A quantitative Pólya’s Theorem with zeros

Let R[X]:=R[X1,…,Xn]. Pólya’s Theorem says that if a form (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. The work in this paper is part of an ongoing project aiming to explain when Pólya’s Theorem holds for forms if the condition “positive on Δn” is relaxed to “nonnegative on Δn”, and to give bounds on N. Schweighofer gave a condition which implies the conclusion of Pólya’s Theorem for polynomials f∈R[X]. We give a quantitative version of this result and use it to settle the case where a form p∈R[X] is positive on Δn, apart from possibly having zeros at the corners of the simplex.