Sums of squares and moment problems in equivariant situations

TL;DR

This paper systematically studies positivity and moment problems in the context of group actions on algebraic varieties, focusing on invariant sums of squares and measures, especially when the acting group is compact.

Contribution

It introduces a framework for analyzing invariant positivity and moment problems, relating quadratic modules of the variety to those of the invariant subring, with new results for the compact group case.

Findings

Characterization of invariant sums of squares representations

Analysis of invariant linear functionals and measures

Results on finite solvability of equivariant moment problems

Abstract

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\R$ acting on an affine $\R$-variety $V$, we consider the induced dual action on the coordinate ring $\R[V]$ and on the linear dual space of $\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\R[V]$ and associated quadratic modules of the (finitely generated) subring $\R[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\R)$ is compact.