Effective Pólya semi-positivity for non-negative polynomials on the simplex

We consider homogeneous polynomials f∈R[x1,…,xn] which are non-negative on the standard simplex in Rn, and we obtain sufficient conditions for such an f to be Pólya semi-positive, that is, all the coefficients of (x1+⋯+xn)Nf are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N under such conditions. Moreover, we also show that any Pólya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Pólya semi-positive polynomials in the low dimensional case when n⩽3 as well as the case (in any dimension) when the zero set of the polynomial in the simplex consists of a finite number of points. We also discuss an application to the representations of non-homogeneous polynomials which are non-negative on a general simplex.