Refined Asymptotics for Multigraded Sums of Squares

TL;DR

This paper refines asymptotic bounds on the proportion of nonnegative polynomials that are sums of squares, considering Newton polytopes, and discusses implications for the efficiency of semidefinite programming in algebraic optimization.

Contribution

It improves Blekherman's bounds by incorporating Newton polytopes, providing sharper asymptotics and insights into when SOS representations are computationally feasible.

Findings

Refined bounds show certain Newton polytopes allow efficient SDP for most polynomials.

Asymptotic analysis indicates SOS polynomials form a small fraction among nonnegative polynomials as n grows.

Incorporating Newton polytopes sharpens understanding of SOS representations in polynomial optimization.

Abstract

To prove that a polynomial is nonnegative on R^n 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 semidefinite programming (SDP) computation 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 R^n, as n tends to infinity. 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.