Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets

We study the problem of representing multivariate polynomials with rational coefficients, which are non-negative and strictly positive on semialgebraic sets, using rational sums of squares. Contrary to the classical Nullstellensatz, the worst case complexity analysis of these representation results, called effective Positivstellensatz, is still for the most part an open area of research.

We obtain existential results and degree bounds for such representations for finite semialgebraic sets, and apply them in the context of polynomial optimization. We retrieve results of Lasserre, Laurent and Parrilo on finite convergence of the moment-sums of squares hierarchy, showing furthermore the existence of sums of squares certificates of optimality over the rational numbers.