A version of Putinar's Positivstellensatz for cylinders

We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type $S \times {\mathbb R}$ with $S = \{x \in {\mathbb R}^n | g_1(x) \ge 0, ..., g_s(x) \ge 0\}$ such that the quadratic module generated by $g_1, ..., g_s$ in ${\mathbb R}[X_1, ..., X_n]$ is archimedean, and we provide a degree bound for the representation of a polynomial $f \in {\mathbb R}[X_1, ..., X_n, Y]$ which is positive on $S \times {\mathbb R}$ as an explicit element of the quadratic module generated by $g_1, ..., g_s$ in ${\mathbb R}[X_1, ..., X_n, Y]$. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type.