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

Abstract We extend previous results about Putinar’s Positivstellensatz for cylinders of type S × ℝ to sets of type S × ℝr in some special cases, taking into account r and the degree of the polynomial with respect to the variables moving in ℝr (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.