A quantitative Polya's Theorem with zeros
Let R[X] := R[X(1),..., X(n)]. Polya's Theorem says that if a form (homogeneous polynomial) P is an element of R[X] is positive on the standard n-simplex Delta(n), then for sufficiently large N all the coefficients of (X(1) + ... + X(n))N p are positive. The work in this paper is part of an ongoing project aiming to explain when Polya's Theorem holds for forms if the condition "positive on Delta(n)" is relaxed to "nonnegative on Delta(n)", and to give bounds on N. Schweighofer gave a condition which implies the conclusion of Polya's Theorem for polynomials f is an element of R[X]. We give a quantitative version of this result and use it to settle the case where a form P is an element of R[X] is positive on Delta(n), apart from possibly having zeros at the corners of the simplex. (C) 2009 Elsevier Ltd. All rights reserved.
polya's theorem positive polynomials sums of squares