Non-representable hyperbolic matroids
| Autor: | Petter Brändén, Nima Amini |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Computer Science::Computer Science and Game Theory General Computer Science General Mathematics 010103 numerical & computational mathematics Positive-definite matrix 01 natural sciences Matroid Theoretical Computer Science Combinatorics Symmetric polynomial Computer Science::Discrete Mathematics Euclidean geometry [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Mathematics - Combinatorics Discrete Mathematics and Combinatorics 0101 mathematics Connection (algebraic framework) Algebraic number Computer Science::Data Structures and Algorithms Mathematics - Optimization and Control Mathematics Mathematics::Combinatorics Conjecture 010102 general mathematics Cone (topology) Optimization and Control (math.OC) Combinatorics (math.CO) Complex number Counterexample |
| Zdroj: | Discrete Mathematics & Theoretical Computer Science. |
| ISSN: | 1365-8050 |
| DOI: | 10.46298/dmtcs.6328 |
| Popis: | The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The V\'amos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the V\'amos matroid and the generalized V\'amos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre-Tur\'an inequality and Jensen's inequality. Finally we explore consequences to algebraic versions of the generalized Lax conjecture. |
| Databáze: | OpenAIRE |
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