On the Generation of Rank 3 Simple Matroids with an Application to Terao’s Freeness Conjecture
Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner
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Submission date: 28. Aug. 2020
MSC-Numbers: 05B35, 52C35, 32S22, 68R05, 68W10
Keywords and phrases: rank $3$ simple matroids, integrally splitting characteristic polynomial, Terao's freeness conjecture, recursive iterator, noSQL database
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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 an integrally 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.