{"id":"https://openalex.org/W1994433543","doi":"https://doi.org/10.1145/1806689.1806780","title":"Tensor-rank and lower bounds for arithmetic formulas","display_name":"Tensor-rank and lower bounds for arithmetic formulas","publication_year":2010,"publication_date":"2010-06-05","ids":{"openalex":"https://openalex.org/W1994433543","doi":"https://doi.org/10.1145/1806689.1806780","mag":"1994433543"},"language":"en","primary_location":{"id":"doi:10.1145/1806689.1806780","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806780","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/A5041545934","display_name":"Ran Raz","orcid":"https://orcid.org/0009-0008-1656-2258"},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Ran Raz","raw_affiliation_strings":["Weizmann Institute, Rehovot, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Weizmann Institute, Rehovot, Israel","institution_ids":["https://openalex.org/I53964585"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041545934"],"corresponding_institution_ids":["https://openalex.org/I53964585"],"apc_list":null,"apc_paid":null,"fwci":6.5748,"has_fulltext":false,"cited_by_count":51,"citation_normalized_percentile":{"value":0.9760651,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"659","last_page":"666"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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":0.9994000196456909,"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/T12303","display_name":"Tensor decomposition and applications","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9937999844551086,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.616628110408783},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.5404866337776184},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5325535535812378},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4304087162017822},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.32358381152153015},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.22816726565361023},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.19555386900901794}],"concepts":[{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.616628110408783},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5404866337776184},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5325535535812378},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4304087162017822},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.32358381152153015},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.22816726565361023},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.19555386900901794}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1806689.1806780","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806780","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":25,"referenced_works":["https://openalex.org/W64890674","https://openalex.org/W201320490","https://openalex.org/W1973342538","https://openalex.org/W1979337672","https://openalex.org/W2007707374","https://openalex.org/W2018202033","https://openalex.org/W2023541349","https://openalex.org/W2026944800","https://openalex.org/W2027044307","https://openalex.org/W2055425281","https://openalex.org/W2081256023","https://openalex.org/W2106221905","https://openalex.org/W2110206454","https://openalex.org/W2116160867","https://openalex.org/W2128510017","https://openalex.org/W2140994271","https://openalex.org/W2150144243","https://openalex.org/W2156776891","https://openalex.org/W2170831059","https://openalex.org/W2171403864","https://openalex.org/W2172394696","https://openalex.org/W2173866771","https://openalex.org/W2611339529","https://openalex.org/W2952058256","https://openalex.org/W3098779275"],"related_works":["https://openalex.org/W2781510240","https://openalex.org/W2950186459","https://openalex.org/W2170114491","https://openalex.org/W2897298721","https://openalex.org/W2242624680","https://openalex.org/W2136127937","https://openalex.org/W4290987221","https://openalex.org/W2216309014","https://openalex.org/W3199841771","https://openalex.org/W3006184558"],"abstract_inverted_index":{"We":[0,91,197],"show":[1,92,198],"that":[2,45,86,93,178,199,272],"any":[3,95,135,201,242],"explicit":[4,30],"example":[5],"for":[6,34,50,63,79,94,115,130,134,155,165,182,185,194,200,221,237,241,258,268,276,279,288],"a":[7,76,106,120,151,160,169,212,226,254,263],"tensor":[8],"A:[n]r":[9],"->":[10],"F":[11],"with":[12],"tensor-rank":[13],"\u2265":[14],"nr":[15],"\u22c5":[16,128,234],"(1-":[17],"o(1)),":[18],"(where":[19],"r":[20,136,144,243],"\u2264":[21,137,145,244],"log":[22,25,26,138,141,142,146,245,248,249],"n":[23,139,246],"/":[24,140,247],"n),":[27],"implies":[28],"an":[29],"super-polynomial":[31,60,179,191,273,285],"lower":[32,48,61,180,192,274,286],"bound":[33],"the":[35,51,64,88],"size":[36,52,65,110,124,153,162,216,230,256,265],"of":[37,53,56,66,72,83,100,109,123,171,187,206,215,229,281],"general":[38,67,195,289],"arithmetic":[39,54,68,84],"formulas":[40,55,184,278],"over":[41],"F.":[42],"This":[43,167,270],"shows":[44,177,271],"strong":[46],"enough":[47],"bounds":[49,62,181,193,275,287],"depth":[57,113,219],"3":[58],"imply":[59,190,284],"formulas.":[69,196,290],"One":[70],"component":[71],"our":[73],"proof":[74],"is":[75],"new":[77],"approach":[78],"homogenization":[80],"and":[81,112,173,176,218],"multilinearization":[82],"formulas,":[85],"gives":[87],"following":[89],"results:":[90],"n-variate":[96,202],"homogenous":[97,121,163,183],"polynomial":[98,152,161,204,255,264],"f":[99,116,156,205,222,259],"degree":[101,189,207,283],"r,":[102,208],"if":[103,148,209,251],"there":[104,118,149,158,210,224,252,261],"exists":[105,119,150,159,211,225,253,262],"(fanin-2)":[107,213],"formula":[108,122,154,164,214,228,257,267],"s":[111,217,235],"d":[114,220],"then":[117,157,223,260],"O":[125,231],"(":[126,232],"d+r+1/r":[127],"s)":[129],"f.":[131,166,238,269],"In":[132,239],"particular,":[133,240],"n,":[143,147,250],"refutes":[168],"conjecture":[170],"Nisan":[172],"Wigderson":[174],"[10]":[175],"polynomials":[186,280],"small":[188,282],"set-multilinear":[203,227,266,277],"(d+2)r":[233],")":[236]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":13},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":7}],"updated_date":"2026-07-29T14:22:42.915294","created_date":"2025-10-10T00:00:00"}