Abstract: Helton and Vinnikov conjectured that any rigidly convex set is a spectrahedron, that is, it can be represented by linear matrix inequalities. To this end they conjectured that any real zero polynomial admits a monic determinantal representation. We disprove the latter conjecture and a relaxation of it where powers are allowed. The disproof of the relaxation uses combinatorial properties of the cone of positive semidefinite matrices which are not satisfied by hyperbolicity cones in general, and we argue that this discrepancy possibly could be used to find rigidly convex sets that are not spectrahedra.
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