On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture
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In this paper we describe a parallel algorithm for generating all nonisomorphic rank 3 simple matroids with a given multiplicity vector. We apply our implementation in the high performance computing 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.
Barakat , M , Behrends , R , Jefferson , C , Kühne , L & Leuner , M 2021 , ' On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture ' , SIAM Journal on Discrete Mathematics , vol. 35 , no. 2 , pp. 1201-1223 . https://doi.org/10.1137/19M1296744
SIAM Journal on Discrete Mathematics
Copyright © 2021 Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kuhne, and Martin Leuner. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the final published version of the work, which was originally published at https://doi.org/10.1137/19M1296744.
DescriptionThis work is a contribution to Project II.1 of SFB-TRR 195 "Symbolic Tools in Mathematics and their Application" funded by Deutsche Forschungsgemeinschaft (DFG). The fourth author was supported by ERC StG 716424 - CASe, a Minerva Fellowship of the Max Planck Society and the Studienstiftung des deutschen Volkes.
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