  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 $[-1, 1]$.
