Abstract
The Gram spectrahedron of a real form f∈R[x_]2d parametrizes all sum of squares representations of f. It is a compact, convex, semi-algebraic set, and we study its facial structure in the case of ternary quartics, i.e. f∈R[x,y,z]4. We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1, thus answering a question in Plaumann et al. (2011) about the existence of positive dimensional faces on such Gram spectrahedra. We complete the proof in Plaumann et al. (2011) showing that the so called Steiner graph of every smooth quartic is isomorphic to K4∐K4. Moreover, we show that the Gram spectrahedron of a generic positive semidefinite ternary quartic contains extreme points of all expected ranks.
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