The link between 1-norm approximation and effective Positivstellensatze for the hypercube

The Schmüdgen's Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial f on a compact, basic, semi-algebraic set 𝐊⊂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 𝐊 = [-1,1]^n and 0 < f_min := min_x ∈𝐊 f(x), then the degrees of the polynomials appearing in the certificate may be bounded by O(√(f_max - f_min/f_min)), where f_max := max_x ∈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.