{"id":"https://openalex.org/W7224818528","doi":"https://doi.org/10.48550/arxiv.2610.38474","title":"On the motivic approach to the Abhyankar rectifiability problem and concrete rectifications","display_name":"On the motivic approach to the Abhyankar rectifiability problem and concrete rectifications","publication_year":2026,"publication_date":"2026-10-28","ids":{"openalex":"https://openalex.org/W7224818528","doi":"https://doi.org/10.48550/arxiv.2610.38474"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.38474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.38474","pdf_url":"https://arxiv.org/pdf/2610.38474","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/2610.38474","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160170973","display_name":"A. S. Frolov","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Frolov, A. S.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5160175070","display_name":"I. S. Dorofeev","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dorofeev, I. S.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9872999787330627,"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":0.9872999787330627,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.008100000210106373,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.002099999925121665,"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/torsors","display_name":"torsors","score":0.8270000219345093},{"id":"https://openalex.org/keywords/coordinate-projection","display_name":"coordinate projection","score":0.6899999976158142},{"id":"https://openalex.org/keywords/normal-maps","display_name":"normal maps","score":0.628000020980835},{"id":"https://openalex.org/keywords/branch-height","display_name":"branch height","score":0.328000009059906}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7965999841690063},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.6692000031471252},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5842999815940857},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.5472999811172485},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5072000026702881},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.4797999858856201},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.4323999881744385},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.412200003862381},{"id":"https://openalex.org/C124194741","wikidata":"https://www.wikidata.org/wiki/Q320809","display_name":"Braid","level":2,"score":0.4036000072956085},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.3707999885082245},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36629998683929443},{"id":"https://openalex.org/C4017995","wikidata":"https://www.wikidata.org/wiki/Q595298","display_name":"Sheaf","level":2,"score":0.3610999882221222},{"id":"https://openalex.org/C128489963","wikidata":"https://www.wikidata.org/wiki/Q1187724","display_name":"Filtration (mathematics)","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.3431999981403351},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3418000042438507},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3382999897003174},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.3312000036239624},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.30480000376701355},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28380000591278076},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.2786000072956085},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.2777999937534332},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.274399995803833},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.27149999141693115},{"id":"https://openalex.org/C2777042112","wikidata":"https://www.wikidata.org/wiki/Q5281658","display_name":"Discontinuity (linguistics)","level":2,"score":0.26980000734329224},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.2621000111103058},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C62623296","wikidata":"https://www.wikidata.org/wiki/Q686291","display_name":"Affine group","level":3,"score":0.2574000060558319},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.2531999945640564}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.38474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.38474","pdf_url":"https://arxiv.org/pdf/2610.38474","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.2610.38474","doi":"https://doi.org/10.48550/arxiv.2610.38474","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.38474","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:2610.38474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.38474","pdf_url":"https://arxiv.org/pdf/2610.38474","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":[{"id":"https://openalex.org/F4320330091","display_name":"Moscow Institute of Physics and Technology","ror":"https://ror.org/00v0z9322"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224818528.pdf","grobid_xml":"https://content.openalex.org/works/W7224818528.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W573948285","https://openalex.org/W1981336002","https://openalex.org/W1991729315","https://openalex.org/W2027864161","https://openalex.org/W2035878227","https://openalex.org/W2047926029","https://openalex.org/W2054909812","https://openalex.org/W2057118977","https://openalex.org/W2073909098","https://openalex.org/W2168588397","https://openalex.org/W2963114466","https://openalex.org/W2972699803","https://openalex.org/W3106408843","https://openalex.org/W4236169745","https://openalex.org/W7169871217"],"related_works":["https://openalex.org/W4226153745","https://openalex.org/W7098300912","https://openalex.org/W2129992497","https://openalex.org/W2008312820","https://openalex.org/W7100549710","https://openalex.org/W1532331629","https://openalex.org/W2232720392","https://openalex.org/W2090241231","https://openalex.org/W1985503783","https://openalex.org/W1980895186"],"abstract_inverted_index":{"We":[0,12,83,97,137,156],"study":[1],"closed":[2],"embeddings":[3],"of":[4,27,79,87,92,110],"the":[5,19,24,59,65,76,80,85,88,111,158,164],"complex":[6,94,166],"affine":[7,10,106],"line":[8],"into":[9],"three-space.":[11],"prove":[13,138],"that,":[14],"for":[15,64],"every":[16,93],"such":[17],"embedding,":[18],"complete-intersection":[20],"normal":[21,50,141,178],"map":[22,142],"identifies":[23],"$\\mathbb{A}^1$-fundamental":[25],"sheaf":[26,91],"its":[28],"complement":[29],"with":[30,120,143],"$\\mathbf{K}_2^{\\mathrm{MW}}$.":[31],"The":[32,124],"proof":[33],"compares":[34],"torsors":[35],"under":[36],"arbitrary":[37,121],"strongly":[38],"$\\mathbb{A}^1$-invariant":[39],"group":[40],"sheaves.":[41],"A":[42],"nodal":[43],"projection":[44],"expresses":[45],"a":[46,49,71,115,140,144,153],"torsor":[47],"by":[48,75],"parameter":[51],"and":[52,105,114,134,147,161,180],"finitely":[53],"many":[54],"node":[55,60],"parameters.":[56],"Extension":[57],"across":[58],"axes":[61],"gives":[62],"equations":[63],"latter.":[66],"Their":[67],"correction":[68],"terms":[69],"increase":[70],"nilpotent":[72],"filtration":[73],"generated":[74],"Milnor-Witt":[77],"symbols":[78],"branch-height":[81],"differences.":[82],"deduce":[84],"vanishing":[86],"first":[89],"Alexander":[90],"embedded":[95],"line.":[96],"also":[98],"construct":[99],"polynomial":[100,145],"rectifications":[101,109],"using":[102],"finite":[103],"inverses":[104],"modifications,":[107],"including":[108],"Shastri":[112],"trefoil":[113],"harmonic":[116],"figure-eight":[117],"over":[118],"$\\mathbb{Z}[a]/(a^2-a+1)$,":[119],"base":[122],"change.":[123],"companion":[125],"repository":[126],"(https://github.com/qft98romanov-hash/abhyankar-moh-trefoil)":[127],"contains":[128],"explicit":[129],"rectifiers,":[130],"their":[131],"inverses,":[132],"proofs,":[133],"symbolic":[135],"checks.":[136],"that":[139],"section":[146],"geometric":[148],"generic":[149],"affine-line":[150],"fiber":[151],"is":[152,169],"coordinate":[154],"projection.":[155],"formulate":[157],"remaining":[159],"rectifiability":[160,167],"descent":[162],"questions;":[163],"general":[165],"problem":[168],"not":[170],"settled":[171],"here.":[172],"An":[173],"appendix":[174],"treats":[175],"other":[176],"fields,":[177],"orientations,":[179],"arithmetic":[181],"descent.":[182]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-31T00:00:00"}