MEGA 2007 Effective Methods in Algebraic Geometry Strobl, Austria, June 25th - 29th http://www.ricam.oeaw.ac.at/mega2007

Abstract

Title	A Quantitative P\'olya's Theorem with Zeros
Keywords	P\'olya's Theorem, positive polynomials
Abstract	P\'olya's Theorem says that if a homogeneous polynomial p in n variables is strictly positive on the standard n-simplex, then for a sufficiently high exponent N, all of the coefficients of the sum of the variables raised to the N-th power times p are positive. In this paper, we prove a "localized" P\'olya's Theorem, with a bound on the N needed, which is a quantitative version of a result of M. Schweighofer. We use our result to characterize forms which are positive on the n-simplex apart from a zero at a corner of the simplex and satisfy the conclusion of P\'olya's Theorem.

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