{"id":"https://openalex.org/W2020713199","doi":"https://doi.org/10.1145/1806689.1806781","title":"Non-commutative circuits and the sum-of-squares problem","display_name":"Non-commutative circuits and the sum-of-squares problem","publication_year":2010,"publication_date":"2010-06-05","ids":{"openalex":"https://openalex.org/W2020713199","doi":"https://doi.org/10.1145/1806689.1806781","mag":"2020713199"},"language":"en","primary_location":{"id":"doi:10.1145/1806689.1806781","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806781","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":"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/A5103278473","display_name":"Pavel Hrube\u0161","orcid":"https://orcid.org/0000-0003-4526-4934"},"institutions":[{"id":"https://openalex.org/I40036882","display_name":"Institute for Advanced Study","ror":"https://ror.org/00f809463","country_code":"US","type":"facility","lineage":["https://openalex.org/I40036882"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Pavel Hrube\u0161","raw_affiliation_strings":["Institute for Advanced Study, Princeton, USA","Institute for Advanced study, Princeton, USA"],"affiliations":[{"raw_affiliation_string":"Institute for Advanced Study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]},{"raw_affiliation_string":"Institute for Advanced study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039043184","display_name":"Avi Wigderson","orcid":"https://orcid.org/0000-0002-1539-1417"},"institutions":[{"id":"https://openalex.org/I40036882","display_name":"Institute for Advanced Study","ror":"https://ror.org/00f809463","country_code":"US","type":"facility","lineage":["https://openalex.org/I40036882"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Avi Wigderson","raw_affiliation_strings":["Institute for Advanced Study, Princeton, USA","Institute for Advanced study, Princeton, USA"],"affiliations":[{"raw_affiliation_string":"Institute for Advanced Study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]},{"raw_affiliation_string":"Institute for Advanced study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041051779","display_name":"Amir Yehudayoff","orcid":"https://orcid.org/0000-0002-0177-1814"},"institutions":[{"id":"https://openalex.org/I40036882","display_name":"Institute for Advanced Study","ror":"https://ror.org/00f809463","country_code":"US","type":"facility","lineage":["https://openalex.org/I40036882"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Amir Yehudayoff","raw_affiliation_strings":["Institute for Advanced Study, Princeton, USA","Institute for Advanced study, Princeton, USA"],"affiliations":[{"raw_affiliation_string":"Institute for Advanced Study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]},{"raw_affiliation_string":"Institute for Advanced study, Princeton, USA","institution_ids":["https://openalex.org/I40036882"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5103278473"],"corresponding_institution_ids":["https://openalex.org/I40036882"],"apc_list":null,"apc_paid":null,"fwci":1.6726,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.83428815,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":null,"issue":null,"first_page":"667","last_page":"676"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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":0.9998000264167786,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9994999766349792,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/commutative-property","display_name":"Commutative property","score":0.8188778162002563},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.739392876625061},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5831242799758911},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.5317320227622986},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.4708383083343506},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4684735834598541},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.4679103493690491},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.467623770236969},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4670257568359375},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.45704323053359985},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4097645878791809},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09870189428329468},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09733042120933533},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06420063972473145}],"concepts":[{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.8188778162002563},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.739392876625061},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5831242799758911},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.5317320227622986},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.4708383083343506},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4684735834598541},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.4679103493690491},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.467623770236969},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4670257568359375},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.45704323053359985},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4097645878791809},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09870189428329468},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09733042120933533},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06420063972473145},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","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/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1806689.1806781","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806781","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W112858668","https://openalex.org/W201320490","https://openalex.org/W593649846","https://openalex.org/W1512291242","https://openalex.org/W1576854535","https://openalex.org/W1671501729","https://openalex.org/W1822909632","https://openalex.org/W1966191871","https://openalex.org/W1973342538","https://openalex.org/W1974331311","https://openalex.org/W1976995585","https://openalex.org/W1979446688","https://openalex.org/W1987465321","https://openalex.org/W1992360103","https://openalex.org/W2002273934","https://openalex.org/W2018202033","https://openalex.org/W2023541349","https://openalex.org/W2026405063","https://openalex.org/W2027044307","https://openalex.org/W2040689926","https://openalex.org/W2049156754","https://openalex.org/W2052168717","https://openalex.org/W2057552658","https://openalex.org/W2064524663","https://openalex.org/W2076574682","https://openalex.org/W2094389960","https://openalex.org/W2116488705","https://openalex.org/W2135189187","https://openalex.org/W2140994271","https://openalex.org/W2150144243","https://openalex.org/W2158111726","https://openalex.org/W2161611531","https://openalex.org/W2170831059","https://openalex.org/W2171403864","https://openalex.org/W2173866771","https://openalex.org/W2321633606","https://openalex.org/W2325839071","https://openalex.org/W2478188808","https://openalex.org/W2611339529"],"related_works":["https://openalex.org/W2165589608","https://openalex.org/W3093557575","https://openalex.org/W2354231916","https://openalex.org/W3182767856","https://openalex.org/W2162140574","https://openalex.org/W2354073393","https://openalex.org/W2361009070","https://openalex.org/W2121457398","https://openalex.org/W2378988399","https://openalex.org/W1973665004"],"abstract_inverted_index":{"We":[0],"initiate":[1],"a":[2,20,33,83],"direction":[3,16],"for":[4],"proving":[5],"lower":[6,23,87,102],"bounds":[7,24],"on":[8,19,25,89,104],"the":[9,26,40,45,77,105,111],"size":[10,27,106],"of":[11,28,107],"non-commutative":[12,29,112],"arithmetic":[13,30,108],"circuits.":[14],"This":[15],"is":[17,68],"based":[18],"connection":[21],"between":[22],"circuits":[31,109],"and":[32,73],"problem":[34],"about":[35],"commutative":[36],"degree":[37],"four":[38],"polynomials,":[39],"classical":[41],"sum-of-squares":[42],"problem:":[43],"find":[44],"smallest":[46],"n":[47,90,93],"such":[48],"that":[49,82],"there":[50],"exists":[51],"an":[52,100],"identity":[53],"(x12+x22+\u2022\u2022":[54],"+":[55,58],"xk2)\u2022":[56],"(y1^2+y22+\u2022\u2022":[57],"yk2)=":[59],"f12+f22+":[60],"...":[61],"+fn2,":[62],"where":[63],"each":[64],"fi":[65],"=":[66],"fi(X,Y)":[67],"bilinear":[69],"in":[70],"X={x1,...":[71],",xk}":[72],"Y={y1,...,":[74],"yk}.":[75],"Over":[76],"complex":[78],"numbers,":[79],"we":[80],"show":[81],"sufficiently":[84],"strong":[85],"super-linear":[86],"bound":[88,103],"in,":[91],"namely,":[92],"\u2265":[94],"k1+\u03b5":[95],"with":[96],"\u03b5":[97],">0,":[98],"implies":[99],"exponential":[101],"computing":[110],"permanent.":[113]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
