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arXiv · 2107.11825

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

Victor Magron·Mohab Safey El Din·Trung-Hieu Vu
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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.

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Victor Magron, Mohab Safey El Din, Trung-Hieu Vu. 2021-07-25. Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients. https://arxiv.org/abs/2107.11825

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