{"id":"https://openalex.org/W1896102805","doi":"https://doi.org/10.1090/mcom/3007","title":"Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields","display_name":"Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields","publication_year":2015,"publication_date":"2015-08-03","ids":{"openalex":"https://openalex.org/W1896102805","doi":"https://doi.org/10.1090/mcom/3007","mag":"1896102805"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3007","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3007","pdf_url":"https://www.ams.org/mcom/2016-85-299/S0025-5718-2015-03007-5/S0025-5718-2015-03007-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2016-85-299/S0025-5718-2015-03007-5/S0025-5718-2015-03007-5.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5033626172","display_name":"Alin Bostan","orcid":"https://orcid.org/0000-0003-3798-9281"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I4210126360","display_name":"Centre Inria de Saclay","ror":"https://ror.org/0315e5x55","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1326498283","https://openalex.org/I4210126360"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Alin Bostan","raw_affiliation_strings":["INRIA Saclay \u00cele-de-France, B\u00e2timent Alan Turing, 1 rue Honor\u00e9 d'Estienne d'Orves, 91120 Palaiseau, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Saclay \u00cele-de-France, B\u00e2timent Alan Turing, 1 rue Honor\u00e9 d'Estienne d'Orves, 91120 Palaiseau, France","institution_ids":["https://openalex.org/I4210126360","https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058294598","display_name":"Guillaume Ch\u00e8ze","orcid":"https://orcid.org/0000-0001-9657-4209"},"institutions":[{"id":"https://openalex.org/I134560555","display_name":"Universit\u00e9 Toulouse III - Paul Sabatier","ror":"https://ror.org/02v6kpv12","country_code":"FR","type":"education","lineage":["https://openalex.org/I134560555"]},{"id":"https://openalex.org/I4210098964","display_name":"Toulouse Mathematics Institute","ror":null,"country_code":"FR","type":null,"lineage":["https://openalex.org/I4210098964"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Guillaume Ch\u00e8ze","raw_affiliation_strings":["Institut de Math\u00e9matiques de Toulouse, Universit\u00e9 Paul Sabatier Toulouse 3, MIP B\u00e2t. 1R3, 31 062 TOULOUSE cedex 9, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de Math\u00e9matiques de Toulouse, Universit\u00e9 Paul Sabatier Toulouse 3, MIP B\u00e2t. 1R3, 31 062 TOULOUSE cedex 9, France","institution_ids":["https://openalex.org/I4210098964","https://openalex.org/I134560555"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021791488","display_name":"Thomas Cluzeau","orcid":"https://orcid.org/0000-0003-0796-2838"},"institutions":[{"id":"https://openalex.org/I4210093397","display_name":"XLIM","ror":"https://ror.org/00f7srh09","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I32881790","https://openalex.org/I4210093397","https://openalex.org/I4210095849","https://openalex.org/I65806277"]},{"id":"https://openalex.org/I65806277","display_name":"Universit\u00e9 de Limoges","ror":"https://ror.org/02cp04407","country_code":"FR","type":"education","lineage":["https://openalex.org/I65806277"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Thomas Cluzeau","raw_affiliation_strings":["Universit\u00e9 de Limoges","DMI, 123 avenue Albert Thomas, 87 060 LIMOGES cedex, France","XLIM UMR 7252"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Limoges","institution_ids":["https://openalex.org/I65806277"]},{"raw_affiliation_string":"DMI, 123 avenue Albert Thomas, 87 060 LIMOGES cedex, France","institution_ids":[]},{"raw_affiliation_string":"XLIM UMR 7252","institution_ids":["https://openalex.org/I4210093397"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108684467","display_name":"Jacques-Arthur Weil","orcid":null},"institutions":[{"id":"https://openalex.org/I4210093397","display_name":"XLIM","ror":"https://ror.org/00f7srh09","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I32881790","https://openalex.org/I4210093397","https://openalex.org/I4210095849","https://openalex.org/I65806277"]},{"id":"https://openalex.org/I65806277","display_name":"Universit\u00e9 de Limoges","ror":"https://ror.org/02cp04407","country_code":"FR","type":"education","lineage":["https://openalex.org/I65806277"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Jacques-Arthur Weil","raw_affiliation_strings":["DMI, 123 avenue Albert Thomas, 87 060 LIMOGES cedex, France","Universit\u00e9 de Limoges","XLIM UMR 7252"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DMI, 123 avenue Albert Thomas, 87 060 LIMOGES cedex, France","institution_ids":[]},{"raw_affiliation_string":"Universit\u00e9 de Limoges","institution_ids":["https://openalex.org/I65806277"]},{"raw_affiliation_string":"XLIM UMR 7252","institution_ids":["https://openalex.org/I4210093397"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.2312,"has_fulltext":true,"cited_by_count":30,"citation_normalized_percentile":{"value":0.93616273,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"85","issue":"299","first_page":"1393","last_page":"1425"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9988999962806702,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9606999754905701,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6780931949615479},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6113771796226501},{"id":"https://openalex.org/keywords/power-series","display_name":"Power series","score":0.5799009203910828},{"id":"https://openalex.org/keywords/formal-power-series","display_name":"Formal power series","score":0.5503427982330322},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.47587814927101135},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.46772515773773193},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.45726239681243896},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.45271196961402893},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.45167627930641174},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.44567161798477173},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4065430462360382},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3864240050315857},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3790912330150604},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22927206754684448},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1471724510192871}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6780931949615479},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6113771796226501},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.5799009203910828},{"id":"https://openalex.org/C151746172","wikidata":"https://www.wikidata.org/wiki/Q1003025","display_name":"Formal power series","level":3,"score":0.5503427982330322},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.47587814927101135},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.46772515773773193},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.45726239681243896},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.45271196961402893},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.45167627930641174},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.44567161798477173},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4065430462360382},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3864240050315857},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3790912330150604},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22927206754684448},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1471724510192871},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1090/mcom/3007","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3007","pdf_url":"https://www.ams.org/mcom/2016-85-299/S0025-5718-2015-03007-5/S0025-5718-2015-03007-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1310.2778","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1310.2778","pdf_url":"https://arxiv.org/pdf/1310.2778","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":"","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.402.9997","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.402.9997","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://hal-unilim.archives-ouvertes.fr/docs/00/87/16/63/PDF/BoChClWe13_web.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.431.7072","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.431.7072","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.univ-toulouse.fr/~cheze/BoChClWe13_web.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.741.7767","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.741.7767","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1310.2778.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-00871663v2","is_oa":true,"landing_page_url":"https://hal.science/hal-00871663","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://www.ams.org/journals/mcom/2016-85-299/S0025-5718-2015-03007-5/","raw_type":"Journal articles"}],"best_oa_location":{"id":"doi:10.1090/mcom/3007","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3007","pdf_url":"https://www.ams.org/mcom/2016-85-299/S0025-5718-2015-03007-5/S0025-5718-2015-03007-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1896102805.pdf","grobid_xml":"https://content.openalex.org/works/W1896102805.grobid-xml"},"referenced_works_count":58,"referenced_works":["https://openalex.org/W4085760","https://openalex.org/W208025404","https://openalex.org/W594548925","https://openalex.org/W1498264388","https://openalex.org/W1515307033","https://openalex.org/W1562183207","https://openalex.org/W1572239187","https://openalex.org/W1573927207","https://openalex.org/W1579005859","https://openalex.org/W1594421220","https://openalex.org/W1963005329","https://openalex.org/W1966102258","https://openalex.org/W1966306237","https://openalex.org/W1978425482","https://openalex.org/W1980267752","https://openalex.org/W1982280614","https://openalex.org/W1990384308","https://openalex.org/W1992696106","https://openalex.org/W1996638762","https://openalex.org/W1997290261","https://openalex.org/W1997435956","https://openalex.org/W2001572296","https://openalex.org/W2002352305","https://openalex.org/W2003189478","https://openalex.org/W2007467867","https://openalex.org/W2009279535","https://openalex.org/W2014684859","https://openalex.org/W2014749888","https://openalex.org/W2017417276","https://openalex.org/W2018818965","https://openalex.org/W2019257511","https://openalex.org/W2049007663","https://openalex.org/W2051154648","https://openalex.org/W2052670035","https://openalex.org/W2059884448","https://openalex.org/W2074539848","https://openalex.org/W2076063965","https://openalex.org/W2077540938","https://openalex.org/W2090093259","https://openalex.org/W2093239577","https://openalex.org/W2108871824","https://openalex.org/W2129139293","https://openalex.org/W2146298493","https://openalex.org/W2168527104","https://openalex.org/W2566427219","https://openalex.org/W2596229482","https://openalex.org/W2626734060","https://openalex.org/W2963215276","https://openalex.org/W2964316084","https://openalex.org/W3102083525","https://openalex.org/W3102593879","https://openalex.org/W4205630623","https://openalex.org/W4242730381","https://openalex.org/W4292170110","https://openalex.org/W4295124794","https://openalex.org/W4301807999","https://openalex.org/W6603094921","https://openalex.org/W6648392953"],"related_works":["https://openalex.org/W2044685772","https://openalex.org/W1999045233","https://openalex.org/W131037327","https://openalex.org/W3181346485","https://openalex.org/W2504047832","https://openalex.org/W3028536017","https://openalex.org/W2089343381","https://openalex.org/W122587815","https://openalex.org/W4300174804","https://openalex.org/W2024906304"],"abstract_inverted_index":{"We":[0,57,91,287,405],"present":[1],"fast":[2,291],"algorithms":[3,421],"for":[4,259],"computing":[5,82],"rational":[6,83,264,297],"first":[7,42,84,97,265,298],"integrals":[8,85],"with":[9,104,435],"bounded":[10],"degree":[11,261],"of":[12,35,40,49,64,88,262,284,417],"a":[13,22,36,41,50,79,101,167,290,296,411,425],"planar":[14],"polynomial":[15,52],"vector":[16],"field.":[17],"Our":[18],"approach":[19],"builds":[20],"upon":[21,95],"method":[23,77,413],"proposed":[24],"by":[25,99],"Ferragut":[26],"and":[27,46,166,267],"Giacomini,":[28],"whose":[29],"main":[30],"ingredients":[31],"are":[32,422],"the":[33,47,62,68,171,256,260,263,282,367,415],"calculation":[34],"power":[37,55,69],"series":[38],"solution":[39],"order":[43],"differential":[44],"equation":[45],"reconstruction":[48],"bivariate":[51],"annihilating":[53],"this":[54,96],"series.":[56,70],"provide":[58,289],"explicit":[59],"bounds":[60],"on":[61],"number":[63],"terms":[65],"needed":[66],"in":[67,173,302,424],"This":[71],"enables":[72],"us":[73],"to":[74,409,414,432],"transform":[75],"their":[76],"into":[78],"certified":[80],"algorithm":[81,98,103,169],"via":[86],"systems":[87],"linear":[89,285],"equations.":[90],"then":[92,406],"significantly":[93],"improve":[94],"building":[100],"probabilistic":[102],"arithmetic":[105,239,363,403],"complexity":[106,371],"<inline-formula":[107,176,242,268,303,373],"content-type=\"math/mathml\">":[108,177,243,269,304,374],"<mml:math":[109,178,244,270,305,375],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[110,179,245,271,306,376],"alttext=\"ModifyingAbove":[111,180,307],"script":[112,181,308],"upper":[113,118,182,187,309,314],"O":[114,183,310],"With":[115,184,311],"tilde":[116,185,312],"left-parenthesis":[117,186,313],"N":[119,188,315,378],"Superscript":[120,189,316,379],"2":[121,190,319],"omega":[122,191,317,381],"Baseline":[123,194,320],"right-parenthesis\">":[124,195,321],"<mml:semantics>":[125,196,248,273,322,384],"<mml:mrow>":[126,197,323],"<mml:mrow":[127,130,146,198,201,217,324,327,343,387],"class=\"MJX-TeXAtom-ORD\">":[128,131,147,199,202,218,325,328,344,388],"<mml:mover>":[129,200,326],"<mml:mi":[132,203,329],"class=\"MJX-tex-caligraphic\"":[133,204,330],"mathvariant=\"script\">O</mml:mi>":[134,205,331],"</mml:mrow>":[135,141,152,156,206,212,225,229,332,338,350,354,395],"<mml:mo":[136,142,154,207,213,227,333,339,352],"stretchy=\"false\">":[137,208,334],"~":[138,209,335],"</mml:mo>":[139,210,336],"</mml:mover>":[140,211,337],"stretchy=\"false\">(</mml:mo>":[143,214,340],"<mml:msup>":[144,215,341,385],"<mml:mi>N</mml:mi>":[145,216,249,342,386],"<mml:mn>2</mml:mn>":[148,219,349],"<mml:mi>":[149,220,274,345,390],"\u03c9":[150,221,275,346,391],"</mml:mi>":[151,222,276,347,392],"</mml:msup>":[153,226,351,396],"stretchy=\"false\">)</mml:mo>":[155,228,353],"<mml:annotation":[157,230,250,277,355,397],"encoding=\"application/x-tex\">\\tilde":[158,231,356],"{\\mathcal":[159,232,357],"{O}}(N^{2":[160,233],"\\omega":[161,234],"})</mml:annotation>":[162],"</mml:semantics>":[163,236,252,279,360,400],"</mml:math>":[164,237,253,280,361,401],"</inline-formula>":[165,238,254,281,362,402],"deterministic":[168],"solving":[170],"problem":[172],"at":[174],"most":[175],"plus":[192,318,382],"1":[193],"<mml:mo>+</mml:mo>":[223,348,393],"<mml:mn>1</mml:mn>":[224],"+1})</mml:annotation>":[235],"operations,":[240],"where":[241],"alttext=\"upper":[246,377],"N\">":[247],"encoding=\"application/x-tex\">N</mml:annotation>":[251],"denotes":[255],"given":[257],"bound":[258],"integral,":[266,299],"alttext=\"omega\">":[272],"encoding=\"application/x-tex\">\\omega</mml:annotation>":[278],"exponent":[283],"algebra.":[286],"also":[288],"heuristic":[292],"variant":[293],"which":[294,429],"computes":[295],"or":[300],"fails,":[301],"{O}}(N^{\\omega":[358],"+2})</mml:annotation>":[359],"operations.":[364,404],"By":[365],"comparison,":[366],"best":[368],"previously":[369],"known":[370],"was":[372],"4":[380],"4\">":[383],"<mml:mn>4</mml:mn>":[389,394],"encoding=\"application/x-tex\">N^{4\\omega":[398],"+4}</mml:annotation>":[399],"show":[407],"how":[408],"apply":[410],"similar":[412],"computation":[416],"Darboux":[418],"polynomials.":[419],"The":[420],"implemented":[423],"Maple":[426],"package":[427],"<sc>RationalFirstIntegrals</sc>":[428],"is":[430],"available":[431],"interested":[433],"readers":[434],"examples":[436],"showing":[437],"its":[438],"efficiency.":[439]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-13T07:54:00.901334","created_date":"2025-10-10T00:00:00"}
