Pólya’s Theorem with zeros

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