A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra

Let R[X]R[x₁,...,x_n] and let and Δ_n denote the simplex {(x₁,...,x_n)|x_i≥0,∑_ix _i=1}. Pólya's Theorem says that if f∈R[X] is homogeneous and positive on Δ_n, then for sufficiently large N all of the coefficients of (x₁++x_n)^Nf(x₁,x_n) are positive. We give an explicit bound for N and an application to some special representations of polynomials positive on polyhedra. In particular, we give a bound for the degree of a representation of a polynomial positive on a convex polyhedron as a positive linear combination of products of the linear polynomials defining the polyhedron.