An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Let S 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].


Posted as arXiv:2109.09528



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