A new bound for Pólyaʹs Theorem with applications to polynomials positive on polyhedra

Let R[X]colon, equalsR[x1,…,xn] and let and Δn denote the simplex {(x1,…,xn)xi≥0,∑ixi=1}. Pólyaʹs Theorem says that if fset membership, variantR[X] is homogeneous and positive on Δn, then for sufficiently large N all of the coefficients of (x1+cdots, three dots, centered+xn)N f(x1,cdots, three dots, centered,xn) 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.