{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T03:15:05Z","timestamp":1791429305553,"version":"4.3.3"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB-TRR 195"],"award-info":[{"award-number":["SFB-TRR 195"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["StG 716424 - CASe"],"award-info":[{"award-number":["StG 716424 - CASe"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/19m1296744","type":"journal-article","created":{"date-parts":[[2021,6,8]],"date-time":"2021-06-08T15:28:44Z","timestamp":1623166124000},"page":"1201-1223","source":"Crossref","is-referenced-by-count":7,"title":["On the Generation of Rank 3 Simple Matroids with an Application to Terao's Freeness Conjecture"],"prefix":"10.1137","volume":"35","author":[{"given":"Mohamed","family":"Barakat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Reimer","family":"Behrends","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christopher","family":"Jefferson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6393-5610","authenticated-orcid":true,"given":"Lukas","family":"K\u00fchne","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Leuner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,6,8]]},"reference":[{"key":"atypb1","first-page":"100","volume":"8","author":"Abe T.","year":"2014","journal-title":"J. Singul."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-015-0615-7"},{"key":"atypb3","first-page":"389","author":"Abe T.","year":"2017","journal-title":"Switzerland"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2015.12.017"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1520-6610(1999)7:2<119::AID-JCD5>3.0.CO;2-W"},{"key":"atypb8","unstructured":"M. Barakat and L. K\u00fchne,\n                      MatroidGeneration\n                      ,https:\/\/homalg-project.github.io\/pkg\/MatroidGeneration(2019)."},{"key":"atypb9","unstructured":"M. Barakat and L. K\u00fchne,\n                      What is the smallest number $n$ of atoms of a representable rank 3 matroid which is divisionally free but not inductively free?\n                      https:\/\/homalg-project.github.io\/nb\/DivFreeNotIndFree(2019)."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498811004562"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1972-0309764-6"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.2140\/iig.2011.12.49"},{"key":"atypb16","first-page":"1277","volume":"51","author":"N.","year":"1948","journal-title":"Proc."},{"key":"atypb17","unstructured":"W. Decker, G.M. Greuel, G. Pfister, and H. Sch\u00f6nemann,\n                      \\sc Singular\n                      ,http:\/\/www.singular.uni-kl.de(2019)."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2019.05.016"},{"key":"atypb19","first-page":"67","volume":"22","author":"Dong F. M.","year":"1998","journal-title":"Bull. Inst. Combin. Appl."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2004.07.040"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(89)90013-3"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdu046"},{"key":"atypb23","unstructured":"The GAP Group,\n                      GAP-Groups, Algorithms, and Programming\n                      , Version 4.9.2,http:\/\/www.gap-system.org(2018)."},{"key":"atypb24","unstructured":"S. Gutsche, S. Posur, and \u00d8. Skarts\\aeterhagen,\n                      On the syntax and semantics of $\\mathtt{CAP}$\n                      , Proceedings of the Workshop Computer Algebra in the Age of Types, Hagenberg, Austria, 2018, O. Hasan, M. Pfeiffer, and G. D. Reis, eds.http:\/\/ceur-ws.org\/Vol-2307\/(2018)."},{"key":"atypb25","unstructured":"homalg project authors,\n                      homalg Project\n                      ,http:\/\/homalg-project.github.io\/prj\/homalg_project(2020)."},{"key":"atypb26","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1515\/crelle-2013-0022","volume":"701","author":"Hoge T.","year":"2015","journal-title":"J. Reine Angew. Math."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2018.11.009"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2017.12.003"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(84)90024-0"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2011.01.010"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80103-4"},{"key":"atypb33","unstructured":"Y. Matsumoto, S. Moriyama, H. Imai, and D. Bremner,\n                      Database of Matroids\n                      ,http:\/\/www-imai.is.s.u-tokyo.ac.jp\/~ymatsu\/matroid\/index.html(2012)."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-011-9388-y"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2016.10.041"},{"key":"atypb36","unstructured":"J. Oxley,\n                      Matroid Theory\n                      , 2nd ed., Oxford Grad. Texts Math. 21, Oxford University Press, Oxford, 2011."},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1007\/BF02945028"},{"key":"atypb38","first-page":"293","volume":"27","author":"Terao H.","year":"1980","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math."},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389197"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1296744","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:44:38Z","timestamp":1787319878000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1296744"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1296744"],"URL":"https:\/\/doi.org\/10.1137\/19m1296744","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}