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Obstructions to determinantal representability
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 2, 1202-1212 p.Article in journal (Refereed) Published
Abstract [en]

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.

Place, publisher, year, edition, pages
2011. Vol. 226, no 2, 1202-1212 p.
Keyword [en]
Linear matrix inequalities, Determinantal representability, Hyperbolic polynomial, Polymatroid, Subspace arrangements, Half-plane property
National Category
URN: urn:nbn:se:su:diva-67329DOI: 10.1016/j.aim.2010.08.003ISI: 000284452900006OAI: diva2:470323
authorCount :1Available from: 2011-12-28 Created: 2011-12-28 Last updated: 2011-12-28Bibliographically approved

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