Gram Spectrahedra of Ternary Quartics

TL;DR

This paper investigates the facial structure of Gram spectrahedra for smooth ternary quartics, revealing their face dimensions, Steiner graph isomorphism, and rank properties of generic forms.

Contribution

It characterizes the faces of Gram spectrahedra for smooth ternary quartics, proving the Steiner graph is isomorphic to two disjoint K4 graphs and analyzing rank distributions.

Findings

Faces of dimension 2 exist in all smooth ternary quartics' spectrahedra.

The Steiner graph of smooth quartics is isomorphic to K4 disjoint union K4.

Generic psd ternary quartics have points of all ranks in the Pataki interval.

Abstract

The Gram spectrahedron of a real form $f\in\mathbb{R}[\underline{x}]_{2d}$ parametrizes all sum of squares representations of $f$. It is a compact, convex, semi-algebraic set, and we study its facial structure in the case of ternary quartics, i.e. $f\in\mathbb{R}[x,y,z]_4$. We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1. We complete the proof that the so called Steiner graph of every smooth quartic is isomorphic to $K_4\coprod K_4$. Moreover, we show that the Gram spectrahedron of a generic psd ternary quartic contains points of all ranks in the Pataki interval.