Sums of squares on reducible real curves

TL;DR

This paper investigates when non-negative polynomials on reducible real algebraic curves can be expressed as sums of squares, providing criteria based on the configuration of components and exploring implications for the moment problem.

Contribution

It extends Scheiderer's classification from irreducible to reducible curves, offering necessary and sufficient conditions for sums of squares representations.

Findings

Criteria for sums of squares on reducible curves

Dependence on the configuration of irreducible components

Partial results for finitely generated preorderings

Abstract

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible curves, we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions. We also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets.