Positivity in power series rings

TL;DR

This paper extends previous results on representing nonnegative polynomials on certain semialgebraic sets, especially those invariant under group actions, by relaxing transversality conditions.

Contribution

It generalizes Scheiderer's 2006 results to include cases without transversality, relevant for semialgebraic sets with symmetry properties.

Findings

Extended representation results for nonnegative polynomials

Applicable to semialgebraic sets invariant under finite groups

Relaxed conditions broaden the scope of polynomial positivity certificates

Abstract

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions.