A quantitative Pólya's Theorem with corner zeros

Pólya's Theorem says that if <i>p</i> is a homogeneous polynomial in <i>n</i> variables which is positive on the standard <i>n</i>-simplex, and <i>F</i> is the sum of the variables, then for a sufficiently large exponent <i>N, F<sup>N</sup> * p</i> 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 <i>p</i> on the simplex. The authors have previously given an explicit bound on <i>N</i>, determined by the data of <i>p</i>; namely, the degree, the size of the coefficients and the minimum value of <i>p</i> 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.

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