{"id":"https://openalex.org/W7204436317","doi":"https://doi.org/10.48550/arxiv.2608.25891","title":"Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes","display_name":"Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes","publication_year":2026,"publication_date":"2026-08-26","ids":{"openalex":"https://openalex.org/W7204436317","doi":"https://doi.org/10.48550/arxiv.2608.25891"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.25891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25891","pdf_url":"https://arxiv.org/pdf/2608.25891","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.25891","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023876779","display_name":"Daniel McGinnis","orcid":"https://orcid.org/0000-0003-3645-6272","observed_orcids":["https://orcid.org/0000-0003-3645-6272"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"McGinnis, Daniel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5023876779"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.9980000257492065,"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":0.9980000257492065,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.00039999998989515007,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/generic-initial-ideals","display_name":"generic initial ideals","score":0.8679999709129333},{"id":"https://openalex.org/keywords/castelnuovo-mumford-regularity","display_name":"Castelnuovo-Mumford regularity","score":0.9039999842643738},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"simplicial complexes","score":0.8859999775886536},{"id":"https://openalex.org/keywords/monomial-orders","display_name":"monomial orders","score":0.6039999723434448}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8867999911308289},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.6815999746322632},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.6417999863624573},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.6180999875068665},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6069999933242798},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6004999876022339},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.589900016784668},{"id":"https://openalex.org/C2777789435","wikidata":"https://www.wikidata.org/wiki/Q19369206","display_name":"Monomial ideal","level":4,"score":0.5218999981880188},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49300000071525574},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.4088999927043915},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.36640000343322754},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.3594000041484833},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.33149999380111694},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.33000001311302185},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.30230000615119934},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.29910001158714294},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29010000824928284},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C46132437","wikidata":"https://www.wikidata.org/wiki/Q6901742","display_name":"Monomial basis","level":3,"score":0.2599000036716461},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2529999911785126}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.25891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25891","pdf_url":"https://arxiv.org/pdf/2608.25891","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.25891","doi":"https://doi.org/10.48550/arxiv.2608.25891","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.25891","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.25891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25891","pdf_url":"https://arxiv.org/pdf/2608.25891","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/G1938303756","display_name":null,"funder_award_id":"IBS\u2010R029\u2010C1","funder_id":"https://openalex.org/F4320326441","funder_display_name":"Institute for Basic Science"},{"id":"https://openalex.org/G3247244596","display_name":null,"funder_award_id":"2402145","funder_id":"https://openalex.org/F4320326441","funder_display_name":"Institute for Basic Science"},{"id":"https://openalex.org/G3601956145","display_name":null,"funder_award_id":"IBS-R029-C1","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8830043004","display_name":"Postdoctoral Fellowship: MSPRF: Discrete Geometry and Geometric Transversal Theory","funder_award_id":"2402145","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320326441","display_name":"Institute for Basic Science","ror":"https://ror.org/00y0zf565"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204436317.pdf","grobid_xml":"https://content.openalex.org/works/W7204436317.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7128488395"],"related_works":["https://openalex.org/W2009820428","https://openalex.org/W1963947382","https://openalex.org/W4411692200","https://openalex.org/W1491300949","https://openalex.org/W2037006863","https://openalex.org/W2065429939","https://openalex.org/W1641620567","https://openalex.org/W1597821535","https://openalex.org/W7099674608","https://openalex.org/W1505679232"],"abstract_inverted_index":{"A":[0],"celebrated":[1],"result":[2,168],"of":[3,16,23,56,74,84,90,123,133,137,141,147,153,173,198],"Bayer":[4],"and":[5,25,66],"Stillman":[6],"from":[7,195],"1987":[8],"states":[9],"that":[10,105],"for":[11,79,188],"a":[12,17,46,61,71,170,192],"homogeneous":[13],"ideal":[14,78,82,118,146],"$I$":[15,83,103,148],"polynomial":[18,47],"ring":[19,48],"$S$,":[20],"the":[21,28,31,39,75,87,91,96,106,114,121,131,142,158,165,176,183,196],"regularities":[22,107],"$S/I$":[24],"$S/\\textrm{GIN}(I)$":[26],"are":[27,51,100,108],"same":[29],"under":[30],"reverse":[32,159],"lexicographic":[33,160],"monomial":[34,124,161],"ordering,":[35,162],"where":[36],"$\\textrm{GIN}(I)$":[37],"is":[38,45,60],"generic":[40,76,116,144],"initial":[41,77,117,145],"ideal.":[42],"If":[43],"$R$":[44],"whose":[49],"variables":[50,57],"subdivided":[52],"into":[53],"disjoint":[54],"blocks":[55],"$X_1,\\dots,X_c$,":[58],"there":[59,99],"natural":[62],"multi-grading":[63],"on":[64,130,175],"$R$,":[65],"one":[67],"can":[68],"analogously":[69],"define":[70],"multi-graded":[72,97,115,143],"version":[73],"any":[80],"multi-homogeneous":[81,101],"$R$.":[85],"However,":[86],"full":[88],"strength":[89],"Bayer--Stillman":[92],"Theorem":[93],"fails":[94],"in":[95,135,200],"setting;":[98],"ideals":[102],"such":[104],"not":[109],"preserved":[110],"after":[111],"passing":[112],"to":[113,150],"no":[119],"matter":[120],"choice":[122,172],"ordering.":[125],"We":[126],"prove":[127,182],"lower":[128,166],"bounds":[129],"regularity":[132],"$R/I$":[134],"terms":[136],"almost":[138],"regular":[139],"sequences":[140],"restricted":[149],"each":[151],"block":[152],"variables.":[154,177],"Again,":[155],"we":[156,181],"use":[157],"but":[163],"interestingly,":[164],"bound":[167],"requires":[169],"particular":[171],"ordering":[174],"As":[178],"an":[179],"application,":[180],"optimal":[184],"fractional":[185],"Helly":[186],"theorem":[187],"$d$-Leray":[189],"simplicial":[190],"complexes,":[191],"problem":[193],"stemming":[194],"work":[197],"Kim":[199],"2017.":[201]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-28T00:00:00"}
