Polyhedral faces in Gram spectrahedra of binary forms

TL;DR

This paper investigates the facial structure of Gram spectrahedra of nonnegative binary forms, establishing bounds on polyhedral face dimensions and ranks, and demonstrating the existence of specific faces for general forms.

Contribution

It provides new bounds and constructions for polyhedral faces in Gram spectrahedra of binary forms, linking face dimension and rank to the degree of the form.

Findings

Polyhedral face dimension k satisfies binomial(k+1, 2) ≤ d.

Existence of k-simplex faces with rank-one extreme points in Hermitian spectrahedra.

Presence of faces with rank 2(k+1) and dimension k in symmetric spectrahedra for large d.

Abstract

We analyze both the facial structure of the Gram spectrahedron $\mathrm{Gram}(f)$ and of the Hermitian Gram spectrahedron $\mathcal{H}^{\scriptscriptstyle+}(f)$ of a nonnegative binary form $f \in \mathbb{R}[x, y]_{2d}$. We show that if $F \subseteq \mathcal{H}^{\scriptscriptstyle+}(f)$ is a polyhedral face of dimension $k$ then $\binom{k+1}{2} \leq d$. Conversely, for all $k \in \mathbb{N}$ and $d \geq \binom{k+1}{2}$ we show that the Hermitian Gram spectrahedron of a general positive binary form $f \in \mathbb{R}[x, y]_{2d}$ with distinct roots contains a face $F$ which is a $k$-simplex and whose extreme points are rank-one tensors. For all $k \in \mathbb{N}$ and $d \geq (k+1)^2$ the (symmetric) Gram spectrahedron of a general positive binary form $f \in \mathbb{R}[x, y]_{2d}$ contains a polyhedral face $F$ with $(\mathrm{rk}(F), \dim(F)) = (2(k+1), k)$.