Representations of non-negative polynomials via critical ideals

TL;DR

This paper provides new conditions under which non-negative polynomials on semi-algebraic sets can be represented as sums of squares modulo their critical ideals, advancing the understanding of polynomial positivity representations.

Contribution

It introduces boundary Hessian conditions and regularity assumptions to represent non-negative polynomials as SOS modulo critical ideals.

Findings

Representation of non-negative polynomials as SOS modulo critical ideals

Conditions under which SOS representations hold on semi-algebraic sets

Extension of SOS representation theory to non-compact sets

Abstract

This paper studies the representations of a non-negative polynomial $f$ on a non-compact semi-algebraic set $K$ modulo its critical ideal. Under the assumptions that the semi-algebraic set $K$ is regular and $f$ satisfies the boundary Hessian conditions (BHC) at each zero of $f$ in $K$, we show that $f$ can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if $f\ge 0$ on $K$. In particular, we focus on the polynomial ring $\mathbb R[x]$.