Extreme Points of Gram Spectrahedra of Binary Forms

Abstract

The Gram spectrahedron mathrm {Gram}(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It 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 mathrm {Gram}(f) has extreme points of all ranks in the Pataki range. 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 mathrm {Gram}(f). The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,rge 1 are integers with left( {begin{array}{c}r+1 2end{array}}right) le 2d+1, there exists a length r sequence f_1,dots ,f_r of binary forms of degree d for which the left( {begin{array}{c}r+1 2end{array}}right) pairwise products f_if_j, ile j, are linearly independent.