Gram Spectrahedra

TL;DR

This paper explores the structure and properties of Gram spectrahedra, convex bodies parametrizing sum-of-squares representations of polynomials, with new results and open questions in real algebraic geometry and polynomial optimization.

Contribution

It provides new insights into the geometry of Gram spectrahedra, including minimal length representations and their relation to Hermitian Gram spectrahedra and toric varieties.

Findings

Analysis of sum-of-squares representations and their minimal length.

Connection between Hermitian Gram spectrahedra and point evaluations.

Presentation of open questions in the study of Gram spectrahedra.

Abstract

Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. This is a fundamental object in polynomial optimization and convex algebraic geometry. We summarize results on sums of squares that fit naturally into the context of Gram spectrahedra, present some new results, and highlight related open questions. We discuss sum-of-squares representations of minimal length and relate them to Hermitian Gram spectrahedra and point evaluations on toric varieties.