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) sumofsquares 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 Schmüdgentype 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 $[1, 1]$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.09528
 arXiv:
 arXiv:2109.09528
 Bibcode:
 2021arXiv210909528L
 Keywords:

 Mathematics  Optimization and Control;
 90C22;
 90C23;
 90C26
 EPrint:
 v2: implemented minor changes based on referee comments. 11 pages, 1 figure