Year: 2018
Type: preprint
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 $\text{Gram}(f)$.
Source: arXiv (Cornell University)
Author: Scheiderer, Claus
Institution: University of Konstanz
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4298936796