An effective version of Schmüdgen's Positivstellensatz for the hypercube

Monique Laurent, Lucas Slot

Abstract: Let \(S \subseteq \mathbb {R}^n\) be a compact semialgebraic set and let f be a polynomial nonnegative on S. Schmüdgen’s Positivstellensatz then states that for any \(\eta > 0\), the nonnegativity of \(f + \eta\) on S can be certified by expressing \(f + \eta\) as a conic combination of products of the polynomials that occur in the inequalities defining S, where the coefficients are (globally nonnegative) sum-of-squares polynomials. It does not, however, provide explicit bounds on the degree of the polynomials required for such an expression. We show that in the special case where \(S = [-1, 1]^n\) is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree \(O(1 / \sqrt{\eta })\). This improves quadratically upon the previously best known estimate in \(O(1/\eta )\). Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval \([-1, 1]\).