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

TL;DR

This paper extends Putinar's Positivstellensatz to certain non-compact sets involving products with unbounded Euclidean spaces, providing degree bounds and conditions for sums of squares representations.

Contribution

It generalizes previous results to higher-dimensional non-compact sets and establishes degree bounds under specific conditions.

Findings

Extended Positivstellensatz to sets of the form S × R^r

Provided degree bounds for polynomial representations

Identified cases where non-negative polynomials equal sums of squares

Abstract

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.