Stability of Quadratic Modules

TL;DR

This paper investigates the stability of quadratic modules in real polynomial rings, providing conditions for stability, exploring implications for the Moment Problem, and extending related results in semi-algebraic geometry.

Contribution

It offers easily checkable sufficient conditions for stability of quadratic modules and generalizes existing results on the Invariant Moment Problem.

Findings

Stability can be guaranteed when nonexistence of bounded polynomials holds.

Stable quadratic modules are often closed and help solve the Membership Problem.

The results extend to certain semi-algebraic sets and generalize previous theorems.

Abstract

A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly by Powers and Scheiderer, is a very useful property. It often implies that the quadratic module is closed; furthermore it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements Schmuedgen's result which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem by Cimpric, Kuhlmann and Scheiderer.