{"id":"https://openalex.org/W7151341253","doi":"https://doi.org/10.48550/arxiv.2604.04252","title":"The Bourbaki degree of the syzygy module of 2 $\\times$ 4 matrices","display_name":"The Bourbaki degree of the syzygy module of 2 $\\times$ 4 matrices","publication_year":2026,"publication_date":"2026-04-05","ids":{"openalex":"https://openalex.org/W7151341253","doi":"https://doi.org/10.48550/arxiv.2604.04252"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2604.04252","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2604.04252","pdf_url":"https://arxiv.org/pdf/2604.04252","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2604.04252","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069991409","display_name":"Marcos Jardim","orcid":"https://orcid.org/0000-0002-8419-7044","observed_orcids":["https://orcid.org/0000-0002-8419-7044"]},"institutions":[{"id":"https://openalex.org/I161028277","display_name":"Institute for Advanced Studies in Basic Sciences","ror":"https://ror.org/00bzsst90","country_code":"IR","type":"education","lineage":["https://openalex.org/I161028277"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Jardim, Marcos","raw_affiliation_strings":["Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran","institution_ids":["https://openalex.org/I161028277"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133147163","display_name":"Felipe Monteiro","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I161028277","display_name":"Institute for Advanced Studies in Basic Sciences","ror":"https://ror.org/00bzsst90","country_code":"IR","type":"education","lineage":["https://openalex.org/I161028277"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Monteiro, Felipe","raw_affiliation_strings":["Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran","institution_ids":["https://openalex.org/I161028277"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066627498","display_name":"Abbas Nasrollah Nejad","orcid":"https://orcid.org/0000-0003-0511-6099","observed_orcids":["https://orcid.org/0000-0003-0511-6099"]},"institutions":[{"id":"https://openalex.org/I161028277","display_name":"Institute for Advanced Studies in Basic Sciences","ror":"https://ror.org/00bzsst90","country_code":"IR","type":"education","lineage":["https://openalex.org/I161028277"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Nejad, Abbas Nasrollah","raw_affiliation_strings":["Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Institute for Advanced Studies in Basic Sciences (IASBS) , Zanjan 45137-66731 , Iran","institution_ids":["https://openalex.org/I161028277"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I161028277"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9984999895095825,"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"}},"topics":[{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9984999895095825,"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"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0010000000474974513,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.00019999999494757503,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/syzygy-modules","display_name":"syzygy modules","score":0.8740000128746033},{"id":"https://openalex.org/keywords/hilbert-coefficients","display_name":"Hilbert coefficients","score":0.9380000233650208},{"id":"https://openalex.org/keywords/perfect-ideals","display_name":"perfect ideals","score":0.7630000114440918},{"id":"https://openalex.org/keywords/plane-curves","display_name":"plane curves","score":0.6039999723434448},{"id":"https://openalex.org/keywords/jacobian-matrix","display_name":"Jacobian matrix","score":0.8100000023841858},{"id":"https://openalex.org/keywords/polynomial-rings","display_name":"polynomial rings","score":0.42500001192092896}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8518999814987183},{"id":"https://openalex.org/C55101346","wikidata":"https://www.wikidata.org/wiki/Q85776755","display_name":"Hilbert's syzygy theorem","level":2,"score":0.8361999988555908},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6467999815940857},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.5638999938964844},{"id":"https://openalex.org/C83979697","wikidata":"https://www.wikidata.org/wiki/Q1778247","display_name":"Codimension","level":2,"score":0.5454999804496765},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4404999911785126},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3547999858856201},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3472000062465668},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.3353999853134155},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.3305000066757202},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31220000982284546},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.31150001287460327},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3012000024318695},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.29120001196861267},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.28299999237060547},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.2734000086784363},{"id":"https://openalex.org/C43809302","wikidata":"https://www.wikidata.org/wiki/Q1060874","display_name":"Plane curve","level":2,"score":0.2624000012874603}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2604.04252","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2604.04252","pdf_url":"https://arxiv.org/pdf/2604.04252","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2604.04252","doi":"https://doi.org/10.48550/arxiv.2604.04252","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.04252","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2604.04252","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2604.04252","pdf_url":"https://arxiv.org/pdf/2604.04252","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1203447851","display_name":null,"funder_award_id":"ANR-21-CE40-0017","funder_id":"https://openalex.org/F4320320997","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo"},{"id":"https://openalex.org/G1593819098","display_name":null,"funder_award_id":"2021/ 04065-6","funder_id":"https://openalex.org/F4320320997","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo"},{"id":"https://openalex.org/G1811992940","display_name":null,"funder_award_id":"305601/2022-9","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G2659019386","display_name":null,"funder_award_id":"ANR-17-EURE-","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G3525068622","display_name":"Brazilian Center for Geometry (CBG)","funder_award_id":"24/00923-6","funder_id":"https://openalex.org/F4320320997","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo"},{"id":"https://openalex.org/G5364903624","display_name":"Brazil-France interplays in Gauge Theory, extremal structures and stability","funder_award_id":"ANR-21-CE40-0017","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G544569310","display_name":"The Aluffi Algebras and Free Divisors","funder_award_id":"22/09853-5","funder_id":"https://openalex.org/F4320320997","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo"},{"id":"https://openalex.org/G6129730045","display_name":"Ing\u00e9nierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarit\u00e9","funder_award_id":"ANR-17-EURE-0002","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G6438247056","display_name":null,"funder_award_id":"2021/","funder_id":"https://openalex.org/F4320320997","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo"},{"id":"https://openalex.org/G6756993358","display_name":null,"funder_award_id":"/2022-9","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G6803667113","display_name":null,"funder_award_id":"ANR-17","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G8841590573","display_name":null,"funder_award_id":"2022/09853-5","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320320997","display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","ror":"https://ror.org/02ddkpn78"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"},{"id":"https://openalex.org/F4320323854","display_name":"Universidade Estadual de Campinas","ror":"https://ror.org/04wffgt70"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7151341253.pdf","grobid_xml":"https://content.openalex.org/works/W7151341253.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4401717167","https://openalex.org/W4406093455","https://openalex.org/W4415937516"],"related_works":["https://openalex.org/W7161526115","https://openalex.org/W4402402735","https://openalex.org/W4386114286","https://openalex.org/W7123514309","https://openalex.org/W7212913416","https://openalex.org/W7214018029","https://openalex.org/W2636819849","https://openalex.org/W7160869638","https://openalex.org/W2183933437","https://openalex.org/W1963843435"],"abstract_inverted_index":{"We":[0,88],"introduce":[1],"and":[2,47,75,130,136,155,202],"study":[3],"the":[4,37,58,64,67,69,76,82,117,146,161,184,206,210],"Bourbaki":[5,34,59,118,153,168,207],"degree":[6,60,71,119,169,208,212],"as":[7],"a":[8,21,33,73,106,127,141,164,173,191,198],"numerical":[9],"invariant":[10],"for":[11,36,44,140,179,194],"\\(2":[12],"\\times":[13],"4\\)":[14],"matrices":[15,97],"$\u0398$":[16],"of":[17,63,66,72,81,163,187],"homogeneous":[18],"polynomials":[19],"over":[20],"polynomial":[22],"ring":[23],"\\(R":[24],"=":[25,86,110],"k[x_1,":[26],"\\dots,":[27],"x_n]\\).":[28],"This":[29,158],"invariant,":[30],"defined":[31],"via":[32],"sequence":[35],"syzygy":[38],"module":[39,84],"\\(\\operatorname{Syz}(\u0398)\\),":[40],"generalizes":[41],"previous":[42],"constructions":[43],"plane":[45],"curves":[46],"Jacobian":[48,180],"matrices.":[49,181],"Our":[50],"main":[51],"result":[52],"is":[53,124,213],"an":[54],"explicit":[55],"formula":[56],"expressing":[57],"in":[61,183],"terms":[62],"degrees":[65,154],"rows,":[68],"initial":[70,211],"syzygy,":[74],"first":[77,100],"two":[78,93],"Hilbert":[79],"coefficients":[80],"cokernel":[83],"\\(\\mathcal{Q}":[85],"\\operatorname{coker}(\u0398)\\).":[87],"apply":[89],"this":[90],"framework":[91],"to":[92,149,171,196],"important":[94],"cases.":[95],"First,":[96],"with":[98,167],"constant":[99],"row,":[101],"which":[102],"are":[103],"determined":[104],"by":[105],"three-equigenerated":[107],"ideal":[108],"\\(J":[109],"(f_1,":[111],"f_2,":[112],"f_3)\\),":[113],"where":[114],"we":[115,131,144,189],"show":[116],"measures":[120],"how":[121],"far":[122],"\\(J\\)":[123],"from":[125],"being":[126],"perfect":[128],"ideal,":[129],"completely":[132],"characterize":[133],"its":[134],"smaller":[135],"larger":[137],"values.":[138],"Second,":[139],"linear":[142,165],"matrix,":[143],"use":[145],"Kronecker--Weierstrass":[147],"classification":[148,159],"determine":[150],"all":[151],"possible":[152],"homological":[156],"types.":[157],"reveals":[160],"existence":[162],"matrix":[166],"equal":[170],"2,":[172],"value":[174],"that":[175],"does":[176],"not":[177],"occur":[178],"Finally,":[182],"geometric":[185],"context":[186],"\\(\\mathbb{P}^3\\),":[188],"provide":[190],"sufficient":[192],"condition":[193],"\\(\\operatorname{Syz}(\u0398)\\)":[195],"define":[197],"codimension":[199],"one":[200],"distribution":[201],"obtain":[203],"bounds":[204],"on":[205],"when":[209],"small.":[214]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-04-08T00:00:00"}