{"id":"https://openalex.org/W2018839402","doi":"https://doi.org/10.1137/s0895480100368050","title":"The Number of Irreducible Polynomials and Lyndon Words with Given Trace","display_name":"The Number of Irreducible Polynomials and Lyndon Words with Given Trace","publication_year":2001,"publication_date":"2001-01-01","ids":{"openalex":"https://openalex.org/W2018839402","doi":"https://doi.org/10.1137/s0895480100368050","mag":"2018839402"},"language":"en","primary_location":{"id":"doi:10.1137/s0895480100368050","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895480100368050","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048017151","display_name":"Frank Ruskey","orcid":"https://orcid.org/0000-0001-5523-7185"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"F. Ruskey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037634492","display_name":"C. Robert Miers","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"C. R. Miers","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5060694274","display_name":"Joe Sawada","orcid":"https://orcid.org/0000-0001-7364-2993"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J. Sawada","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":2.7027,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.90927017,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"14","issue":"2","first_page":"240","last_page":"245"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9840999841690063,"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/T11567","display_name":"semigroups and automata theory","score":0.98089998960495,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8618003129959106},{"id":"https://openalex.org/keywords/monic-polynomial","display_name":"Monic polynomial","score":0.7612912654876709},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7580255270004272},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.6500579714775085},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6335641145706177},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.43804931640625},{"id":"https://openalex.org/keywords/cyclotomic-polynomial","display_name":"Cyclotomic polynomial","score":0.4291120767593384},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42088571190834045},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4196022152900696},{"id":"https://openalex.org/keywords/irreducible-polynomial","display_name":"Irreducible polynomial","score":0.4125065803527832},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.13905510306358337}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8618003129959106},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.7612912654876709},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7580255270004272},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.6500579714775085},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6335641145706177},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.43804931640625},{"id":"https://openalex.org/C76253118","wikidata":"https://www.wikidata.org/wiki/Q1051983","display_name":"Cyclotomic polynomial","level":3,"score":0.4291120767593384},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42088571190834045},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4196022152900696},{"id":"https://openalex.org/C165030954","wikidata":"https://www.wikidata.org/wiki/Q1476663","display_name":"Irreducible polynomial","level":4,"score":0.4125065803527832},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.13905510306358337},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s0895480100368050","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895480100368050","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.18.9280","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.9280","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.csr.uvic.ca/~fruskey/Publications/Traceq/tqSIAM.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.23.1312","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.1312","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.usyd.edu.au/~sawada/papers/traceq.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.4.7409","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.7409","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cis.uoguelph.ca/~sawada/papers/traceq.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1559375430","https://openalex.org/W2026532517","https://openalex.org/W2752885492","https://openalex.org/W2888871578","https://openalex.org/W2914659449"],"related_works":["https://openalex.org/W1985839497","https://openalex.org/W4200010336","https://openalex.org/W2003536029","https://openalex.org/W2021016070","https://openalex.org/W4296465811","https://openalex.org/W2937328185","https://openalex.org/W3042751680","https://openalex.org/W70023776","https://openalex.org/W2011142789","https://openalex.org/W2006352939"],"abstract_inverted_index":{"The":[0],"trace":[1,42],"of":[2,13,32,38,71,89,120,132],"a":[3,45,50,129],"degree":[4,39],"n":[5,40,72,121],"polynomial":[6],"f(x)":[7],"over":[8,36,67,116],"GF(q)":[9,37],"is":[10,66,82,102,115],"the":[11,30,64,87,113],"coefficient":[12],"xn-1":[14],".":[15],"Carlitz":[16],"[Proc.":[17],"Amer.":[18],"Math.":[19],"Soc.,":[20],"3":[21],"(1952),":[22],"pp.":[23],"693--700]":[24],"obtained":[25],"an":[26],"expression":[27,53],"Iq(n,t)":[28,58],"for":[29,54,122],"number":[31,88,101],"monic":[33],"irreducible":[34],"polynomials":[35],"and":[41],"t.":[43],"Using":[44],"different":[46],"approach,":[47],"we":[48],"derive":[49],"simple":[51],"explicit":[52],"Iq(n,t).":[55],"If":[56],"t>0,":[57],"=":[59,106],"(\\sum":[60,107],"\\mu(d)":[61,110],"q^{n/d})/(qn)$,":[62,111],"where":[63,112],"sum":[65,95,114],"all":[68,117],"divisors":[69,118],"d":[70,119],"which":[73,123],"are":[74],"relatively":[75],"prime":[76],"to":[77,84,96],"q.":[78,99],"This":[79,100],"same":[80],"approach":[81],"used":[83],"count":[85],"Lq(n,t),":[86],"q-ary":[90],"Lyndon":[91],"words":[92],"whose":[93],"characters":[94],"t":[97],"mod":[98],"given":[103],"by":[104],"Lq(n,t)":[105],"{\\rm":[108],"gcd}(d,q)":[109],"gcd(d,q)|t.":[124],"Both":[125],"results":[126],"rely":[127],"on":[128],"new":[130],"form":[131],"M\u00f6bius":[133],"inversion.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
