{"id":"https://openalex.org/W7222274384","doi":"https://doi.org/10.48550/arxiv.2610.20025","title":"On the Topology of Fibers of Rational Functions","display_name":"On the Topology of Fibers of Rational Functions","publication_year":2026,"publication_date":"2026-10-14","ids":{"openalex":"https://openalex.org/W7222274384","doi":"https://doi.org/10.48550/arxiv.2610.20025"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.20025","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20025","pdf_url":"https://arxiv.org/pdf/2610.20025","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/2610.20025","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025035712","display_name":"Kiyoshi Takeuchi","orcid":"https://orcid.org/0000-0002-8392-1029","observed_orcids":["https://orcid.org/0000-0002-8392-1029"]},"institutions":[{"id":"https://openalex.org/I201537933","display_name":"Tohoku University","ror":"https://ror.org/01dq60k83","country_code":"JP","type":"education","lineage":["https://openalex.org/I201537933"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Takeuchi, Kiyoshi","raw_affiliation_strings":["Mathematical Institute , Tohoku University , Aramaki Aza-Aoba 6-3 , Aobaku , Sendai , 980-8578 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Institute , Tohoku University , Aramaki Aza-Aoba 6-3 , Aobaku , Sendai , 980-8578 , Japan","institution_ids":["https://openalex.org/I201537933"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5025035712"],"corresponding_institution_ids":["https://openalex.org/I201537933"],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9829000234603882,"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":0.9829000234603882,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.006500000134110451,"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/T12351","display_name":"Geometry and complex manifolds","score":0.003700000001117587,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rational-functions","display_name":"rational functions","score":0.9160000085830688},{"id":"https://openalex.org/keywords/homotopy-type","display_name":"homotopy type","score":0.625},{"id":"https://openalex.org/keywords/cohomology-groups","display_name":"cohomology groups","score":0.6869999766349792},{"id":"https://openalex.org/keywords/meromorphic-functions","display_name":"meromorphic functions","score":0.9660000205039978}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7803000211715698},{"id":"https://openalex.org/C75190567","wikidata":"https://www.wikidata.org/wiki/Q41237","display_name":"Rational function","level":2,"score":0.6531000137329102},{"id":"https://openalex.org/C190333341","wikidata":"https://www.wikidata.org/wiki/Q217616","display_name":"Meromorphic function","level":2,"score":0.6078000068664551},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.5831000208854675},{"id":"https://openalex.org/C78606066","wikidata":"https://www.wikidata.org/wiki/Q1198376","display_name":"Cohomology","level":2,"score":0.5655999779701233},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5461999773979187},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5450999736785889},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.5066999793052673},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.49549999833106995},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4072999954223633},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.3395000100135803},{"id":"https://openalex.org/C519885992","wikidata":"https://www.wikidata.org/wiki/Q161","display_name":"Fiber","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.328000009059906},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29679998755455017},{"id":"https://openalex.org/C147688034","wikidata":"https://www.wikidata.org/wiki/Q1626416","display_name":"Homotopy group","level":3,"score":0.287200003862381},{"id":"https://openalex.org/C35234519","wikidata":"https://www.wikidata.org/wiki/Q3983973","display_name":"Homotopy lifting property","level":4,"score":0.27379998564720154},{"id":"https://openalex.org/C156330621","wikidata":"https://www.wikidata.org/wiki/Q3393476","display_name":"Rational point","level":3,"score":0.2727000117301941},{"id":"https://openalex.org/C137212723","wikidata":"https://www.wikidata.org/wiki/Q1948412","display_name":"Morphism","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.20025","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20025","pdf_url":"https://arxiv.org/pdf/2610.20025","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.2610.20025","doi":"https://doi.org/10.48550/arxiv.2610.20025","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.20025","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:2610.20025","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20025","pdf_url":"https://arxiv.org/pdf/2610.20025","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222274384.pdf","grobid_xml":"https://content.openalex.org/works/W7222274384.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2895826685","https://openalex.org/W4308670811","https://openalex.org/W7196956581"],"related_works":["https://openalex.org/W7096787659","https://openalex.org/W7196956581","https://openalex.org/W2119862054","https://openalex.org/W1997281768","https://openalex.org/W4408677450","https://openalex.org/W2027922085","https://openalex.org/W2008746042","https://openalex.org/W7097011118","https://openalex.org/W2062743852","https://openalex.org/W30037198"],"abstract_inverted_index":{"In":[0,43,109],"1988":[1],"Broughton":[2],"proved":[3],"that":[4,87],"if":[5],"a":[6,79,92,102],"complex":[7],"polynomial":[8,33],"function":[9,81],"on":[10,78,82],"${\\mathbb":[11,83],"C}^n$":[12,84],"is":[13],"tame":[14],"at":[15,64,69],"infinity":[16,65],"its":[17],"fibers":[18],"are":[19,97],"homotopy":[20],"equivalent":[21],"to":[22,57,99],"bouquets":[23],"of":[24,32,53,72,91,95,101,104,112,117],"some":[25,48,61,76,107],"spheres.":[26,108],"Since":[27],"then":[28],"several":[29],"basic":[30],"properties":[31],"functions":[34,59,126],"were":[35],"studied":[36],"by":[37],"many":[38],"mathematicians":[39],"in":[40],"various":[41],"situations.":[42],"this":[44],"paper,":[45],"we":[46,85],"show":[47],"evidence":[49],"for":[50,124],"the":[51,67,88,110,113,115,118],"possibility":[52],"extending":[54],"Broughton's":[55],"result":[56],"rational":[58,80],"satisfying":[60],"tameness":[62],"condition":[63],"and":[66,106,120],"one":[68],"their":[70],"points":[71],"indeterminacy.":[73],"Specifically,":[74],"assuming":[75],"conditions":[77],"prove":[86],"cohomology":[89],"groups":[90],"generic":[93],"fiber":[94],"it":[96],"isomorphic":[98],"those":[100],"bouquet":[103],"$S^1$":[105],"course":[111],"proof,":[114],"theory":[116],"nearby":[119],"vanishing":[121],"cycle":[122],"sheaves":[123],"meromorphic":[125],"will":[127],"play":[128],"an":[129],"essential":[130],"role.":[131]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-16T00:00:00"}