Generic Spectrahedral Shadows

Spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices. We characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic.Comment: version to be publishe

Dimension of gram spectrahedra of univariate polynomials

The Gram Spectrahedron of a polynomial parametrizes its sums-of-squares representations. In this note, we determine the dimension of Gram Spectrahedra of univariate polynomials.Comment: 4 page

Schottky Algorithms: Classical meets Tropical

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.Comment: 17 page

Random spectrahedra

Spectrahedra are affine-linear sections of the cone Pn of positive semidefinite symmetric n 7 n-matrices. We consider random spectrahedra that are obtained by intersecting Pn with the affine-linear space 1 + V , where 1 is the identity matrix and V is an `-dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of `-planes in the space of n 7 n real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for ` = 3 we relate the average number E\u3c3n of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra (n = 4) we show that E\u3c34 = 6 12 1a43 . Moreover, we prove that the average number E \u3c1n of singular points on the real variety of singular matrices in 1 + V is n(n 12 1). This quantity is related to the volume of the variety of real symmetric matrices with repeated eigenvalues. Furthermore, we compute the asymptotics of the volume and the volume of the boundary of a random spectrahedron