Volumes of certain loci of polynomials and their applicatoins

To prove that a polynomial is nonnegative on Rn, one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semi-definite programming (SDP) computation that is much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables with k = 4 fixed those that are SOS occupy a vanishingly small fraction of those that are nonnegative on Rn, as n -> 1. With an eye toward the case of small n, we refine Blekherman'[s bounds by incorporating the underlying Newton polytope, simultaneously sharpening some of his older bounds along the way. Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs.