Exponential lower bounds on spectrahedral representations of hyperbolicity cones
| Autor: | Raghavendra, Prasad, Ryder, Nick, Srivastava, Nikhil, Weitz, Benjamin |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables contains $(n/d)^{\Omega(d)}$ pairwise distant cones in a certain metric, and therefore that any semidefinite representation of such cones must have dimension at least $(n/d)^{\Omega(d)}$ (even if a small approximation is allowed). The proof contains several ingredients of independent interest, including the identification of a large subspace in which the elementary symmetric polynomials lie in the relative interior of the set of hyperbolic polynomials, and quantitative versions of several basic facts about real rooted polynomials. Comment: Fixed a mistake in the proof of Lemma 6. The statement is unchanged except for constant factors, and the main theorem is unaffected. Wrote a slightly stronger statement for the main theorem, emphasizing approximate representations (the proof is the same). Added one figure |
| Databáze: | arXiv |
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