Positive Gorenstein Ideals

TL;DR

This paper introduces positive Gorenstein ideals, a new class of ideals linked to nonnegative polynomials and sums of squares, with applications in algebraic geometry, analysis, and optimization.

Contribution

It defines positive Gorenstein ideals and demonstrates their utility in studying sums of squares representations and nonnegative forms.

Findings

Provided a simple proof of Hilbert's result on ternary nonnegative forms.

Connected positive Gorenstein ideals to sums of squares and nonnegative polynomials.

Applied convex geometry and Cayley-Bacharach duality in the analysis.

Abstract

We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $\RR[x]$ with socle in degree 2d, which when viewed as a linear functional on $\RR[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work, our main tools are Cayley-Bacharach duality and elementary convex geometry.