{"id":"https://openalex.org/W7215120981","doi":"https://doi.org/10.48550/arxiv.2609.40123","title":"Galois groups of random polynomials of large degree","display_name":"Galois groups of random polynomials of large degree","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7215120981","doi":"https://doi.org/10.48550/arxiv.2609.40123"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.40123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40123","pdf_url":"https://arxiv.org/pdf/2609.40123","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/2609.40123","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071160828","display_name":"Guy Blachar","orcid":"https://orcid.org/0009-0001-6829-5461","observed_orcids":["https://orcid.org/0009-0001-6829-5461"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Blachar, Guy","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024332688","display_name":"Emmanuel Breuillard","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Breuillard, Emmanuel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5061776834","display_name":"Gady Kozma","orcid":"https://orcid.org/0000-0003-3033-9235","observed_orcids":["https://orcid.org/0000-0003-3033-9235"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kozma, Gady","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/T11166","display_name":"Analytic Number Theory Research","score":0.37869998812675476,"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":0.37869998812675476,"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":0.18639999628067017,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.10520000010728836,"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/galois-groups","display_name":"Galois groups","score":0.8450000286102295},{"id":"https://openalex.org/keywords/random-polynomials","display_name":"random polynomials","score":0.9869999885559082},{"id":"https://openalex.org/keywords/renyi-entropy","display_name":"R\u00e9nyi entropy","score":0.8309999704360962},{"id":"https://openalex.org/keywords/high-probability-asymptotics","display_name":"high probability asymptotics","score":0.4650000035762787}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8855999708175659},{"id":"https://openalex.org/C67536143","wikidata":"https://www.wikidata.org/wiki/Q730384","display_name":"Galois group","level":2,"score":0.4927000105381012},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.47870001196861267},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4406000077724457},{"id":"https://openalex.org/C94398972","wikidata":"https://www.wikidata.org/wiki/Q92552","display_name":"Galois theory","level":2,"score":0.427700012922287},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.41769999265670776},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.3846000134944916},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.37929999828338623},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36230000853538513},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3472000062465668},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.33889999985694885},{"id":"https://openalex.org/C35435516","wikidata":"https://www.wikidata.org/wiki/Q5370016","display_name":"Embedding problem","level":3,"score":0.3375999927520752},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.3197999894618988},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.30149999260902405},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.30140000581741333},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.29989999532699585},{"id":"https://openalex.org/C130402806","wikidata":"https://www.wikidata.org/wiki/Q5361768","display_name":"Random field","level":2,"score":0.28700000047683716},{"id":"https://openalex.org/C13929819","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Random function","level":3,"score":0.28209999203681946},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.2782000005245209},{"id":"https://openalex.org/C52627884","wikidata":"https://www.wikidata.org/wiki/Q4424101","display_name":"Random compact set","level":4,"score":0.2741999924182892},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.2727999985218048},{"id":"https://openalex.org/C205567592","wikidata":"https://www.wikidata.org/wiki/Q3527199","display_name":"Normal basis","level":3,"score":0.26840001344680786},{"id":"https://openalex.org/C52386014","wikidata":"https://www.wikidata.org/wiki/Q166211","display_name":"Random element","level":3,"score":0.25450000166893005},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.40123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40123","pdf_url":"https://arxiv.org/pdf/2609.40123","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.2609.40123","doi":"https://doi.org/10.48550/arxiv.2609.40123","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.40123","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:2609.40123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40123","pdf_url":"https://arxiv.org/pdf/2609.40123","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215120981.pdf","grobid_xml":"https://content.openalex.org/works/W7215120981.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4415111106","https://openalex.org/W4416057642","https://openalex.org/W7115596567"],"related_works":["https://openalex.org/W2179018518","https://openalex.org/W4405701308","https://openalex.org/W7160339236","https://openalex.org/W4416057642","https://openalex.org/W4225526748","https://openalex.org/W4389040504","https://openalex.org/W3045689250","https://openalex.org/W2767106737","https://openalex.org/W3194468933","https://openalex.org/W4400607074"],"abstract_inverted_index":{"We":[0,25],"study":[1],"random":[2,15],"polynomials":[3],"of":[4,33],"the":[5,30],"form":[6],"$R(x)=x^n+\u03c9_{n-1}x^{n-1}+\\cdots+\u03c9_0$,":[7],"where":[8],"$\u03c9_0,\\dots,\u03c9_{n-1}$":[9],"are":[10],"independent,":[11],"uniformly":[12],"bounded":[13],"integer-valued":[14],"variables,":[16],"and":[17,55],"$\u03c9_1,\\dots,\u03c9_{n-1}$":[18],"have":[19],"a":[20,41],"fixed":[21],"common":[22],"law":[23],"$\u03bc$.":[24],"prove":[26],"(unconditionally)":[27],"that,":[28,50],"if":[29],"R\u00e9nyi":[31],"entropy":[32],"order":[34],"$2$":[35],"satisfies":[36],"$H_2(\u03bc)=-\\log\\|\u03bc\\|_2^2>12$,":[37],"then":[38],"$\\mathbb{P}(\\operatorname{disc}(R)\\text{":[39],"is":[40],"square})=O_\u03bc(1/\\log":[42],"n)$.":[43],"Combined":[44],"with":[45,65],"previous":[46],"results,":[47],"this":[48],"shows":[49],"for":[51],"such":[52],"measures":[53],"$\u03bc$":[54],"under":[56],"additional":[57],"hypotheses,":[58],"$R$":[59],"has":[60],"full":[61],"Galois":[62],"group":[63],"$\\mathrm{Sym}(n)$":[64],"high":[66],"probability,":[67],"when":[68],"conditioned":[69],"on":[70],"$\u03c9_0\\ne":[71],"0$.":[72]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-02T00:00:00"}
