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

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Barakat_2021_SIAM_DM_generation_rank_3_VoR.pdf (1.249Mb)
Date
08/06/2021
Author
Barakat, M.
Behrends, R.
Jefferson, C.
Kühne, L.
Leuner, M.
Keywords
ArangoDB
Integrally splitting characteristic polynomial
Iterator of leaves of rooted tree
Leaf-iterator
NoSQL database
Parallel evaluation of recursive iterator
Priority queue
Rank 3 simple matroids
Recursive iterator
Terao's freeness conjecture
Tree-iterator
QA Mathematics
T-NDAS
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Abstract
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.
Citation
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
Publication
SIAM Journal on Discrete Mathematics
Status
Peer reviewed
DOI
https://doi.org/10.1137/19M1296744
ISSN
0895-4801
Type
Journal article
Rights
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.
Description
This 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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  • University of St Andrews Research
URL
https://arxiv.org/pdf/1907.01073
URI
http://hdl.handle.net/10023/23563

Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.

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