Extreme points of Gram spectrahedra of binary forms

2018/02/15 by Claus Scheiderer, Scheiderer, Claus · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Control (math.OC) #math.CO #math.OC

paper · pdf · doi:10.48550/arxiv.1802.05513

The proof of Proposition 4.2 in v1, more specifically the argument in 4.8, was incorrect. It got replaced by a different proof in v3

arxiv created 2019/02/17 · arxiv updated 2019/02/19

Abstract

The Gram spectrahedron 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 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 Gram(f).

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