An effective version of Schm\"udgen's Positivstellensatz for the hypercube

TL;DR

This paper provides an explicit degree bound for Schmüdgen's Positivstellensatz certificates on the hypercube, showing they can be achieved with polynomials of degree proportional to 1 over the square root of the approximation parameter.

Contribution

The paper establishes a quadratic improvement in the degree bounds for Schmüdgen's certificates specifically on the hypercube, using the polynomial kernel method.

Findings

Degree of certificates is O(1/√η) for hypercube

Improves previous bound of O(1/η)

Uses polynomial kernel method and Jackson kernel

Abstract

Let $S \subseteq \mathbb{R}^n$ be a compact semialgebraic set and let $f$ be a polynomial nonnegative on $S$. Schm\"udgen'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 Schm\"udgen-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 $[-1, 1]$.