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MiS Preprint

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


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.

MSC Codes:
05B35, 52C35, 32S22, 68R05, 68W10
rank $3$ simple matroids, integrally splitting characteristic polynomial, Terao's freeness conjecture, recursive iterator, noSQL database

Related publications

2021 Repository Open Access
Mohamed Barakat, Reimer Behrends, Christoper Jefferson, Lukas Kühne and Martin Leuner

On the generation of rank 3 simple matroids with an application to Terao' freeness conjecture

In: SIAM journal on discrete mathematics, 35 (2021) 2, pp. 1201-1223