The Positivstellensätze of Putinar and Schmüdgen show that any polynomial f positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellensätze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube B n = [−1, 1] n . We show an upper degree bound for Putinar-type representations on B n of the order O(fmax/f min ), where fmax, f min are the maximum and minimum of f on B n , respectively. Previously, specialized results of this kind were available only for Schmüdgentype representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in Ω( 8 fmax/f min ). This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar Positivstellensatz over a compact basic closed semialgebraic set S, with new polynomial bounds on the degree of the positivity certificates. These bounds involve a Łojasiewicz exponent associated to the description of S. We show that under Constraint Qualification Conditions, this Łojasiewicz exponent is equal to 1. We deduce new bounds on the convergence rate of the optima in Lasserre Sum-of-Squares hierarchy to the global optimum of a polynomial function on S and new bounds on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated moment sequences, which are positive on the quadratic module of S.

We present a new algorithm for computing the real radical of an ideal and, more generally, the -radical of , which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of . We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over .The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new efficient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, effectively performed with an exact MOP. Finally we illustrate our approach in some examples. CCS CONCEPTS• Theory of computation → Semidefinite programming; • Mathematics of computing → Grobner bases and other special bases; • Computing methodologies → Hybrid symbolic-numeric methods.