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

M. Barakat; R. Behrends; C. Jefferson; L. Kühne; M. Leuner

doi:10.1137/19M1296744

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

M. Barakat, R. Behrends, C. Jefferson, L. Kühne, M. Leuner

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1201-1223
Number of pages	23
Journal	SIAM Journal on Discrete Mathematics
Volume	35
Issue number	2
DOIs	https://doi.org/10.1137/19M1296744
Publication status	Published - 8 Jun 2021

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

Access to Document

10.1137/19M1296744

Barakat_2021_SIAM-DM_generation-rank-3_VoR
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.
Final published version, 1.25 MB

Cite this

@article{14604745cba842049812082bc2581dae,

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

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. ",

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",

author = "M. Barakat and R. Behrends and C. Jefferson and L. K{\"u}hne and M. Leuner",

note = "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. ",

year = "2021",

month = jun,

day = "8",

doi = "10.1137/19M1296744",

language = "English",

volume = "35",

pages = "1201--1223",

journal = "SIAM Journal on Discrete Mathematics",

issn = "0895-4801",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "2",

}

TY - JOUR

T1 - On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture

AU - Barakat, M.

AU - Behrends, R.

AU - Jefferson, C.

AU - Kühne, L.

AU - Leuner, M.

N1 - 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.

PY - 2021/6/8

Y1 - 2021/6/8

N2 - 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.

AB - 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.

KW - ArangoDB

KW - Integrally splitting characteristic polynomial

KW - Iterator of leaves of rooted tree

KW - Leaf-iterator

KW - NoSQL database

KW - Parallel evaluation of recursive iterator

KW - Priority queue

KW - Rank 3 simple matroids

KW - Recursive iterator

KW - Terao's freeness conjecture

KW - Tree-iterator

UR - https://arxiv.org/pdf/1907.01073

U2 - 10.1137/19M1296744

DO - 10.1137/19M1296744

M3 - Article

SN - 0895-4801

VL - 35

SP - 1201

EP - 1223

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this