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.

中文翻译：

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具有有理系数的多项式在其梯度理想上的平方和分解

通过计算 {\em 非负证明}，评估实数上多元多项式的非负性是多项式优化中的一个热门话题。这通常通过计算 {\em sums-of-squares 分解} 来解决，该分解依赖于有效的数值求解器进行半定规划。这种方法面临两个困难。第一个是通过这种方式获得的证书是{\em 近似}，然后是不精确的。第二个原因是并非所有非负多项式都是平方和。在本文中，我们以 Parrilo、Nie、Demmel 和 Sturmfels 之前的工作为基础，他们介绍了非负模{\em 梯度理想}的证明。我们证明，实际上，这样的证书可以{\em 精确}，如果所考虑的多项式具有有理系数，并且我们提供 {\em 精确} 算法来计算它们，则超过有理数。我们分析这些算法的位复杂度并推导出此类证书的位大小界限。