{"id":"https://openalex.org/W2060589377","doi":"https://doi.org/10.1145/1145768.1145798","title":"Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields","display_name":"Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields","publication_year":2006,"publication_date":"2006-07-09","ids":{"openalex":"https://openalex.org/W2060589377","doi":"https://doi.org/10.1145/1145768.1145798","mag":"2060589377"},"language":"en","primary_location":{"id":"doi:10.1145/1145768.1145798","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1145768.1145798","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2006 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-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/A5056874049","display_name":"Erich Kaltofen","orcid":"https://orcid.org/0000-0003-2739-3230"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Erich Kaltofen","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts","MIT - Massachusetts Institute of Technology (77 Massachusetts Ave, Cambridge, MA 02139 - United States)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"MIT - Massachusetts Institute of Technology (77 Massachusetts Ave, Cambridge, MA 02139 - United States)","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5105512963","display_name":"Pascal Koiran","orcid":"https://orcid.org/0000-0003-3405-7155"},"institutions":[{"id":"https://openalex.org/I113428412","display_name":"\u00c9cole Normale Sup\u00e9rieure de Lyon","ror":"https://ror.org/04zmssz18","country_code":"FR","type":"education","lineage":["https://openalex.org/I113428412","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I4210144566","display_name":"Laboratoire de l'Informatique du Parall\u00e9lisme","ror":"https://ror.org/04msnz457","country_code":"FR","type":"facility","lineage":["https://openalex.org/I100532134","https://openalex.org/I113428412","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I203339264","https://openalex.org/I203339264","https://openalex.org/I4210144566"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Pascal Koiran","raw_affiliation_strings":["\u00c9cole Normale Sup\u00e9rieure de Lyon, Lyon, France","LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)","MC2 - Mod\u00e8les de calcul, Complexit\u00e9, Combinatoire (ENS Lyon 46 all\u00e9e d'Italie 69364 Lyon Cedex 07 - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\u00c9cole Normale Sup\u00e9rieure de Lyon, Lyon, France","institution_ids":["https://openalex.org/I113428412"]},{"raw_affiliation_string":"LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)","institution_ids":["https://openalex.org/I4210144566","https://openalex.org/I113428412"]},{"raw_affiliation_string":"MC2 - Mod\u00e8les de calcul, Complexit\u00e9, Combinatoire (ENS Lyon 46 all\u00e9e d'Italie 69364 Lyon Cedex 07 - France)","institution_ids":["https://openalex.org/I113428412"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5056874049"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":4.1943,"has_fulltext":false,"cited_by_count":33,"citation_normalized_percentile":{"value":0.93769064,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"162","last_page":"168"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998000264167786,"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":0.9998000264167786,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9995999932289124,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9991000294685364,"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/mathematics","display_name":"Mathematics","score":0.7996691465377808},{"id":"https://openalex.org/keywords/lacunary-function","display_name":"Lacunary function","score":0.7177156805992126},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.6502946615219116},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5535680055618286},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.541570782661438},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5270609855651855},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.52633136510849},{"id":"https://openalex.org/keywords/irreducible-polynomial","display_name":"Irreducible polynomial","score":0.5219161510467529},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5076733827590942},{"id":"https://openalex.org/keywords/univariate","display_name":"Univariate","score":0.5026414394378662},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.4867255985736847},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.48026132583618164},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.44959086179733276},{"id":"https://openalex.org/keywords/factorization-of-polynomials","display_name":"Factorization of polynomials","score":0.44129207730293274},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4291796088218689},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.32312101125717163},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22949844598770142},{"id":"https://openalex.org/keywords/multivariate-statistics","display_name":"Multivariate statistics","score":0.21908986568450928},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.19370803236961365},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07307910919189453}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7996691465377808},{"id":"https://openalex.org/C63356602","wikidata":"https://www.wikidata.org/wiki/Q1986013","display_name":"Lacunary function","level":2,"score":0.7177156805992126},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.6502946615219116},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5535680055618286},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.541570782661438},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5270609855651855},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.52633136510849},{"id":"https://openalex.org/C165030954","wikidata":"https://www.wikidata.org/wiki/Q1476663","display_name":"Irreducible polynomial","level":4,"score":0.5219161510467529},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5076733827590942},{"id":"https://openalex.org/C199163554","wikidata":"https://www.wikidata.org/wiki/Q1681619","display_name":"Univariate","level":3,"score":0.5026414394378662},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.4867255985736847},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.48026132583618164},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.44959086179733276},{"id":"https://openalex.org/C108741685","wikidata":"https://www.wikidata.org/wiki/Q392477","display_name":"Factorization of polynomials","level":4,"score":0.44129207730293274},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4291796088218689},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.32312101125717163},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22949844598770142},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.21908986568450928},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.19370803236961365},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07307910919189453},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/1145768.1145798","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1145768.1145798","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2006 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"},{"id":"pmh:oai:HAL:hal-04655377v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04655377","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSAC 2006, Jul 2006, Genoa, Italy. pp.162-168, ⟨10.1145/1145768.1145798⟩","raw_type":"Conference papers"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W178528876","https://openalex.org/W1524052280","https://openalex.org/W1535047043","https://openalex.org/W1567200859","https://openalex.org/W1590003630","https://openalex.org/W1599026900","https://openalex.org/W1974321539","https://openalex.org/W1976166549","https://openalex.org/W1980267752","https://openalex.org/W1980469175","https://openalex.org/W2005567558","https://openalex.org/W2028346139","https://openalex.org/W2031046967","https://openalex.org/W2111028162","https://openalex.org/W2126318803","https://openalex.org/W2155376493","https://openalex.org/W2487822328","https://openalex.org/W2740986359","https://openalex.org/W4229633778"],"related_works":["https://openalex.org/W2288130547","https://openalex.org/W4249593543","https://openalex.org/W2068749660","https://openalex.org/W1576380577","https://openalex.org/W2012320470","https://openalex.org/W1997080184","https://openalex.org/W1507937674","https://openalex.org/W4220797632","https://openalex.org/W3083856079","https://openalex.org/W4280617141"],"abstract_inverted_index":{"We":[0,105],"present":[1],"algorithms":[2],"that":[3,88],"compute":[4],"all":[5],"irreducible":[6],"factors":[7,59,68,96,114],"of":[8,12,35,50,97,115,132,140],"degree":[9],"\u2264":[10],"d":[11],"supersparse":[13,40,64,99,116],"(lacunary)":[14],"multivariate":[15],"polynomials":[16,100,118],"in":[17,25,28,83],"n":[18],"variables":[19],"over":[20,101,119],"an":[21],"algebraic":[22,102,133],"number":[23,103],"field":[24],"deterministic":[26],"polynomial-time":[27,82],"(l+d)n,":[29],"where":[30],"l":[31],"is":[32],"the":[33,36,42,54,67,120,130,141],"size":[34,55],"input":[37],"polynomial.":[38],"In":[39],"polynomials,":[41,77],"term":[43],"degrees":[44],"enter":[45],"logarithmically":[46],"as":[47,63,71],"their":[48],"numbers":[49,122,134],"binary":[51],"digits":[52],"into":[53],"measure":[56],"l.":[57],"The":[58],"are":[60,69],"again":[61],"represented":[62,70],"polynomials.":[65],"If":[66],"straight-line":[72],"programs":[73],"or":[74],"black":[75],"box":[76],"we":[78],"can":[79],"achieve":[80],"randomized":[81],"(l+d)O(1).":[84],"Our":[85],"approach":[86],"follows":[87],"by":[89,123],"H.":[90],"W.":[91],"Lenstra,":[92],"Jr.,":[93],"on":[94,129],"computing":[95,112],"univariate":[98],"fields.":[104],"generalize":[106],"our":[107],"ISSAC":[108],"2005":[109],"results":[110],"for":[111],"linear":[113],"bivariate":[117],"rational":[121],"appealing":[124],"to":[125,136],"recent":[126],"lower":[127],"bounds":[128],"height":[131],"and":[135],"a":[137],"special":[138],"case":[139],"former":[142],"Lang":[143],"conjecture.":[144]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-07T13:39:58.223016","created_date":"2025-10-10T00:00:00"}