{"id":"https://openalex.org/W7206198355","doi":"https://doi.org/10.48550/arxiv.2609.01399","title":"The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions","display_name":"The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions","publication_year":2026,"publication_date":"2026-09-01","ids":{"openalex":"https://openalex.org/W7206198355","doi":"https://doi.org/10.48550/arxiv.2609.01399"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.01399","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01399","pdf_url":"https://arxiv.org/pdf/2609.01399","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/2609.01399","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010080149","display_name":"Thomas W. Cusick","orcid":"https://orcid.org/0000-0002-0087-8855","observed_orcids":["https://orcid.org/0000-0002-0087-8855"]},"institutions":[{"id":"https://openalex.org/I63190737","display_name":"University at Buffalo, State University of New York","ror":"https://ror.org/01y64my43","country_code":"US","type":"education","lineage":["https://openalex.org/I63190737"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Cusick, Thomas W.","raw_affiliation_strings":["Department of Mathematics , University at Buffalo 244 Mathematics Bldg. , Buffalo , NY 14260"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University at Buffalo 244 Mathematics Bldg. , Buffalo , NY 14260","institution_ids":["https://openalex.org/I63190737"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5010080149"],"corresponding_institution_ids":["https://openalex.org/I63190737"],"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/T11130","display_name":"Coding theory and cryptography","score":0.29440000653266907,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.29440000653266907,"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"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.20250000059604645,"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"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.17149999737739563,"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/rotation-symmetric-boolean-functions","display_name":"rotation symmetric Boolean functions","score":0.9399999976158142},{"id":"https://openalex.org/keywords/dynamical-zeta-functions","display_name":"dynamical zeta functions","score":0.9039999842643738},{"id":"https://openalex.org/keywords/symbolic-dynamics","display_name":"symbolic dynamics","score":0.6980000138282776},{"id":"https://openalex.org/keywords/characteristic-polynomial","display_name":"characteristic polynomial","score":0.7350000143051147},{"id":"https://openalex.org/keywords/nonsingularity","display_name":"nonsingularity","score":0.7269999980926514}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.850600004196167},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.54830002784729},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5386000275611877},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5056999921798706},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.4607999920845032},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45489999651908875},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.42669999599456787},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42410001158714294},{"id":"https://openalex.org/C170320093","wikidata":"https://www.wikidata.org/wiki/Q1953457","display_name":"De Bruijn sequence","level":2,"score":0.42340001463890076},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.42309999465942383},{"id":"https://openalex.org/C80695182","wikidata":"https://www.wikidata.org/wiki/Q864134","display_name":"Binary quadratic form","level":4,"score":0.4074999988079071},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.4065000116825104},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3926999866962433},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.37610000371932983},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.37610000371932983},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.3587000072002411},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.3580000102519989},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.35589998960494995},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.351500004529953},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35109999775886536},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.33970001339912415},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.29660001397132874},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C12817185","wikidata":"https://www.wikidata.org/wiki/Q878259","display_name":"Quadratic residue","level":2,"score":0.27709999680519104},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.26739999651908875},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.2630999982357025},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.26260000467300415},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.2574999928474426},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.2554999887943268},{"id":"https://openalex.org/C2776175482","wikidata":"https://www.wikidata.org/wiki/Q1195816","display_name":"Transfer (computing)","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.01399","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01399","pdf_url":"https://arxiv.org/pdf/2609.01399","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.2609.01399","doi":"https://doi.org/10.48550/arxiv.2609.01399","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.01399","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:2609.01399","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01399","pdf_url":"https://arxiv.org/pdf/2609.01399","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7206198355.pdf","grobid_xml":"https://content.openalex.org/works/W7206198355.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W2324802422","https://openalex.org/W2581081884","https://openalex.org/W2978496444","https://openalex.org/W4390535921","https://openalex.org/W4412449681"],"related_works":["https://openalex.org/W4390535921","https://openalex.org/W2978496444","https://openalex.org/W6966939331","https://openalex.org/W2282920058","https://openalex.org/W4361801847","https://openalex.org/W2969407382","https://openalex.org/W2730223687","https://openalex.org/W2090259599","https://openalex.org/W2330106245","https://openalex.org/W7084063610"],"abstract_inverted_index":{"We":[0,58,157],"prove":[1,159],"the":[2,42,46,65,70,74,89,102,107,118,128,136,139,143,153,163,168],"Quadratic":[3],"Easy":[4],"Coefficients":[5],"Conjecture":[6,9],"stated":[7],"as":[8],"1":[10],"in":[11,73,83],"T.":[12],"W.":[13],"Cusick,":[14],"\\emph{Recursions":[15],"for":[16],"quadratic":[17,34],"rotation":[18,36],"symmetric":[19,37],"functions":[20],"weights},":[21],"Discrete":[22],"Applied":[23],"Mathematics":[24],"378":[25],"(2026),":[26],"93--101.":[27],"For":[28],"an":[29,84,96],"arbitrary":[30],"finite":[31],"sum":[32],"of":[33,45,64,77,92,138,142,155,167],"monomial":[35],"Boolean":[38,71],"functions,":[39],"we":[40],"identify":[41],"recurrent":[43],"part":[44],"rules":[47,119,144],"matrix":[48,56,91],"with":[49,69,124,135],"a":[50,61],"signed":[51,103],"binary":[52],"de":[53,98],"Bruijn":[54,99],"transfer":[55],"$B$.":[57,105],"then":[59],"give":[60],"one-step":[62],"presentation":[63],"finite-type":[66],"shift":[67,94],"associated":[68],"function":[72,110],"symbolic-dynamics":[75],"construction":[76],"Chirvasitu":[78],"and":[79,101,161],"Cusick.":[80],"Fourier":[81],"transformation":[82],"auxiliary":[85],"$\\F_2$":[86],"coordinate":[87],"decomposes":[88],"adjacency":[90],"this":[93],"into":[95],"unsigned":[97],"block":[100,104],"Consequently":[106],"dynamical":[108],"zeta":[109],"is":[111,117],"\\[":[112],"\u03b6_{X_f}(z)=\\frac{1}{\\det(I-z\\cR(f))},":[113],"\\]":[114],"where":[115],"$\\cR(f)$":[116],"matrix.":[120,145],"This":[121],"equality":[122],"identifies,":[123],"their":[125],"algebraic":[126],"multiplicities,":[127],"characteristic":[129,140],"values,":[130],"supplied":[131],"by":[132],"symbolic":[133],"dynamics,":[134],"roots":[137],"polynomial":[141],"The":[146],"desired":[147],"easy":[148],"coefficients":[149],"formula":[150],"follows":[151],"from":[152],"trace":[154],"$B^n$.":[156],"also":[158],"nonsingularity":[160],"justify":[162],"unique":[164],"backward":[165],"extension":[166],"weight":[169],"recurrence.":[170]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-03T00:00:00"}