{"id":"https://openalex.org/W2067476801","doi":"https://doi.org/10.1137/070694879","title":"Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates","display_name":"Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2067476801","doi":"https://doi.org/10.1137/070694879","mag":"2067476801"},"language":"en","primary_location":{"id":"doi:10.1137/070694879","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070694879","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":null,"display_name":"Amir Shpilka","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Amir Shpilka","raw_affiliation_strings":["shpilka@cs.technion.ac.il#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"shpilka@cs.technion.ac.il#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7774,"has_fulltext":false,"cited_by_count":29,"citation_normalized_percentile":{"value":0.90068818,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"38","issue":"6","first_page":"2130","last_page":"2161"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9993000030517578,"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/T10720","display_name":"Complexity and Algorithms in Graphs","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/T11130","display_name":"Coding theory and cryptography","score":0.9986000061035156,"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"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9986000061035156,"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/multilinear-map","display_name":"Multilinear map","score":0.7247108817100525},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6979055404663086},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.6179118752479553},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5677233934402466},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.49121612310409546},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4879903197288513},{"id":"https://openalex.org/keywords/adder","display_name":"Adder","score":0.466006338596344},{"id":"https://openalex.org/keywords/arithmetic-circuit-complexity","display_name":"Arithmetic circuit complexity","score":0.45523786544799805},{"id":"https://openalex.org/keywords/factorization-of-polynomials","display_name":"Factorization of polynomials","score":0.45261603593826294},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.43168407678604126},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3817020058631897},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.33757728338241577},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.28810590505599976},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.20202791690826416},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17913714051246643},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.11922454833984375},{"id":"https://openalex.org/keywords/arbitrary-precision-arithmetic","display_name":"Arbitrary-precision arithmetic","score":0.09552383422851562}],"concepts":[{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.7247108817100525},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6979055404663086},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.6179118752479553},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5677233934402466},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.49121612310409546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4879903197288513},{"id":"https://openalex.org/C164620267","wikidata":"https://www.wikidata.org/wiki/Q376953","display_name":"Adder","level":3,"score":0.466006338596344},{"id":"https://openalex.org/C155920423","wikidata":"https://www.wikidata.org/wiki/Q4791120","display_name":"Arithmetic circuit complexity","level":3,"score":0.45523786544799805},{"id":"https://openalex.org/C108741685","wikidata":"https://www.wikidata.org/wiki/Q392477","display_name":"Factorization of polynomials","level":4,"score":0.45261603593826294},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.43168407678604126},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3817020058631897},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33757728338241577},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.28810590505599976},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.20202791690826416},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17913714051246643},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.11922454833984375},{"id":"https://openalex.org/C83581934","wikidata":"https://www.wikidata.org/wiki/Q527381","display_name":"Arbitrary-precision arithmetic","level":2,"score":0.09552383422851562},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","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},{"id":"https://openalex.org/C82876162","wikidata":"https://www.wikidata.org/wiki/Q17096504","display_name":"Latency (audio)","level":2,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/070694879","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070694879","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1974321539","https://openalex.org/W1980267752","https://openalex.org/W1991584875","https://openalex.org/W1995511496","https://openalex.org/W2003523975","https://openalex.org/W2030296711","https://openalex.org/W2033625452","https://openalex.org/W2050570613","https://openalex.org/W2094462297","https://openalex.org/W2094893810","https://openalex.org/W2113204179","https://openalex.org/W2117362057","https://openalex.org/W2120217745","https://openalex.org/W2124218043","https://openalex.org/W2142399242","https://openalex.org/W2148132824","https://openalex.org/W2912080987","https://openalex.org/W2914183794","https://openalex.org/W3142725347","https://openalex.org/W4210639400","https://openalex.org/W4255348961"],"related_works":["https://openalex.org/W4302086638","https://openalex.org/W2090100016","https://openalex.org/W4250880453","https://openalex.org/W2949163092","https://openalex.org/W2113600343","https://openalex.org/W2258315229","https://openalex.org/W2044077194","https://openalex.org/W3148150683","https://openalex.org/W2569466879","https://openalex.org/W4289744365"],"abstract_inverted_index":{"In":[0,128,174],"this":[1],"paper":[2],"we":[3,18,72,109,168,182],"consider":[4],"the":[5,43,56,63,82,90,96,104,126,131,135,146,152,156,170,175,178,210,235,246,261],"problem":[6],"of":[7,49,95,125,139,151,191,248,263],"constructing":[8],"a":[9,14,29,111,122,161,184,188,258],"small":[10,140,164,181],"arithmetic":[11,36],"circuit":[12,64,105,133,172,186],"for":[13,16,134,205,238],"polynomial":[15,136],"which":[17],"have":[19,34,200],"oracle":[20],"access.":[21],"Our":[22,227],"focus":[23],"is":[24,65,98,106,137,180,230],"on":[25,234,242,245,257],"n-variate":[26],"polynomials,":[27,241],"over":[28],"finite":[30],"field":[31],"$\\mathbb{F}$,":[32],"that":[33,76,117,177],"depth-3":[35,83,132,141,171,185,219,266],"circuits":[37,84,220,267],"(with":[38],"an":[39,74],"addition":[40],"gate":[41],"at":[42,51],"top)":[44],"with":[45,78,85,119,187,218,221,253,268],"two":[46,86,153,224,254],"multiplication":[47,87,154,192,225,270],"gates":[48,68,88],"degree":[50],"most":[52],"d.":[53],"We":[54],"obtain":[55],"following":[57],"results:":[58],"1.":[59],"Multilinear":[60],"case.":[61,102],"When":[62,103],"multilinear":[66,70,108],"(multiplication":[67],"compute":[69],"polynomials)":[71],"give":[73,110],"algorithm":[75,97,116,237],"outputs,":[77,118],"probability":[79,120],"$1-o(1)$,":[80,121],"all":[81,209],"computing":[89],"polynomial.":[91,127],"The":[92],"running":[93],"time":[94,115],"$\\operatorname{poly}(n,|\\mathbb{F}|)$.":[99],"2.":[100],"General":[101],"not":[107,138,162],"quasi-polynomial":[112,189],"(in":[113],"$n,d,|\\mathbb{F}|$)":[114],"succinct":[123],"representation":[124],"particular,":[129],"if":[130],"rank":[142,179],"(namely,":[143],"after":[144],"removing":[145],"g.c.d.":[147],"(greatest":[148],"common":[149],"divisor)":[150],"gates,":[155],"remaining":[157],"linear":[158,165,249],"functions":[159],"span":[160],"too":[163],"space),":[166],"then":[167],"output":[169,183],"itself.":[173],"case":[176],"number":[190],"gates.":[193,226,271],"$\\diamond$":[194],"Prior":[195],"to":[196],"our":[197],"work":[198],"there":[199,213],"been":[201],"several":[202],"interpolation":[203],"algorithms":[204],"restricted":[206],"models.":[207],"However,":[208],"techniques":[211],"used":[212],"completely":[214],"fail":[215],"when":[216],"dealing":[217],"even":[222],"just":[223],"proof":[228],"technique":[229],"new":[231],"and":[232,256],"relies":[233],"factorization":[236],"multivariate":[239],"black-box":[240],"lower":[243],"bounds":[244],"length":[247],"locally":[250],"decodable":[251],"codes":[252],"queries,":[255],"theorem":[259],"regarding":[260],"structure":[262],"identically":[264],"zero":[265],"four":[269]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}