  The Schm\"udgen's 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.
