at time of publication: General Education Unit, Office of the Provost, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077
Email: colinwytan@gmail.com

Wing-Keung To
Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
Email: mattowk@nus.edu.sg

DOI: https://doi.org/10.1090/proc/13709
Keywords: Polynomials, positive coefficients, strongly pseudoconvex
Received by editor(s): January 8, 2017
Received by editor(s) in revised form: February 20, 2017
Published electronically: October 23, 2017
Additional Notes: The second author was partially supported by the research grant R-146-000-142-112 from the National University of Singapore and the Ministry of Education
Communicated by: Franc Forstnerič
Article copyright: © Copyright 2017 American Mathematical Society