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, 1ⁿ and 0 < f_: = ₗ ₊ f (x), then the degrees of the polynomials appearing in the certificate may be bounded by O (f_ - f₅_), where f_: = ₗ ₊ 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.