{"id":"https://openalex.org/W2011374998","doi":"https://doi.org/10.1137/0218024","title":"On the Worst-Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials","display_name":"On the Worst-Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials","publication_year":1989,"publication_date":"1989-04-01","ids":{"openalex":"https://openalex.org/W2011374998","doi":"https://doi.org/10.1137/0218024","mag":"2011374998"},"language":"en","primary_location":{"id":"doi:10.1137/0218024","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0218024","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-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/A5015511691","display_name":"James Renegar","orcid":"https://orcid.org/0000-0002-6171-204X"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"James Renegar","raw_affiliation_strings":["Cornell University (Ithaca, NY);"],"affiliations":[{"raw_affiliation_string":"Cornell University (Ithaca, NY);","institution_ids":["https://openalex.org/I205783295"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5015511691"],"corresponding_institution_ids":["https://openalex.org/I205783295"],"apc_list":null,"apc_paid":null,"fwci":3.8273,"has_fulltext":false,"cited_by_count":60,"citation_normalized_percentile":{"value":0.94050669,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"18","issue":"2","first_page":"350","last_page":"370"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9993000030517578,"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":0.9993000030517578,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9984999895095825,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.753103494644165},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7291697859764099},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6026870012283325},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.5413271188735962},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5394147038459778},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4319693148136139},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08449509739875793},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07607454061508179}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.753103494644165},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7291697859764099},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6026870012283325},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.5413271188735962},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5394147038459778},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4319693148136139},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08449509739875793},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07607454061508179},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/0218024","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0218024","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"pmh:oai:ecommons.cornell.edu:1813/8631","is_oa":false,"landing_page_url":"https://hdl.handle.net/1813/8631","pdf_url":null,"source":{"id":"https://openalex.org/S4306400757","display_name":"eCommons (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"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":"technical report"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1551631489","https://openalex.org/W1590407118","https://openalex.org/W1977574397","https://openalex.org/W1982522946","https://openalex.org/W2055909024","https://openalex.org/W2082387287","https://openalex.org/W2124521536"],"related_works":["https://openalex.org/W2306553012","https://openalex.org/W4251417359","https://openalex.org/W2381059884","https://openalex.org/W4229993731","https://openalex.org/W2372756075","https://openalex.org/W1979193983","https://openalex.org/W2220405813","https://openalex.org/W1821795854","https://openalex.org/W2374111230","https://openalex.org/W4367190712"],"abstract_inverted_index":{"Let":[0,9,44],"$d_1":[1,55],",":[2,56],"\\cdots":[3,57],",d_n":[4,58],"$":[5,21,40,83],"be":[6],"positive":[7],"integers.":[8],"$\\mathcal{P}$":[10],"denote":[11],"the":[12,69,118],"set":[13],"of":[14,16,67,73,125],"systems":[15],"polynomials":[17],"$f:\\mathbb{C}^n":[18],"\\to":[19],"\\mathbb{C}^n":[20],"that":[22,33,60,103],"have":[23],"only":[24],"finitely":[25],"many":[26],"zeros,":[27],"including":[28],"those":[29,80],"\u201cat":[30],"infinity,\u201d":[31],"and":[32,109],"satisfy":[34],"degree":[35],"$(f_i":[36],")":[37],"=":[38,162,164],"d_i":[39],"for":[41,53,89,116],"all":[42],"i.":[43],"$0":[45],"<":[46],"\\varepsilon":[47,101,141],"\\leqq":[48,87],"R$.":[49],"It":[50],"is":[51,94,104,121,131],"shown":[52],"fixed":[54],"$,":[59],"with":[61],"respect":[62],"to":[63,77,105],"a":[64],"certain":[65],"model":[66],"computation,":[68],"worst-case":[70],"computational":[71],"complexity":[72],"obtaining":[74],"$\\varepsilon":[75],"$-approximations":[76],"at":[78],"least":[79],"zeros":[81],"$\\xi":[82],"satisfying":[84],"$|\\xi":[85],"|":[86],"R$":[88],"arbitrary":[90],"$f":[91],"\\in":[92],"\\mathcal{P}$":[93],"$\\Theta":[95],"(\\log":[96,134],"\\log":[97,136],"(({":[98],"R":[99,139],"/":[100,140],"}))$;":[102],"say,":[106],"both":[107],"upper":[108,119],"lower":[110],"bounds":[111],"are":[112],"proved.":[113],"An":[114],"algorithm":[115,130],"proving":[117],"bound":[120],"introduced.":[122],"The":[123],"number":[124],"operations":[126],"required":[127],"by":[128],"this":[129],"\\[O\\left[":[132],"{n\\mathcal{D}^4":[133],"\\mathcal{D})(\\log":[135],"(":[137],"{":[138],"}":[142,158],"))":[143],"+":[144,150],"n^2":[145],"\\mathcal{D}^4":[146],"\\left(":[147],"\\begin{array}{*{20}c}":[148],"{1":[149],"\\Sigma":[151],"d_i}":[152],"\\\\":[153,155],"n":[154],"\\end{array}":[156],"\\right)^4":[157],"\\right],\\quad":[159],"{\\text{where":[160],"}}\\mathcal{D}":[161],"\\prod\\limits_{i":[163],"1}^n":[165],"{d_i":[166],".}":[167],"\\]":[168]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}