Extreme Points of Gram Spectrahedra of Binary Forms

Abstract The Gram spectrahedron $$\mathrm {Gram}(f)$$ Gram ( f ) of a form f with real coefficients is compact affine-linear section the cone psd symmetric matrices. It parametrizes sum squares decompositions , modulo orthogonal equivalence. For sufficiently general positive binary arbitrary degree, we show that has extreme points all ranks in Pataki range. We also calculate dimension set rank r points, for any . Moreover, determine pairs two which connecting line segment an edge proof main result relies on purely algebraic fact independent interest: Whenever $$d,r\ge 1$$ d , r ≥ 1 are integers $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) \le 2d+1$$ + 2 ≤ there exists length sequence $$f_1,\dots ,f_r$$ ⋯ forms degree d $$ pairwise products $$f_if_j$$ i j $$i\le j$$ linearly independent.