{"id":"https://openalex.org/W4417539502","doi":"https://doi.org/10.48550/arxiv.2506.16937","title":"On the Characteristic Polynomial of Linearized Polynomials","display_name":"On the Characteristic Polynomial of Linearized Polynomials","publication_year":2025,"publication_date":"2025-06-20","ids":{"openalex":"https://openalex.org/W4417539502","doi":"https://doi.org/10.48550/arxiv.2506.16937"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2506.16937","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2506.16937","pdf_url":"https://arxiv.org/pdf/2506.16937","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2506.16937","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Bastioni, Luca","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bastioni, Luca","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Micheli, Giacomo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Micheli, Giacomo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Zhao, Shujun","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhao, Shujun","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/T11130","display_name":"Coding theory and cryptography","score":0.45980000495910645,"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":0.45980000495910645,"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":0.29019999504089355,"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"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.23270000517368317,"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/polynomial","display_name":"Polynomial","score":0.6031000018119812},{"id":"https://openalex.org/keywords/stable-polynomial","display_name":"Stable polynomial","score":0.5975000262260437},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.589900016784668},{"id":"https://openalex.org/keywords/reciprocal-polynomial","display_name":"Reciprocal polynomial","score":0.5701000094413757},{"id":"https://openalex.org/keywords/characteristic-polynomial","display_name":"Characteristic polynomial","score":0.5436999797821045},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.49540001153945923},{"id":"https://openalex.org/keywords/square-free-polynomial","display_name":"Square-free polynomial","score":0.47699999809265137},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.47350001335144043},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.4675999879837036},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.46059998869895935}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8259999752044678},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6031000018119812},{"id":"https://openalex.org/C45025165","wikidata":"https://www.wikidata.org/wiki/Q13424738","display_name":"Stable polynomial","level":5,"score":0.5975000262260437},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.589900016784668},{"id":"https://openalex.org/C97395012","wikidata":"https://www.wikidata.org/wiki/Q1422356","display_name":"Reciprocal polynomial","level":4,"score":0.5701000094413757},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.5436999797821045},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.49540001153945923},{"id":"https://openalex.org/C101525915","wikidata":"https://www.wikidata.org/wiki/Q7581979","display_name":"Square-free polynomial","level":4,"score":0.47699999809265137},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.47350001335144043},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4675999879837036},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.46059998869895935},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4553000032901764},{"id":"https://openalex.org/C30597210","wikidata":"https://www.wikidata.org/wiki/Q1163608","display_name":"Minimal polynomial (linear algebra)","level":5,"score":0.4442000091075897},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.435699999332428},{"id":"https://openalex.org/C170412648","wikidata":"https://www.wikidata.org/wiki/Q4736415","display_name":"Alternating polynomial","level":4,"score":0.42239999771118164},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4194999933242798},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.41780000925064087},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.40450000762939453},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.40450000762939453},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.3961000144481659},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3840999901294708},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.3772999942302704},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.3693999946117401},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.34389999508857727},{"id":"https://openalex.org/C170907076","wikidata":"https://www.wikidata.org/wiki/Q2613931","display_name":"Wilkinson's polynomial","level":5,"score":0.3384999930858612},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C108741685","wikidata":"https://www.wikidata.org/wiki/Q392477","display_name":"Factorization of polynomials","level":4,"score":0.30399999022483826},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.30329999327659607},{"id":"https://openalex.org/C194777461","wikidata":"https://www.wikidata.org/wiki/Q1085860","display_name":"Root-finding algorithm","level":3,"score":0.2969000041484833},{"id":"https://openalex.org/C173047080","wikidata":"https://www.wikidata.org/wiki/Q6400676","display_name":"Kharitonov's theorem","level":5,"score":0.2964000105857849},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.29510000348091125},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.28029999136924744},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27459999918937683},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2644999921321869},{"id":"https://openalex.org/C6077466","wikidata":"https://www.wikidata.org/wiki/Q5156597","display_name":"Complex quadratic polynomial","level":3,"score":0.2632000148296356},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C123450019","wikidata":"https://www.wikidata.org/wiki/Q5532677","display_name":"Generic polynomial","level":5,"score":0.25949999690055847}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2506.16937","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2506.16937","pdf_url":"https://arxiv.org/pdf/2506.16937","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2506.16937","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2506.16937","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"pmh:oai:arXiv.org:2506.16937","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2506.16937","pdf_url":"https://arxiv.org/pdf/2506.16937","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"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],"$k$":[1,14,62],"be":[2,8],"a":[3,9,29,35,80],"finite":[4],"field,":[5],"and":[6,63,69],"$L$":[7,33],"$q$-linearized":[10],"polynomial":[11,31],"defined":[12],"over":[13,34,84],"of":[15,32,47,51],"$q$-degree":[16],"$r$":[17],"($L=\\sum^r_{i=0}a_iZ^{q^i}$,":[18],"with":[19,55],"$a_i\\in":[20],"k$).":[21],"This":[22],"paper":[23],"provides":[24,79],"an":[25],"algorithm":[26,43],"to":[27,66],"compute":[28],"characteristic":[30],"large":[36],"extension":[37],"field":[38],"$\\mathbb":[39,52],"F_{q^n}\\supseteq":[40],"k$.":[41],"Our":[42],"has":[44],"computational":[45],"complexity":[46],"$O(n(\\log(n))^4)$":[48],"in":[49],"terms":[50],"F_q$":[53],"operations":[54],"the":[56],"implied":[57],"constant":[58],"depending":[59],"only":[60],"on":[61],"$r$.":[64],"Up":[65],"logarithmic":[67],"factors,":[68],"for":[70],"linear":[71],"maps":[72],"represented":[73],"by":[74],"low":[75],"degree":[76],"polynomials,":[77],"this":[78],"square":[81],"root":[82],"improvement":[83],"generic":[85],"algorithms.":[86]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2025-10-10T00:00:00"}