Obstructions to determinantal representability
| Autor: | Petter Brändén |
|---|---|
| Rok vydání: | 2010 |
| Předmět: |
Mathematics(all)
Polynomial Conjecture Hyperbolic polynomial Determinantal representability General Mathematics Zero (complex analysis) Linear matrix inequalities Subspace arrangements Mathematics - Rings and Algebras Matroid Convexity Half-plane property Functional Analysis (math.FA) Combinatorics Mathematics - Functional Analysis Rings and Algebras (math.RA) FOS: Mathematics Symmetric matrix Polymatroid Complex number Mathematics |
| DOI: | 10.48550/arxiv.1004.1382 |
| Popis: | 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. Comment: 10 pages. To appear in Advances in Mathematics |
| Databáze: | OpenAIRE |
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