Dimensions of faces of Gram spectrahedra

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.