An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Abstract

AbstractLet $$S \subseteq \mathbb {R}^n$$ S ⊆ R n be a compact semialgebraic set and let f be a polynomial nonnegative on S. Schmüdgen’s Positivstellensatz then states that for any $$\eta > 0$$ η > 0 , the nonnegativity of $$f + \eta$$ f + η on S can be certified by expressing $$f + \eta$$ f + η 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 $$S = [-1, 1]^n$$ S = [ - 1 , 1 ] n is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree $$O(1 / \sqrt{\eta })$$ O ( 1 / η ) . This improves quadratically upon the previously best known estimate in $$O(1/\eta )$$ O ( 1 / η ) . Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval $$[-1, 1]$$ [ - 1 , 1 ] .