A quantitative Pólya's theorem with corner zeros

Victoria Powers, Bruce Reznick

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Pólya's Theorem says that if p is a homogeneous polynomial in n variables which is positive on the standard n-simplex, and F is the sum of the variables, then for a sufficiently large exponent N, FN * p has positive coefficients. Pólya's Theorem has had many applications in both pure and applied mathematics; for example it provides a certificate for the positivity of p on the simplex. The authors have previously given an explicit bound on N, determined by the data of p; namely, the degree, the size of the coefficients and the minimum value of p on the simplex. In this paper, we extend this quantitative Pólya's Theorem to non-negative polynomials which are allowed to have simple zeros at the corners of the simplex.

Original languageEnglish (US)
Title of host publicationProceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006
PublisherAssociation for Computing Machinery
Pages285-289
Number of pages5
ISBN (Print)1595932763, 9781595932761
DOIs
StatePublished - 2006
EventInternational Symposium on Symbolic and Algebraic Computation, ISSAC 2006 - Genova, Italy
Duration: Jul 9 2006Jul 12 2006

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
Volume2006

Other

OtherInternational Symposium on Symbolic and Algebraic Computation, ISSAC 2006
Country/TerritoryItaly
CityGenova
Period7/9/067/12/06

Keywords

  • Positive polynomials
  • Pólya's theorem
  • Sums of squares

ASJC Scopus subject areas

  • General Mathematics

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