A quantitative Pólya's Theorem with corner zeros

  • Authors:

  • Victoria Powers;Bruce Reznick

  • Affiliations:

  • Emory University, Atlanta, GA;University of Illinois at Urbana-Champaign, Urbana, IL

  • Venue:
  • Proceedings of the 2006 international symposium on Symbolic and algebraic computation
  • Year:
  • 2006

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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.