Dimensions of faces of Gram spectrahedra

TL;DR

This paper investigates the geometric structure of Gram spectrahedra associated with sum of squares polynomials, providing combinatorial bounds on face dimensions and exploring the impact of smoothness on these bounds.

Contribution

It establishes combinatorial upper bounds for the dimensions of faces of Gram spectrahedra and analyzes how smoothness of the polynomial affects these bounds.

Findings

Upper bounds for face dimensions are combinatorially determined.

Faces of maximal dimension relate to singular forms for large degrees.

Smooth forms allow for improved bounds on face dimensions.

Abstract

Let $f\in\Sigma_{n,2d}$ be a sum of squares. The Gram spectrahedron of $f$ is a compact, convex set that parametrizes all sum of squares representations of $f$. Let $F\subseteq\mathrm{Gram}(f)$ be a face of its Gram spectrahedron. We are interested in upper bounds for the dimension of $F$. We show that this upper bound can be determined combinatorially. As it turns out, if the degree is large enough, a face realizing this bound, is a face of a Gram spectrahedron such that the form $f$ is singular. Thus we are also interested in finding better bounds whenever the form $f$ is smooth.