An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Let S⊆ Rn be a compact semialgebraic set and let f be a polynomial nonnegative on S. Schmüdgen’s Positivstellensatz then states that for any η> 0 , the nonnegativity of f+ η on S can be certified by expressing 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 is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree O(1/η). This improves quadratically upon the previously best known estimate in 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]