1. On the Generation of Rank 3 Simple Matroids with an Application to Terao's Freeness Conjecture
- Author
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Mohamed Barakat, Martin Leuner, Reimer Behrends, Lukas Kühne, Christopher Jefferson, University of St Andrews. School of Computer Science, University of St Andrews. Centre for Research into Equality, Diversity & Inclusion, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and University of St Andrews. St Andrews GAP Centre
- Subjects
Integrally splitting characteristic polynomial ,Rank (linear algebra) ,ArangoDB ,General Mathematics ,T-NDAS ,Parallel algorithm ,Terao's freeness conjecture ,0102 computer and information sciences ,01 natural sciences ,Matroid ,Iterator of leaves of rooted tree ,Combinatorics ,Recursive iterator ,Priority queue ,Simple (abstract algebra) ,FOS: Mathematics ,Mathematics - Combinatorics ,QA Mathematics ,Parallel evaluation of recursive iterator ,QA ,Leaf-iterator ,Mathematics ,Discrete mathematics ,05B35, 52C35, 32S22, 68R05, 68W10 ,Conjecture ,Multiplicity (mathematics) ,010201 computation theory & mathematics ,Rank 3 simple matroids ,NoSQL database ,Combinatorics (math.CO) ,Tree-iterator - Abstract
In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics distinct from $2$ and $5$. Another database query proves that Terao's freeness conjecture is true for rank $3$ arrangements with $14$ hyperplanes in any characteristic., Improved exposition
- Published
- 2021
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