Pólya's Theorem with zeros

Let R[X] be the real polynomial ring in n variables. Pólya's Theorem says that if a homogeneous polynomial p∈R[X] is positive on the standard n-simplex Δn, then for sufficiently large N all the coefficients of (X1+⋯+Xn)Np are positive. We give a complete characterization of forms, possibly with zeros on Δn, for which there exists N so that all coefficients of (X1+⋯+Xn)Np have only nonnegative coefficients, along with a bound on the N needed.