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Smooth hyperbolicity cones are spectrahedral shadows

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Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow.

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We would like to thank Daniel Plaumann for many inspiring discussions on the topic and the referees for helpful suggestions.

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Correspondence to Tim Netzer.

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Raman Sanyal has been supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 247029.

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Netzer, T., Sanyal, R. Smooth hyperbolicity cones are spectrahedral shadows. Math. Program. 153, 213–221 (2015).

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