{"id":"https://openalex.org/W3006679116","doi":"https://doi.org/10.1145/3371506","title":"On the Power of Border of Depth-3 Arithmetic Circuits","display_name":"On the Power of Border of Depth-3 Arithmetic Circuits","publication_year":2020,"publication_date":"2020-02-11","ids":{"openalex":"https://openalex.org/W3006679116","doi":"https://doi.org/10.1145/3371506","mag":"3006679116"},"language":"en","primary_location":{"id":"doi:10.1145/3371506","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3371506","pdf_url":null,"source":{"id":"https://openalex.org/S205019203","display_name":"ACM Transactions on Computation Theory","issn_l":"1942-3454","issn":["1942-3454","1942-3462"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Computation Theory","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/A5100713973","display_name":"Mrinal Kumar","orcid":"https://orcid.org/0000-0002-6430-0219"},"institutions":[{"id":"https://openalex.org/I162827531","display_name":"Indian Institute of Technology Bombay","ror":"https://ror.org/02qyf5152","country_code":"IN","type":"education","lineage":["https://openalex.org/I162827531"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Mrinal Kumar","raw_affiliation_strings":["IIT Bombay, Mumbai, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IIT Bombay, Mumbai, India","institution_ids":["https://openalex.org/I162827531"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100713973"],"corresponding_institution_ids":["https://openalex.org/I162827531"],"apc_list":null,"apc_paid":null,"fwci":0.8288,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.78007246,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"12","issue":"1","first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":1.0,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":1.0,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9987000226974487,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9983000159263611,"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/degree","display_name":"Degree (music)","score":0.8228365182876587},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.741976797580719},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.6925172805786133},{"id":"https://openalex.org/keywords/arithmetic-circuit-complexity","display_name":"Arithmetic circuit complexity","score":0.648109495639801},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6320781111717224},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5174408555030823},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.5066016912460327},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4908662438392639},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4875590205192566},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4348067343235016},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18577489256858826},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1334349513053894},{"id":"https://openalex.org/keywords/arbitrary-precision-arithmetic","display_name":"Arbitrary-precision arithmetic","score":0.10066315531730652}],"concepts":[{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.8228365182876587},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.741976797580719},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.6925172805786133},{"id":"https://openalex.org/C155920423","wikidata":"https://www.wikidata.org/wiki/Q4791120","display_name":"Arithmetic circuit complexity","level":3,"score":0.648109495639801},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6320781111717224},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5174408555030823},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.5066016912460327},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4908662438392639},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4875590205192566},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4348067343235016},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18577489256858826},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1334349513053894},{"id":"https://openalex.org/C83581934","wikidata":"https://www.wikidata.org/wiki/Q527381","display_name":"Arbitrary-precision arithmetic","level":2,"score":0.10066315531730652},{"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/3371506","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3371506","pdf_url":null,"source":{"id":"https://openalex.org/S205019203","display_name":"ACM Transactions on Computation Theory","issn_l":"1942-3454","issn":["1942-3454","1942-3462"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Computation Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1508771701","https://openalex.org/W2009998423","https://openalex.org/W2015676876","https://openalex.org/W2093103267","https://openalex.org/W2162761802","https://openalex.org/W2171403864","https://openalex.org/W2467589244","https://openalex.org/W2590079450","https://openalex.org/W2805951067","https://openalex.org/W2922071740","https://openalex.org/W3139272858"],"related_works":["https://openalex.org/W3161230432","https://openalex.org/W1487201914","https://openalex.org/W280458463","https://openalex.org/W1170131723","https://openalex.org/W4301429470","https://openalex.org/W2168900540","https://openalex.org/W2125310826","https://openalex.org/W2372627074","https://openalex.org/W4367190712","https://openalex.org/W2052788392"],"abstract_inverted_index":{"We":[0],"show":[1,103],"that":[2,51,104],"over":[3],"the":[4,19,77,88,135,157,162,166,170,186],"field":[5],"of":[6,12,28,47,87,93,119,137,141,159,173,185],"complex":[7],"numbers,":[8],"every":[9,127],"homogeneous":[10,41,109],"polynomial":[11,110],"degree":[13,48,82,107,150],"d":[14,108,123,130],"can":[15,112],"be":[16,113],"approximated":[17],"(in":[18],"border":[20,136],"complexity":[21],"sense)":[22],"by":[23,115],"a":[24,73,106,138],"depth-3":[25,53,139],"arithmetic":[26,54,117],"circuit":[27,55,118,140],"top":[29,60,78,142,167],"fan-in":[30,61,79,143,168],"at":[31,62,178],"most":[32],"2.":[33],"This":[34],"is":[35,133,176],"quite":[36],"surprising,":[37],"since":[38],"there":[39],"exist":[40],"polynomials":[42],"P":[43,57,111,132],"on":[44,165],"n":[45,66],"variables":[46],"2,":[49],"such":[50],"any":[52],"computing":[56],"must":[58],"have":[59],"least":[63,179],"\u03a9":[64,181],"(":[65],").":[67],"As":[68],"an":[69,84,116],"application,":[70],"we":[71,102],"get":[72],"new":[74],"tradeoff":[75],"between":[76],"and":[80,97,148],"formal":[81,149,187],"in":[83,134,169],"approximate":[85],"analog":[86],"celebrated":[89],"depth":[90],"reduction":[91],"result":[92],"Gupta,":[94],"Kamath,":[95],"Kayal,":[96],"Saptharishi":[98],"[7,":[99],"10].":[100],"Formally,":[101],"if":[105],"computed":[114],"size":[120],"s":[121,144,151,180],"\u2265":[122],",":[124,131,183],"then":[125],"for":[126],"t":[128,146],"\u2264":[129],"O(":[145,152],")":[147,154],"d/t":[153],".":[155],"To":[156],"best":[158],"our":[160],"knowledge,":[161],"upper":[163],"bound":[164],"original":[171],"proof":[172],"Reference":[174],"[7]":[175],"always":[177],"(\u221ad)":[182],"regardless":[184],"degree.":[188]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
