  Let $g_1,\dots, g_s \in \mathbb{R}[X_1,\dots, X_n,Y]$ and $S =
\{(\bar{x},y)\in \mathbb{R}^{n+1} \mid g_1(\bar{x},y) \ge 0, \dots,
g_s(\bar{x}, y) \ge 0\}$ be a non-empty, possibly unbounded, subset of a
cylinder in $\mathbb{R}^{n+1}$. Let $f \in \mathbb{R}[X_1, \dots, X_n, Y]$ be a
polynomial which is positive on $S$. We prove that, under certain additional
assumptions, for any non-constant polynomial $q \in \mathbb{R}[Y]$ which is
positive on $\mathbb{R}$, there is a certificate of the non-negativity of $f$
on $S$ given by a rational function having as numerator a polynomial in the
quadratic module generated by $g_1, \dots, g_s$ and as denominator a power of
$q$.
