Strong nonnegativity and sums of squares on real varieties

TL;DR

This paper introduces the concept of strong nonnegativity on real varieties, linking algebraic properties to nonnegativity and analyzing the convergence of theta body hierarchies for convex hull approximations.

Contribution

It defines strong nonnegativity, proves its equivalence to nonnegativity on nonsingular varieties, and generalizes obstructions to theta body hierarchy convergence for singular varieties.

Findings

Strong nonnegativity is equivalent to nonnegativity on nonsingular varieties.

Obstructions to theta body hierarchy convergence are extended to singular varieties.

Sum of squares are strongly nonnegative on real varieties.

Abstract

Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real varieties. Moreover, for singular varieties, we reprove and generalize obstructions of Gouveia and Netzer to the convergence of the theta body hierarchy of convex bodies approximating the convex hull of a real variety.