An effective version of Schmüdgen’s Positivstellensatz for the hypercube

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].