{"id":"https://openalex.org/W2076927517","doi":"https://doi.org/10.1145/1806689.1806759","title":"Solving polynomial equations in smoothed polynomial time and a near solution to smale's 17th problem","display_name":"Solving polynomial equations in smoothed polynomial time and a near solution to smale's 17th problem","publication_year":2010,"publication_date":"2010-06-05","ids":{"openalex":"https://openalex.org/W2076927517","doi":"https://doi.org/10.1145/1806689.1806759","mag":"2076927517"},"language":"en","primary_location":{"id":"doi:10.1145/1806689.1806759","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806759","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-second ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"conference-paper","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/A5029244957","display_name":"Peter B\u00fcrgisser","orcid":"https://orcid.org/0000-0001-8169-0514"},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Peter B\u00fcrgisser","raw_affiliation_strings":["University of Paderborn, Paderborn, Germany","University of Paderborn,, Paderborn, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Paderborn, Paderborn, Germany","institution_ids":["https://openalex.org/I206945453"]},{"raw_affiliation_string":"University of Paderborn,, Paderborn, Germany","institution_ids":["https://openalex.org/I206945453"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079608792","display_name":"Felipe Cucker","orcid":"https://orcid.org/0000-0002-4569-3248"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Felipe Cucker","raw_affiliation_strings":["City University of Hong Kong, Hong Kong, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"City University of Hong Kong, Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I168719708"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"503","last_page":"512"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"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"}},"topics":[{"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994000196456909,"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/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.6395560503005981},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6194067597389221},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49534574151039124},{"id":"https://openalex.org/keywords/running-time","display_name":"Running time","score":0.444132000207901},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.43201518058776855},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.43109434843063354},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.4236880838871002},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35975944995880127},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.35266169905662537},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.347648948431015},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3390467166900635},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33449646830558777},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11673840880393982},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06453973054885864}],"concepts":[{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.6395560503005981},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6194067597389221},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49534574151039124},{"id":"https://openalex.org/C3017489831","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Running time","level":2,"score":0.444132000207901},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.43201518058776855},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.43109434843063354},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.4236880838871002},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35975944995880127},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35266169905662537},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.347648948431015},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3390467166900635},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33449646830558777},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11673840880393982},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06453973054885864},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/1806689.1806759","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806759","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-second ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.409.744","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.409.744","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math-www.upb.de/agpb/work/stoc149-buergisser.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W75046271","https://openalex.org/W1550331971","https://openalex.org/W1571614612","https://openalex.org/W1590743980","https://openalex.org/W1965654736","https://openalex.org/W2011374998","https://openalex.org/W2024090003","https://openalex.org/W2044413922","https://openalex.org/W2061860086","https://openalex.org/W2062461628","https://openalex.org/W2063917062","https://openalex.org/W2081045838","https://openalex.org/W2087831729","https://openalex.org/W2113361400","https://openalex.org/W2120280171","https://openalex.org/W2129280636","https://openalex.org/W2138739352","https://openalex.org/W2170078634","https://openalex.org/W2478857688","https://openalex.org/W2592503220","https://openalex.org/W3101584966","https://openalex.org/W6632740934","https://openalex.org/W6661262005","https://openalex.org/W6672845998"],"related_works":["https://openalex.org/W2089134982","https://openalex.org/W4299305030","https://openalex.org/W2040347389","https://openalex.org/W2952081610","https://openalex.org/W1983383464","https://openalex.org/W961142965","https://openalex.org/W4252842032","https://openalex.org/W2113270940","https://openalex.org/W2900627138","https://openalex.org/W3103742520"],"abstract_inverted_index":{"The":[0],"17th":[1,180,210],"of":[2,17,25,28,45,94,98,120,124,134,146,151,170],"the":[3,10,15,39,42,46,92,110,118,121,125,147,168],"problems":[4],"proposed":[5],"by":[6,57],"Steve":[7],"Smale":[8],"for":[9,14,142,185],"21st":[11],"century":[12],"asks":[13],"existence":[16],"a":[18,26,66,88,131,140,206],"deterministic":[19,186],"algorithm":[20,99,192],"computing":[21],"an":[22,191],"approximate":[23],"solution":[24,51,207],"system":[27,144],"n":[29],"complex":[30],"polynomials":[31],"in":[32,35,41,82,109],"$n$":[33],"unknowns":[34],"time":[36,150],"polynomial,":[37],"on":[38,161,167],"average,":[40],"size":[43,112,119],"N":[44,162],"input":[47,111,126,154],"system.":[48],"A":[49],"partial":[50],"to":[52,158,174,178,208],"this":[53,75,80,163],"problem":[54,181],"was":[55],"given":[56],"Carlos":[58],"Beltran":[59],"and":[60,96,101,113,173,193],"Luis":[61],"Miguel":[62],"Pardo":[63],"who":[64],"exhibited":[65],"randomized":[67],"algorithm,":[68],"call":[69],"it":[70],"LV,":[71],"doing":[72],"so.":[73],"In":[74,156],"paper":[76],"we":[77,86,129,138,176],"further":[78],"extend":[79],"result":[81],"several":[83],"directions.":[84],"Firstly,":[85],"perform":[87,130],"smoothed":[89,105],"analysis":[90,133],"(in":[91],"sense":[93],"Spielman":[95],"Teng)":[97],"LV":[100,152],"prove":[102],"that":[103,195],"its":[104,159,196],"complexity":[106,198],"is":[107,199,204],"polynomial":[108],"\u03c3-1,":[114],"where":[115],"\u03c3":[116],"controls":[117],"random":[122],"perturbation":[123],"systems.":[127],"Secondly,":[128],"condition-based":[132],"LV.":[135],"That":[136],"is,":[137],"give":[139],"bound,":[141],"each":[143],"f,":[145],"expected":[148],"running":[149],"with":[153],"f.":[155,171],"addition":[157],"dependence":[160],"bound":[164],"also":[165],"depends":[166],"condition":[169],"Thirdly,":[172],"conclude,":[175],"return":[177],"Smale's":[179,209],"as":[182],"originally":[183],"formulated":[184],"algorithms.":[187],"We":[188],"exhibit":[189],"such":[190],"show":[194],"average":[197],"NO(log":[200],"log":[201],"N).":[202],"This":[203],"nearly":[205],"problem.":[211]},"counts_by_year":[{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-14T23:27:15.235271","created_date":"2025-10-10T00:00:00"}