A few more extensions of Putinar's Positivstellensatz to non-compact sets

We extend previous results about Putinar’s Positivstellensatz for cylinders of type $S\times \mathbb{R}$ to sets of type $S\times {\mathbb{R}}^r$ in some special cases taking into account $r$ and the degree of the polynomial with respect to the variables moving in ${\mathbb{R}}^r$ (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials and the cone of sums of squares holds. Degree bounds are provided.