Refined Asymptotics for Multigraded Sums of Squares
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.
Keywords
Cite
@article{arxiv.0901.3786,
title = {Refined Asymptotics for Multigraded Sums of Squares},
author = {J. Maurice Rojas and Swaminathan Sethuraman},
journal= {arXiv preprint arXiv:0901.3786},
year = {2010}
}
Comments
There is an error invalidating the bounds of our main theorem. We are withdrawing this paper until we can replace it with a correct new version