Dimensional Differences between Nonnegative Polynomials and Sums of Squares

Abstract

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. As either the degree or the number of variables grows the gaps become very large, asymptotically the gaps approach the full dimension of the vector space of polynomials in $n$ variables of degree 2d. The gaps occur generically, they are not a product of selecting special faces of the cones. Using these dimensional differences we show how to derive inequalities that separate nonnegative polynomials from sums of squares; the inequalities will hold for all sums of squares, but will fail for some nonnegative polynomials.