The link between <inline-formula><tex-math id="M1">$ 1 $</tex-math></inline-formula>-norm approximation and effective Positivstellensätze for the hypercube

The Schmüdgen Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial $ f $ on a compact, basic, semi-algebraic set $ \mathbf{K} \subset {\mathbb R}^n $. A Positivstellensatz of this type is called effective if one may bound the degrees of the polynomials appearing in the certificate in terms of properties of $ f $. If $ \mathbf{K} = [-1, 1]^n $ and $ 0 < f_\min : = \min_{x \in \mathbf{K}} f(x) $, then the degrees of the polynomials appearing in the certificate may be bounded by $ O\left(\sqrt{\frac{f_\max - f_\min}{f_\min}}\right) $, where $ f_\max : = \max_{x \in \mathbf{K}} f(x) $, as was recently shown by Laurent and Slot [Optimization Letters 17:515-530, 2023]. The big-O notation suppresses dependence on $ n $ and the degree $ d $ of $ f $. In this paper we show a similar result, but with a better dependence on $ n $ and $ d $. In particular, our bounds depend on the $ 1 $-norm of the coefficients of $ f $, that may readily be calculated.