{"id":"https://openalex.org/W3192642756","doi":"https://doi.org/10.1137/21m1404417","title":"Geometry of the Minimal Solutions of a Linear Diophantine Equation","display_name":"Geometry of the Minimal Solutions of a Linear Diophantine Equation","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3192642756","doi":"https://doi.org/10.1137/21m1404417","mag":"3192642756"},"language":"en","primary_location":{"id":"doi:10.1137/21m1404417","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1404417","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2108.05886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5035763020","display_name":"Papa A. Sissokho","orcid":"https://orcid.org/0000-0001-8897-0201"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Papa A. Sissokho","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5035763020"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.19812126,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"35","issue":"3","first_page":"1952","last_page":"1963"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9909999966621399,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9909999966621399,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9908999800682068,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.98580002784729,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/diophantine-equation","display_name":"Diophantine equation","score":0.802377462387085},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.7761539220809937},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.7468394041061401},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7168603539466858},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6262317895889282},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6012454032897949},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5237522125244141},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4347638487815857},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.31644952297210693},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07513740658760071}],"concepts":[{"id":"https://openalex.org/C206530611","wikidata":"https://www.wikidata.org/wiki/Q905896","display_name":"Diophantine equation","level":2,"score":0.802377462387085},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.7761539220809937},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7468394041061401},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7168603539466858},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6262317895889282},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6012454032897949},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5237522125244141},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4347638487815857},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.31644952297210693},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07513740658760071},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/21m1404417","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1404417","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2108.05886","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2108.05886","pdf_url":"https://arxiv.org/pdf/2108.05886","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:3192642756","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2108.05886","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2108.05886","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2108.05886","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2108.05886","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2108.05886","pdf_url":"https://arxiv.org/pdf/2108.05886","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3192642756.pdf","grobid_xml":"https://content.openalex.org/works/W3192642756.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W23064316","https://openalex.org/W1604413132","https://openalex.org/W2083860925","https://openalex.org/W2482388004","https://openalex.org/W2963085974"],"related_works":["https://openalex.org/W2922195034","https://openalex.org/W2942762972","https://openalex.org/W2982236222","https://openalex.org/W2802144288","https://openalex.org/W2081923385","https://openalex.org/W3083552369","https://openalex.org/W2069918834","https://openalex.org/W2220460597","https://openalex.org/W84581126","https://openalex.org/W2952713872","https://openalex.org/W3101549238","https://openalex.org/W2165906884","https://openalex.org/W204965768","https://openalex.org/W2363840875","https://openalex.org/W2094535391","https://openalex.org/W2079316362","https://openalex.org/W2396735709","https://openalex.org/W2036864977","https://openalex.org/W300546274","https://openalex.org/W2008384575"],"abstract_inverted_index":{"Let":[0],"$a_1,\\ldots,a_n$":[1],"and":[2,8,58,70,91],"$b_1,\\ldots,b_m$":[3],"be":[4,38],"fixed":[5],"positive":[6],"integers,":[7],"let":[9],"${\\mathcal":[10,30,48],"S}$":[11,31],"denote":[12],"the":[13,21,41,62,89,92],"set":[14],"of":[15,20,43,88,98],"all":[16],"nonnegative":[17],"integer":[18],"solutions":[19,46],"equation":[22],"$x_1a_1+\\ldots":[23],"+x_na_n=y_1b_1+\\ldots":[24],"+y_mb_m$.":[25],"A":[26],"solution":[27,63,83],"$(x_1,\\ldots,x_n,y_1,\\ldots,y_m)$":[28],"in":[29,47],"is":[32,72,84],"called":[33,73],"$\\textit{minimal}$":[34],"if":[35],"it":[36],"cannot":[37],"expressed":[39],"as":[40],"sum":[42],"two":[44],"nonzero":[45,66],"S}$.":[49],"For":[50],"each":[51],"pair":[52],"$(i,j)$":[53],"with":[54],"$1\\leq":[55,59],"i\\leq":[56],"n$":[57],"j\\leq":[60],"m$,":[61],"whose":[64],"only":[65],"coordinates":[67],"are":[68],"$x_i=b_j$":[69],"$y_j=a_i$":[71],"a":[74,85,96],"$\\textit{generator}$.":[75],"Our":[76],"main":[77],"result":[78],"shows":[79],"that":[80],"every":[81],"minimal":[82],"convex":[86],"combination":[87],"generators":[90],"zero-solution.":[93],"This":[94],"proves":[95],"conjecture":[97],"Henk-Weismantel":[99],"and,":[100],"independently,":[101],"Ho\u015ften-Sturmfels.":[102]},"counts_by_year":[],"updated_date":"2026-03-11T14:59:36.786465","created_date":"2025-10-10T00:00:00"}