THE ARVESON BOUNDARY OF A FREE QUADRILATERAL IS GIVEN BY A NONCOMMUTATIVE VARIETY

Let SMn(R)(g) denote the set of g-tuples of nxn real symmetric matrices and set SM(R)(g) = boolean OR nSMn(R)(g). A free quadrilateral is the collection of tuples X is an element of SM(R)(2) which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under a generalized type of convex combination called amatrix convex combination. That is, given elements X =(X-1, ... , X-g) is an element of SMn1 (R)(g) and Y = (Y-1, ... ,Y-g) is an element of SMn2 (R)(g) of a free quadrilateral Q, one has (V1XV1)-X-T+(V2YV2)-Y-T is an element of Q for any contractions V-1 : R-n -> R-n1 and V-2 : R-n -> R-n2 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. Elements of a free quadrilateral which cannot be expressed as a nontrivial matrix convex combination of other elements of the free quadrilateral are called free extreme points. Free extreme points serve as the minimal set which recovers a free quadrilateral through matrix convex combinations. In this way, free extreme points are the natural type of extreme point for a free quadrilateral. 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 Q, we construct noncommutative polynomials p(1), p(2), p(3), p(4) such that a tuple X is an element of SM(R)(2) is a free extreme point of Q if and only if X is an element of 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. In particular, we show that that the image of a free extreme point under an invertible projective map is again a free extreme point. We also extend a kernel condition for a tuple to be a free extreme point to the setting of homogeneous free spectrahedra.