Extreme points of Gram spectrahedra of binary forms

TL;DR

This paper studies the structure of Gram spectrahedra of binary forms, showing they have extreme points of all ranks in the Pataki range and analyzing their geometric properties.

Contribution

It provides the first example of spectrahedra with extreme points of all ranks in the Pataki range for large dimensions and computes the dimension of rank $r$ extreme points.

Findings

Spectrahedra have extreme points of all Pataki range ranks.

Calculated the dimension of the set of rank $r$ extreme points.

Identified pairs of rank two extreme points forming edges.

Abstract

The Gram spectrahedron $\text{Gram}(f)$ of a form $f$ with real coefficients parametrizes the sum of squares decompositions of $f$, modulo orthogonal equivalence. For $f$ a sufficiently general positive binary form of arbitrary degree, we show that $\text{Gram}(f)$ has extreme points of all ranks in the Pataki range. This is the first example of a family of spectrahedra of arbitrarily large dimensions with this property. We also calculate the dimension of the set of rank $r$ extreme points, for any $r$. Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of $\text{Gram}(f)$.