{"id":"https://openalex.org/W7167069651","doi":"https://doi.org/10.48550/arxiv.2607.00313","title":"Fast Deterministic Normal Bases and Circulant Polynomial Determinants","display_name":"Fast Deterministic Normal Bases and Circulant Polynomial Determinants","publication_year":2026,"publication_date":"2026-07-01","ids":{"openalex":"https://openalex.org/W7167069651","doi":"https://doi.org/10.48550/arxiv.2607.00313"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.00313","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00313","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,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2607.00313","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5139882027","display_name":"Mark Giesbrecht","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Giesbrecht, Mark","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087450791","display_name":"Armin Jamshidpey","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jamshidpey, Armin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5041679295","display_name":"\u00c9ric Schost","orcid":"https://orcid.org/0000-0002-8638-9357"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Schost, \u00c9ric","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":false,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.8027999997138977,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.8027999997138977,"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/T11130","display_name":"Coding theory and cryptography","score":0.06989999860525131,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.066600002348423,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/circulant-matrix","display_name":"Circulant matrix","score":0.8355000019073486},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5968999862670898},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.5385000109672546},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4805000126361847},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.4584999978542328},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.41940000653266907},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.3982999920845032},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.37880000472068787}],"concepts":[{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.8355000019073486},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7789999842643738},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5968999862670898},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.5385000109672546},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4805000126361847},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.4584999978542328},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43529999256134033},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4124000072479248},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3982999920845032},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.37880000472068787},{"id":"https://openalex.org/C89290136","wikidata":"https://www.wikidata.org/wiki/Q550791","display_name":"Algebraic extension","level":5,"score":0.3702000081539154},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.3634999990463257},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.34929999709129333},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.34540000557899475},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.3431999981403351},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.34139999747276306},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3407999873161316},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.3149000108242035},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.29739999771118164},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2847999930381775},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.28439998626708984},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.27720001339912415},{"id":"https://openalex.org/C69044650","wikidata":"https://www.wikidata.org/wiki/Q2739329","display_name":"Square matrix","level":4,"score":0.2549000084400177}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.00313","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00313","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,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2607.00313","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00313","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,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Let":[0],"$\\mathsf{E}=\\mathbb":[1],"F_q[x]/(\u0393)$":[2],"be":[3],"an":[4,32,46,144,183,230],"algebraic":[5],"extension":[6,235],"of":[7,49,96,166,224],"degree":[8,114,194],"$n$":[9],"over":[10,38,108,136,150,188],"the":[11,42,85,126,137,213],"finite":[12,222],"field":[13],"$\\mathbb":[14,39,47,109,151,189],"F_q$,":[15,40,84],"given":[16],"by":[17,77,101,143,215],"a":[18,56,62,81,103,159,179,233],"$\u0393\\in\\mathbb":[19],"F_q[x]$":[20],"monic":[21],"and":[22,124,173],"irreducible.":[23],"It":[24],"is":[25,36,88,99,121,128,158],"classical":[26],"that":[27,35,119],"any":[28,72],"such":[29,61,118],"$\\mathsf{E}$":[30],"contains":[31],"element":[33,64,86],"$\u03b2\\in\\mathsf{E}$":[34],"normal":[37,63,89,122],"i.e.,":[41],"conjugates":[43],"$\u03b2,\u03b2^q,\\ldots,\u03b2^{q^{n-1}}$":[44],"form":[45],"F_q$-basis":[48],"$\\mathsf{E}$.":[50],"In":[51],"this":[52,142,220],"paper":[53],"we":[54,140,198],"give":[55],"deterministic":[57],"algorithm":[58,75,162],"which":[59,168],"finds":[60],"using":[65],"$O_\u03b5((n^2\\log":[66],"q)^{1+\u03b5})+O\\,\\tilde{}\\,(n\\log^2":[67],"q)$":[68],"bit":[69,209],"operations,":[70],"for":[71,80,91,163],"$\u03b5>0$.":[73],"The":[74],"works":[76],"showing":[78,216],"that,":[79],"parameter":[82],"$t\\in\\mathbb":[83,131],"$\u03b2_t=(\u03b8-t)^{-1}$":[87],"except":[90],"at":[92,115,130,195,237],"most":[93,116,196],"$n(n-1)$":[94],"values":[95],"$t$.":[97],"This":[98],"established":[100],"constructing":[102],"\"cleared":[104],"Moore\"":[105],"circulant":[106,148,164,186],"matrix":[107,149,187],"F_{q^n}[\\mathcal":[110],"T]$,":[111],"whose":[112,191],"determinant":[113,127,161,204],"$n(n-1)$,":[117,228],"$\u03b2_t$":[120],"if":[123],"only":[125],"non-zero":[129],"F_q$.":[132],"For":[133],"faster":[134],"computation":[135],"base":[138],"field,":[139,236],"replace":[141],"equivalent":[145],"trace":[146],"Gram":[147],"F_q[\\mathcal":[152],"T]$.":[153],"A":[154],"main":[155],"algorithmic":[156],"contribution":[157],"fast":[160],"matrices":[165],"polynomials,":[167],"uses":[169],"triangular":[170],"set":[171],"projection":[172],"modular":[174],"composition":[175],"techniques":[176],"to":[177,201,218,221],"achieve":[178],"near-linear":[180],"cost.":[181,240],"Given":[182],"$n\\times":[184],"n$":[185],"F_q[t]$":[190],"entries":[192],"have":[193],"$m>0$,":[197],"show":[199],"how":[200,217],"compute":[202],"its":[203],"deterministically":[205],"with":[206],"$O_\u03b5((nm\\log":[207],"q)^{1+\u03b5})$":[208],"operations.":[210],"We":[211],"complete":[212],"solution":[214],"extend":[219],"fields":[223],"size":[225],"less":[226],"than":[227],"through":[229],"embedding":[231],"in":[232],"low-degree":[234],"poly-logarithmic":[238],"additional":[239]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-07-03T00:00:00"}