{"id":"https://openalex.org/W2364754288","doi":"https://doi.org/10.1145/2930889.2930929","title":"Critical Point Computations on Smooth Varieties","display_name":"Critical Point Computations on Smooth Varieties","publication_year":2016,"publication_date":"2016-07-19","ids":{"openalex":"https://openalex.org/W2364754288","doi":"https://doi.org/10.1145/2930889.2930929","mag":"2364754288"},"language":"en","primary_location":{"id":"doi:10.1145/2930889.2930929","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2930889.2930929","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","raw_type":"proceedings-article"},"type":"preprint","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1605.02518","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048254079","display_name":"Mohab Safey El Din","orcid":"https://orcid.org/0000-0001-9463-1257"},"institutions":[{"id":"https://openalex.org/I39804081","display_name":"Sorbonne Universit\u00e9","ror":"https://ror.org/02en5vm52","country_code":"FR","type":"education","lineage":["https://openalex.org/I39804081"]},{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Mohab Safey El DIn","raw_affiliation_strings":["Sorbonne Universit\u00e9s/UPMC/Univ. Paris 6, Paris, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Sorbonne Universit\u00e9s/UPMC/Univ. Paris 6, Paris, France","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I39804081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056341371","display_name":"Pierre-Jean Spaenlehauer","orcid":"https://orcid.org/0000-0001-7906-0505"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I4210127166","display_name":"Centre Inria de l'Universit\u00e9 de Lorraine","ror":"https://ror.org/03fcjvn64","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1326498283","https://openalex.org/I4210127166"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Pierre-Jean Spaenlehauer","raw_affiliation_strings":["INRIA Nancy - Grand Est, Nancy, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Nancy - Grand Est, Nancy, France","institution_ids":["https://openalex.org/I4210127166","https://openalex.org/I1326498283"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5048254079"],"corresponding_institution_ids":["https://openalex.org/I204730241","https://openalex.org/I39804081"],"apc_list":null,"apc_paid":null,"fwci":1.442,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.83407756,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":null,"issue":null,"first_page":"183","last_page":"190"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9998999834060669,"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9970999956130981,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.986299991607666,"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":0.7903579473495483},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6621449589729309},{"id":"https://openalex.org/keywords/locus","display_name":"Locus (genetics)","score":0.5550280213356018},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5511142015457153},{"id":"https://openalex.org/keywords/critical-point","display_name":"Critical point (mathematics)","score":0.542163610458374},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5178830623626709},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4989349842071533},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.48288655281066895},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.4779624938964844},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.443937748670578},{"id":"https://openalex.org/keywords/algebraic-geometry","display_name":"Algebraic geometry","score":0.4435754716396332},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4309559166431427},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3839026987552643},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3465481996536255},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2152431607246399},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.20464131236076355},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.17430660128593445}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7903579473495483},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6621449589729309},{"id":"https://openalex.org/C84597430","wikidata":"https://www.wikidata.org/wiki/Q106227","display_name":"Locus (genetics)","level":3,"score":0.5550280213356018},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5511142015457153},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.542163610458374},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5178830623626709},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4989349842071533},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.48288655281066895},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4779624938964844},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.443937748670578},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.4435754716396332},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4309559166431427},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3839026987552643},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3465481996536255},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2152431607246399},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.20464131236076355},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.17430660128593445},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"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":3,"locations":[{"id":"doi:10.1145/2930889.2930929","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2930889.2930929","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:1605.02518","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.02518","pdf_url":"https://arxiv.org/pdf/1605.02518","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:HAL:hal-01312750v1","is_oa":true,"landing_page_url":"https://inria.hal.science/hal-01312750","pdf_url":"https://inria.hal.science/hal-01312750/document","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":"International Symposium on Symbolic and Algebraic Computation (ISSAC), Jul 2016, Waterloo, Canada. pp.183--190, ⟨10.1145/2930889.2930929⟩","raw_type":"Conference papers"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1605.02518","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.02518","pdf_url":"https://arxiv.org/pdf/1605.02518","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"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322893","display_name":"Institut Universitaire de France","ror":"https://ror.org/055khg266"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W647687692","https://openalex.org/W1495410784","https://openalex.org/W1496428155","https://openalex.org/W1504054803","https://openalex.org/W1537258143","https://openalex.org/W1984254060","https://openalex.org/W1985924702","https://openalex.org/W1988709433","https://openalex.org/W1991675728","https://openalex.org/W1991813962","https://openalex.org/W2001588532","https://openalex.org/W2006394656","https://openalex.org/W2015508922","https://openalex.org/W2018980874","https://openalex.org/W2032831657","https://openalex.org/W2032925102","https://openalex.org/W2033016032","https://openalex.org/W2058786070","https://openalex.org/W2063194632","https://openalex.org/W2065276731","https://openalex.org/W2066979034","https://openalex.org/W2072992931","https://openalex.org/W2082387287","https://openalex.org/W2104223485","https://openalex.org/W2118254039","https://openalex.org/W2124706092","https://openalex.org/W2130918732","https://openalex.org/W2134386365","https://openalex.org/W2151054383","https://openalex.org/W2164838307","https://openalex.org/W2279788372","https://openalex.org/W2572028317","https://openalex.org/W2592887901","https://openalex.org/W4206272148"],"related_works":["https://openalex.org/W2952185452","https://openalex.org/W2798106756","https://openalex.org/W2046736294","https://openalex.org/W2980815195","https://openalex.org/W4293584968","https://openalex.org/W2032964960","https://openalex.org/W2384758770","https://openalex.org/W4382701036","https://openalex.org/W2081086357","https://openalex.org/W3141524051"],"abstract_inverted_index":{"Let":[0],"V":[1,32,60],"\u2282":[2],"Cn":[3],"be":[4,11],"an":[5,12,145],"equidimensional":[6],"algebraic":[7,41],"set":[8],"and":[9,43,54,80,154,181,188],"g":[10,27,123],"n-variate":[13],"polynomial":[14,182],"with":[15],"rational":[16],"coefficients.":[17],"Computing":[18],"the":[19,23,29,46,49,56,69,72,81,84,97,100,107,113,119,127,174,178,184,189,193],"critical":[20,50,73,116,158],"points":[21,30,117,159],"of":[22,31,36,59,71,83,99,115,118,122,139,144,176,186,192],"map":[24,121],"that":[25,48,55,112],"evaluates":[26],"at":[28],"is":[33,52,61,124,163],"a":[34,136,141],"cornerstone":[35],"several":[37],"algorithms":[38],"in":[39,92,160,165,173,183],"real":[40],"geometry":[42],"optimization.":[44],"Under":[45],"assumption":[47],"locus":[51,74],"finite":[53],"projective":[57],"closure":[58],"smooth,":[62],"we":[63],"provide":[64],"sharp":[65],"upper":[66],"bounds":[67],"on":[68,78],"degree":[70,130,191],"which":[75,162],"depend":[76],"only":[77],"deg(g)":[79],"degrees":[82,98],"generic":[85,101],"polar":[86,102],"varieties":[87,103],"associated":[88],"to":[89,148,169],"V.":[90],"Hence,":[91],"some":[93],"special":[94],"cases":[95],"where":[96],"do":[104],"not":[105],"reach":[106],"worst-case":[108],"bounds,":[109],"this":[110,166],"implies":[111],"number":[114,185],"evaluation":[120],"less":[125],"than":[126],"currently":[128],"known":[129],"bounds.":[131],"We":[132],"show":[133],"that,":[134],"given":[135],"lifting":[137],"fiber":[138],"V,":[140],"slight":[142],"variant":[143],"algorithm":[146],"due":[147],"Bank,":[149],"Giusti,":[150],"Heintz,":[151],"Lecerf,":[152],"Matera":[153],"Solerno":[155],"computes":[156],"these":[157],"time":[161],"quadratic":[164],"bound":[167],"up":[168],"logarithmic":[170],"factors,":[171],"linear":[172],"complexity":[175],"evaluating":[177],"input":[179,194],"system":[180],"variables":[187],"maximum":[190],"polynomials.":[195]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2016-06-24T00:00:00"}