Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients

TL;DR

This paper develops exact algorithms for sum of squares decompositions of polynomials over their gradient ideals with rational coefficients, addressing approximation issues and expanding the scope of certificates of non-negativity in polynomial optimization.

Contribution

It introduces exact methods for sum of squares decompositions over gradient ideals with rational coefficients, improving upon previous approximate approaches.

Findings

Certificates can be obtained exactly over the rationals for polynomials with rational coefficients.

The paper provides algorithms with analyzed bit complexity and size bounds for these certificates.

Addresses limitations of approximate sum of squares methods in polynomial non-negativity certification.

Abstract

Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em sums-of-squares decompositions} which rely on efficient numerical solvers for semi-definite programming. This method faces two difficulties. The first one is that the certificates obtained this way are {\em approximate} and then non-exact. The second one is due to the fact that not all non-negative polynomials are sums-of-squares. In this paper, we build on previous works by Parrilo, Nie, Demmel and Sturmfels who introduced certificates of non-negativity modulo {\em gradient ideals}. We prove that, actually, such certificates can be obtained {\em exactly}, over the rationals if the polynomial under consideration has rational coefficients and we provide {\em exact} algorithms to compute them. We analyze the bit complexity of these algorithms and deduce bit size bounds of such certificates.