Extreme Points of Gram Spectrahedra of Binary Forms
Claus Scheiderer · Discrete & Computational Geometry · 2022
Abstract The Gram spectrahedron $$\mathrm {Gram}(f)$$ 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 $$\mathrm {Gram}(f)$$ 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 $$\mathrm {Gram}(f)$$ Gram ( f ) . The proof of the main result relies on a purely algebraic fact of independent interest: Whenever $$d,r\ge 1$$ d , r ≥ 1 are integers with $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) \le 2d+1$$ r + 1 2 ≤ 2 d + 1 , there exists a length r sequence $$f_1,\dots ,f_r$$ f 1 , ⋯ , f r of binary forms of degree d for which the $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) $$ r + 1 2 pairwise products $$f_if_j$$ f i f j , $$i\le j$$ i ≤ j , are linearly independent.