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

The Schm\"udgen's Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial f on a compact, basic, semi-algebraic set K ⊂ Rⁿ. 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 < fₘin := minx ∈ K f(x), then the degrees of the polynomials appearing in the certificate may be bounded by O(√fₘax - fₘin/fₘin), where fₘax := maxx ∈ 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.