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.

翻译：评估实际中多变量多元数的不增强性是多元优化的一个时下的问题。 通常通过计算半确定性编程中依赖高效数字解析器的 {em sume- game of squares decomposition} 来解决这个问题。 这种方法面临两个困难。 第一个是, 通过计算非非否定性证书来评估实际中多变量多元数的不增强性。 第二个原因是, 并非所有非否定性多元数都是方数值。 在本文中, 我们根据Parrilo、 Nie、 Demmel 和 Sturmfels 先前的作品, 后者引入了非增强性调制调制 { em 梯度理想} 。 我们证明, 实际上, 如果所考虑的多元数具有合理的系数, 并且我们提供了精确的算法, 来分析这些模型的复杂度。 我们分析了这些模型的复杂度。