{"id":"https://openalex.org/W3013848491","doi":"https://doi.org/10.1145/3543685","title":"A Polynomial Degree Bound on Equations for Non-rigid Matrices and Small Linear Circuits","display_name":"A Polynomial Degree Bound on Equations for Non-rigid Matrices and Small Linear Circuits","publication_year":2022,"publication_date":"2022-06-30","ids":{"openalex":"https://openalex.org/W3013848491","doi":"https://doi.org/10.1145/3543685","mag":"3013848491"},"language":"en","primary_location":{"id":"doi:10.1145/3543685","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3543685","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/A5073175433","display_name":"Ben Lee Volk","orcid":"https://orcid.org/0000-0002-7143-7280"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ben Lee Volk","raw_affiliation_strings":["Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel"],"raw_orcid":"https://orcid.org/0000-0002-7143-7280","affiliations":[{"raw_affiliation_string":"Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel","institution_ids":[]}]},{"author_position":"last","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":false,"raw_author_name":"Mrinal Kumar","raw_affiliation_strings":["Department of Computer Science &amp; Engineering, IIT Bombay Mumbai, Mumbai, India"],"raw_orcid":"https://orcid.org/0000-0002-6430-0219","affiliations":[{"raw_affiliation_string":"Department of Computer Science &amp; Engineering, IIT Bombay Mumbai, Mumbai, India","institution_ids":["https://openalex.org/I162827531"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.00613657,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"14","issue":"2","first_page":"1","last_page":"14"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9994999766349792,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8115943670272827},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6781779527664185},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6521360278129578},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6028052568435669},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5764434337615967},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5273391008377075},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5214471220970154},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5194389820098877},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44732803106307983},{"id":"https://openalex.org/keywords/monic-polynomial","display_name":"Monic polynomial","score":0.443335622549057},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.43053358793258667},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.4149111211299896},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4118097126483917},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18587356805801392}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8115943670272827},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6781779527664185},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6521360278129578},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6028052568435669},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5764434337615967},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5273391008377075},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5214471220970154},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5194389820098877},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44732803106307983},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.443335622549057},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.43053358793258667},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.4149111211299896},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4118097126483917},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18587356805801392},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/3543685","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3543685","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1530008367","https://openalex.org/W1978055063","https://openalex.org/W1981241857","https://openalex.org/W1984256153","https://openalex.org/W1988738948","https://openalex.org/W2026036943","https://openalex.org/W2026944800","https://openalex.org/W2053068069","https://openalex.org/W2074207216","https://openalex.org/W2077143506","https://openalex.org/W2081192873","https://openalex.org/W2087664562","https://openalex.org/W2137442467","https://openalex.org/W2171197316","https://openalex.org/W2181660280","https://openalex.org/W2576114839","https://openalex.org/W2611339529","https://openalex.org/W2904278047","https://openalex.org/W2962908760","https://openalex.org/W2971503262","https://openalex.org/W2986343958","https://openalex.org/W3029679829","https://openalex.org/W3129128072","https://openalex.org/W4210445513","https://openalex.org/W4288027245","https://openalex.org/W4293468793"],"related_works":["https://openalex.org/W2010641749","https://openalex.org/W2951225764","https://openalex.org/W1985103827","https://openalex.org/W3172879256","https://openalex.org/W2257297152","https://openalex.org/W1980475358","https://openalex.org/W1968689469","https://openalex.org/W936569904","https://openalex.org/W1976750632","https://openalex.org/W1986901684"],"abstract_inverted_index":{"We":[0,137],"show":[1,28,139,248,266],"that":[2,29,64,68,157,249,267],"there":[3,36],"is":[4,37,205],"an":[5,168],"equation":[6],"of":[7,18,21,56,75,78,87,102,149,152,202,219,270],"degree":[8,57,143],"at":[9,58,80,89,172],"most":[10,59,81,90,173],"poly(":[11,60,134],"n":[12,40,54,61,82,91,125,135,174],")":[13,62,127],"for":[14,30,118,145,231],"the":[15,22,113,146,158,182,200,203,260,271],"(Zariski":[16,147],"closure":[17,148],"the)":[19,150],"set":[20,151],"non-rigid":[23,232],"matrices:":[24],"That":[25],"is,":[26],"we":[27,187,265],"every":[31,65],"large":[32],"enough":[33],"field":[34],"\ud835\udd3d,":[35],"a":[38,73,76,85,100,140,216,242,255,263],"non-zero":[39],"2":[41,92,126,175],"-variate":[42],"polynomial":[43,142,217,252,258,272],"P":[44],"\u03b5":[45],"\ud835\udd3d[":[46],"x":[47,52],"1,":[48],"1":[49],",":[50,130],"...,":[51],"n,":[53],"]":[55,110,132,225],"such":[63,156,191,251],"matrix":[66,77,86],"M":[67,155,163],"can":[69,164],"be":[70,165],"written":[71],"as":[72,186],"sum":[74],"rank":[79],"/100":[83,93],"and":[84,106,111,211,213,221,234,291],"sparsity":[88],"satisfies":[94],"P(M)":[95],"=":[96],"0.":[97],"This":[98],"confirms":[99],"conjecture":[101],"Gesmundo,":[103],"Hauenstein,":[104],"Ikenmeyer,":[105],"Landsberg":[107],"[":[108,128,223,293],"9":[109,129],"improves":[112],"best":[114],"upper":[115],"bound":[116,144,192],"known":[117,194],"this":[119,197,239],"problem":[120,275],"down":[121],"from":[122],"exp":[123],"(":[124],"12":[131],"to":[133,196,226,247],").":[136],"also":[138],"similar":[141,283],"all":[153],"matrices":[154,233],"linear":[159,236],"transformation":[160],"represented":[161],"by":[162,167,289],"computed":[166],"algebraic":[169],"circuit":[170,279],"with":[171,241],"/200":[176],"edges":[177],"(without":[178],"any":[179,250,268],"restriction":[180],"on":[181,215],"depth).":[183],"As":[184,262],"far":[185],"are":[188,209],"aware,":[189],"no":[190],"was":[193,287],"prior":[195],"work":[198],"when":[199],"depth":[201],"circuits":[204],"unbounded.":[206],"Our":[207],"methods":[208],"elementary":[210],"short":[212],"rely":[214],"map":[218,253],"Shpilka":[220],"Volkovich":[222],"21":[224],"construct":[227],"low-degree":[228,256],"\u201cuniversal\u201d":[229],"maps":[230],"small":[235],"circuits.":[237],"Combining":[238],"construction":[240],"simple":[243],"dimension":[244],"counting":[245],"argument":[246],"has":[254],"annihilating":[257],"completes":[259],"proof.":[261],"corollary,":[264],"derandomization":[269],"identity":[273],"testing":[274],"will":[276],"imply":[277],"new":[278],"lower":[280],"bounds.":[281],"A":[282],"(but":[284],"incomparable)":[285],"theorem":[286],"proved":[288],"Kabanets":[290],"Impagliazzo":[292],"11":[294],"].":[295]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2020,"cited_by_count":2}],"updated_date":"2026-08-08T07:41:36.138363","created_date":"2025-10-10T00:00:00"}
