{"id":"https://openalex.org/W7204194257","doi":"https://doi.org/10.48550/arxiv.2608.22297","title":"The strong form of Van der Waerden's conjecture via twisted Chowla","display_name":"The strong form of Van der Waerden's conjecture via twisted Chowla","publication_year":2026,"publication_date":"2026-08-23","ids":{"openalex":"https://openalex.org/W7204194257","doi":"https://doi.org/10.48550/arxiv.2608.22297"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.22297","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22297","pdf_url":"https://arxiv.org/pdf/2608.22297","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","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/2608.22297","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079190763","display_name":"Theresa C. Anderson","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Anderson, Theresa C.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5125712406","display_name":"Sam Chow","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chow, Sam","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080350415","display_name":"Rainer Dietmann","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dietmann, Rainer","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071115111","display_name":"David Hokken","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hokken, David","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007525723","display_name":"Peter Koymans","orcid":"https://orcid.org/0000-0002-7582-4149","observed_orcids":["https://orcid.org/0000-0002-7582-4149"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Koymans, Peter","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102849268","display_name":"Robert J. Lemke Oliver","orcid":"https://orcid.org/0000-0002-7170-051X","observed_orcids":["https://orcid.org/0000-0002-7170-051X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Oliver, Robert J. Lemke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066955363","display_name":"Will Sawin","orcid":"https://orcid.org/0000-0003-0344-5438","observed_orcids":["https://orcid.org/0000-0003-0344-5438"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sawin, Will","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5090576518","display_name":"Mark Shusterman","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shusterman, Mark","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":1,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9970999956130981,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9970999956130981,"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":0.0012000000569969416,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.000699999975040555,"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/random-polynomials","display_name":"random polynomials","score":0.6190000176429749},{"id":"https://openalex.org/keywords/monic-irreducible-polynomials","display_name":"monic irreducible polynomials","score":0.890999972820282},{"id":"https://openalex.org/keywords/function-fields","display_name":"function fields","score":0.5490000247955322},{"id":"https://openalex.org/keywords/chowla-s-conjecture","display_name":"Chowla's conjecture","score":0.6809999942779541},{"id":"https://openalex.org/keywords/counting-irreducible-polynomials","display_name":"counting irreducible polynomials","score":0.5040000081062317}],"concepts":[{"id":"https://openalex.org/C116493835","wikidata":"https://www.wikidata.org/wiki/Q764287","display_name":"Van der Waerden's theorem","level":2,"score":0.9546999931335449},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8310999870300293},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8302000164985657},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6032000184059143},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.5706999897956848},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.5644000172615051},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5001999735832214},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.4814000129699707},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4359999895095825},{"id":"https://openalex.org/C173808221","wikidata":"https://www.wikidata.org/wiki/Q50705","display_name":"Square number","level":2,"score":0.4081000089645386},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4047999978065491},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.358599990606308},{"id":"https://openalex.org/C12657307","wikidata":"https://www.wikidata.org/wiki/Q616608","display_name":"Algebraic number field","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.32429999113082886},{"id":"https://openalex.org/C17732976","wikidata":"https://www.wikidata.org/wiki/Q7511782","display_name":"Sieve (category theory)","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C143732363","wikidata":"https://www.wikidata.org/wiki/Q485520","display_name":"Goldbach's conjecture","level":3,"score":0.27300000190734863},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26159998774528503},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.2599000036716461}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.22297","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22297","pdf_url":"https://arxiv.org/pdf/2608.22297","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.22297","doi":"https://doi.org/10.48550/arxiv.2608.22297","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.22297","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.22297","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22297","pdf_url":"https://arxiv.org/pdf/2608.22297","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204194257.pdf","grobid_xml":"https://content.openalex.org/works/W7204194257.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W79184385","https://openalex.org/W3015805420","https://openalex.org/W4367000173","https://openalex.org/W4388685164","https://openalex.org/W4389775193","https://openalex.org/W4405095188","https://openalex.org/W4415521157","https://openalex.org/W4415958999","https://openalex.org/W4416016454","https://openalex.org/W4416239195","https://openalex.org/W6929382730","https://openalex.org/W7140588827"],"related_works":["https://openalex.org/W4362500383","https://openalex.org/W3212401890","https://openalex.org/W4312122038","https://openalex.org/W2028072114","https://openalex.org/W7155094365","https://openalex.org/W2316015595","https://openalex.org/W7214633812","https://openalex.org/W4400141808","https://openalex.org/W3176320575","https://openalex.org/W12649081"],"abstract_inverted_index":{"Determining":[0],"the":[1,12,37,116,123,126,191],"properties":[2],"of":[3,18,25,39,64,75,110,128,158,190,194],"a":[4,22,73,153],"random":[5],"polynomial":[6],"has":[7],"fuelled":[8],"significant":[9,179],"investigation":[10],"over":[11],"past":[13],"century.":[14],"One":[15],"driving":[16,146],"force":[17],"this":[19,97,139,170,182],"research":[20],"is":[21,135],"1936":[23],"paper":[24],"Van":[26,76,118,164,195],"der":[27,77,119,165,196],"Waerden.":[28],"Fix":[29],"$n":[30,102,173,201],"\\ge":[31,103,174],"3$":[32],"and":[33,112,161],"let":[34],"$E_n(B)$":[35],"be":[36],"number":[38,127],"monic,":[40],"irreducible,":[41],"non-$S_n$":[42],"polynomials":[43,129],"$f":[44],"=":[45,88],"X^n":[46],"+":[47,50,52],"a_1":[48],"X^{n-1}":[49],"\\cdots":[51],"a_n$":[53],"with":[54,108,131],"$|a_j|":[55],"\\leq":[56],"B$":[57],"for":[58,100,172,199],"all":[59,101,200],"$j$.":[60],"A":[61],"recent":[62,92],"breakthrough":[63],"Bhargava":[65],"bounds":[66],"$E_n(B)":[67,87],"\\ll":[68],"B^{n-1}$.":[69],"This":[70],"spectacularly":[71],"resolves":[72],"conjecture":[74,121,167,171,198],"Waerden,":[78],"but":[79],"leaves":[80],"open":[81],"its":[82],"stronger":[83,98],"form,":[84],"namely":[85],"that":[86,125],"o(B^{n-1})$.":[89],"Inspired":[90],"by":[91],"progress,":[93],"we":[94,151,177],"now":[95],"prove":[96],"form":[99,193],"40$.":[104,203],"Bhargava's":[105],"result,":[106],"together":[107],"work":[109],"Chow":[111],"Dietmann,":[113],"essentially":[114],"reduces":[115,168],"strong":[117,192],"Waerden":[120],"to":[122,141,169,187],"claim":[124,140],"$f$":[130],"Galois":[132],"group":[133],"$A_n$":[134],"$o(B^{n-1})$.":[136],"We":[137],"connect":[138],"Chowla's":[142,159],"conjecture,":[143,160,185],"an":[144],"area":[145],"major":[147],"mathematical":[148],"progress.":[149],"Indeed,":[150],"formulate":[152],"twisted":[154,183],"function":[155],"field":[156],"version":[157],"show":[162],"how":[163],"Waerden's":[166,197],"8$.":[175],"Finally,":[176],"make":[178],"progress":[180],"towards":[181],"Chowla":[184],"leading":[186],"our":[188],"resolution":[189],"\\geq":[202]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-26T00:00:00"}
