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

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.