Effective Pólya semi-positivity for non-negative polynomials on the simplex

  • Authors:

  • Hoi-Nam Mok;Wing-Keung To

  • Affiliations:

  • Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore;Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore

  • Venue:
  • Journal of Complexity
  • Year:
  • 2008

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Abstract

We consider homogeneous polynomials f@?R[x"1,...,x"n] which are non-negative on the standard simplex in R^n, and we obtain sufficient conditions for such an f to be Polya semi-positive, that is, all the coefficients of (x"1+...+x"n)^Nf are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N under such conditions. Moreover, we also show that any Polya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Polya semi-positive polynomials in the low dimensional case when n=