Polya's Theorem with zeros

Let R[X] be the real polynomial ring in n variables. Polya's Theorem says that if a 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. We give a complete characterization of forms, possibly with zeros on Delta(n), for which there exists N so that all coefficients of (X-1 + ... + X-n)(N) p have only nonnegative coefficients, along with a bound on the N needed. (C) 2011 Elsevier Ltd. All rights reserved.