A quantitative Pólya's Theorem with zeros

Let R [X] : = R [X₁, ..., X_n]. Pólya's Theorem says that if a form (homogeneous polynomial) p ∈ R [X] is positive on the standard n-simplex Δ_n, then for sufficiently large N all the coefficients of (X₁ + ⋯ + X_n)^N p 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 Δ_n" is relaxed to "nonnegative on Δ_n", 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 Δ_n, apart from possibly having zeros at the corners of the simplex.