The Arveson boundary of a Free Quadrilateral is given by a noncommutative variety

TL;DR

This paper characterizes the Arveson boundary of a free quadrilateral using noncommutative polynomials, linking free extreme points to algebraic varieties within the framework of matrix convex sets.

Contribution

It introduces a novel description of free extreme points of free quadrilaterals via zero sets of specific noncommutative polynomials, advancing the understanding of their boundary structure.

Findings

Free extreme points are characterized by noncommutative polynomial equations.

The set of free extreme points is determined by the zero set of four noncommutative polynomials.

Results include properties of projective maps of free spectrahedra and homogeneous free spectrahedra.

Abstract

Let $SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices and set $SM(\mathbb{R})^g = \cup_n SM_n(\mathbb{R})^g$. A free quadrilateral is the collection of tuples $X \in SM(\mathbb{R})^2$ which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under a rich class of convex combinations called matrix convex combination. That is, given elements $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$ of a free quadrilateral $\mathcal{Q}$, one has \[ V_1^T X V_1+V_2^T Y V_2 \in \mathcal{Q} \] for any contractions $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ satisfying $V_1^T V_1+V_2^T V_2=I_n$. These matrix convex combinations are a natural analogue of convex combinations in the dimension free setting. A natural class of extreme point for free quadrilaterals is free extreme points: elements of a free quadrilateral which cannot be expressed as a nontrivial matrix convex combination of elements of the free quadrilateral. These free extreme points serve as the minimal set which recovers a free quadrilateral through matrix convex combinations. In this article we show that the set of free extreme points of a free quadrilateral is determined by the zero set of a collection of noncommutative polynomials. More precisely, given a free quadrilateral $\mathcal{Q}$, we construct noncommutative polynomials $p_1,p_2,p_3,p_4$ such that a tuple $X \in SM (\mathbb{R})^2$ is a free extreme point of a $\mathcal{Q}$ if and only if $X \in \mathcal{Q}$ and $p_i(X) =0 $ for $i=1,2,3,4$ and $X$ is irreducible. In addition we establish several basic results for projective maps of free spectrahedra and for homogeneous free spectrahedra.