{"id":"https://openalex.org/W2407882290","doi":"https://doi.org/10.1137/151002423","title":"An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas","display_name":"An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2407882290","doi":"https://doi.org/10.1137/151002423","mag":"2407882290"},"language":"en","primary_location":{"id":"doi:10.1137/151002423","is_oa":false,"landing_page_url":"https://doi.org/10.1137/151002423","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/A5079701530","display_name":"Neeraj Kayal","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Neeraj Kayal","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083072245","display_name":"Nutan Limaye","orcid":"https://orcid.org/0000-0002-0238-1674"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nutan Limaye","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101655060","display_name":"Chandan Saha","orcid":"https://orcid.org/0009-0003-7496-7396"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chandan Saha","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5025305994","display_name":"Srikanth Srinivasan","orcid":"https://orcid.org/0000-0001-6491-124X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Srikanth Srinivasan","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5079701530"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00375563,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"46","issue":"1","first_page":"307","last_page":"335"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9987999796867371,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9987999796867371,"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":0.9983000159263611,"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/monomial","display_name":"Monomial","score":0.8842951059341431},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7357869744300842},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7121973633766174},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.6713421940803528},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6682705879211426},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.634113073348999},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.591550886631012},{"id":"https://openalex.org/keywords/homogeneous-polynomial","display_name":"Homogeneous polynomial","score":0.5569420456886292},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5525816679000854},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4220760762691498},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.4205129146575928},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12242287397384644},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11348986625671387},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.09987163543701172}],"concepts":[{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.8842951059341431},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7357869744300842},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7121973633766174},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.6713421940803528},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6682705879211426},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.634113073348999},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.591550886631012},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.5569420456886292},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5525816679000854},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4220760762691498},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4205129146575928},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12242287397384644},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11348986625671387},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.09987163543701172},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/151002423","is_oa":false,"landing_page_url":"https://doi.org/10.1137/151002423","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:eprints.iisc.ac.in:53069","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4377196309","display_name":"NOT FOUND REPOSITORY (Indian Institute of Science Bangalore)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I59270414","host_organization_name":"Indian Institute of Science Bangalore","host_organization_lineage":["https://openalex.org/I59270414"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Conference Proceedings"},{"id":"pmh:oai:dsapce.library.iitb.ac.in:123456789/21331","is_oa":false,"landing_page_url":"http://doi.org/10.1137/151002423","pdf_url":null,"source":{"id":"https://openalex.org/S4306400899","display_name":"DSpace (IIT Bombay)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I162827531","host_organization_name":"Indian Institute of Technology Bombay","host_organization_lineage":["https://openalex.org/I162827531"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"mag:2407882290","is_oa":false,"landing_page_url":"https://dblp.uni-trier.de/db/journals/eccc/eccc21.html#KayalLSS14","pdf_url":null,"source":{"id":"https://openalex.org/S11189987","display_name":"Electronic colloquium on computational complexity","issn_l":"1433-8092","issn":["1433-8092"],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":"Electronic colloquium on computational complexity","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4745074546","display_name":null,"funder_award_id":"4702-1","funder_id":"https://openalex.org/F4320321027","funder_display_name":"Indo-French Centre for the Promotion of Advanced Research"}],"funders":[{"id":"https://openalex.org/F4320321027","display_name":"Indo-French Centre for the Promotion of Advanced Research","ror":"https://ror.org/017aem598"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W2016576580","https://openalex.org/W2023541349","https://openalex.org/W2026944800","https://openalex.org/W2081256023","https://openalex.org/W2089857781","https://openalex.org/W2110206454","https://openalex.org/W2149617935","https://openalex.org/W2171403864","https://openalex.org/W2172394696","https://openalex.org/W2313549751","https://openalex.org/W4256705525","https://openalex.org/W4293765205"],"related_works":["https://openalex.org/W3106203092","https://openalex.org/W2405068759","https://openalex.org/W1988738948","https://openalex.org/W2582672610","https://openalex.org/W3083374753","https://openalex.org/W3021197982","https://openalex.org/W1983449088","https://openalex.org/W3124934252","https://openalex.org/W2162153693","https://openalex.org/W2762239886","https://openalex.org/W2153575169","https://openalex.org/W3205049429","https://openalex.org/W1487480175","https://openalex.org/W1611937190","https://openalex.org/W96597593","https://openalex.org/W2172544626","https://openalex.org/W87866823","https://openalex.org/W2982504013","https://openalex.org/W2964181019","https://openalex.org/W1484588855"],"abstract_inverted_index":{"We":[0],"show":[1],"here":[2],"a":[3,53,77],"$2^{\\Omega(\\sqrt{d}":[4],"\\cdot":[5,107],"\\log":[6,108,172],"N)}$":[7,173],"size":[8],"lower":[9,136,150,174],"bound":[10,137,144,151,175],"for":[11,49],"homogeneous":[12,73,83],"depth":[13],"four":[14],"arithmetic":[15],"formulas":[16],"over":[17],"fields":[18],"of":[19,29,31,52,59,100,123,152,161,180],"characteristic":[20],"zero.":[21],"That":[22],"is,":[23],"we":[24,115],"give":[25],"an":[26,141],"explicit":[27],"family":[28,58,122],"polynomials":[30,74],"degree":[32],"$d$":[33],"on":[34,134,159],"$N$":[35],"variables":[36],"(with":[37],"$N":[38],"=":[39,64],"d^3$":[40],"in":[41,56,126,157,176,186],"our":[42],"case)":[43],"with":[44],"0,":[45],"1-coefficients":[46],"such":[47],"that":[48,76,95],"any":[50],"representation":[51],"polynomial":[54,78],"$f$":[55],"this":[57],"the":[60,70,89,119,127,148,170,177],"form":[61],"$":[62,68,96,110],"f":[63],"\\sum_{i}":[65],"\\prod_{j}":[66],"Q_{ij},":[67],"where":[69],"$Q_{ij}$'s":[71],"are":[72],"(recall":[75],"is":[79,125],"said":[80],"to":[81,117,147],"be":[82],"if":[84],"all":[85],"its":[86],"monomials":[87,101],"have":[88],"same":[90],"degree),":[91],"it":[92],"must":[93],"hold":[94],"\\sum_{i,":[97],"j}":[98],"(\\text{number":[99],"of~}":[102],"Q_{ij})":[103],"\\geq":[104],"2^{\\Omega":[105],"(\\sqrt{d}":[106],"N)}.":[109],"The":[111],"abovementioned":[112],"family,":[113],"which":[114],"refer":[116],"as":[118,145],"Nisan--Wigderson":[120],"design-based":[121],"polynomials,":[124],"complexity":[128],"class":[129],"$\\mathsf{VNP}$.":[130],"Our":[131],"work":[132,179],"builds":[133],"recent":[135],"results":[138],"and":[139,169,183,189],"yields":[140],"improved":[142],"quantitative":[143],"compared":[146],"quasi-polynomial":[149],"[N.":[153],"Kayal":[154],"et":[155],"al.,":[156],"Symposium":[158],"Theory":[160],"Computing,":[162],"ACM,":[163],"New":[164],"York,":[165],"2014,":[166,195],"pp.":[167,196],"119--127]":[168],"$N^{\\Omega(\\log":[171],"independent":[178],"[M.":[181],"Kumar":[182],"S.":[184],"Saraf,":[185],"Automata,":[187],"Languages,":[188],"Programming,":[190],"Part":[191],"I,":[192],"Springer,":[193],"Berlin,":[194],"751--762].":[197]},"counts_by_year":[],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}
