{"id":"https://openalex.org/W7170140596","doi":"https://doi.org/10.4230/lipics.ccc.2026.17","title":"Rank Bounds and Polynomial-Time PIT for \u03a3^k \u03a0 \u03a3 \u03a0\u00b2 Circuits","display_name":"Rank Bounds and Polynomial-Time PIT for \u03a3^k \u03a0 \u03a3 \u03a0\u00b2 Circuits","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7170140596","doi":"https://doi.org/10.4230/lipics.ccc.2026.17"},"language":"en","primary_location":{"id":"doi:10.4230/lipics.ccc.2026.17","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.ccc.2026.17","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"ConferencePaper"},"type":"conference-paper","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.4230/lipics.ccc.2026.17","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091058113","display_name":"Abhibhav Garg","orcid":"https://orcid.org/0000-0001-9084-7499"},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Garg, Abhibhav","raw_affiliation_strings":["University of Waterloo, Canada"],"raw_orcid":"https://orcid.org/0000-0001-9084-7499","affiliations":[{"raw_affiliation_string":"University of Waterloo, Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101874933","display_name":"Rafael Oliveira","orcid":"https://orcid.org/0000-0001-8917-8689"},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Oliveira, Rafael","raw_affiliation_strings":["University of Waterloo, Canada"],"raw_orcid":"https://orcid.org/0000-0001-8917-8689","affiliations":[{"raw_affiliation_string":"University of Waterloo, Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008550694","display_name":"Akash Kumar Sengupta","orcid":"https://orcid.org/0000-0002-0459-4411"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sengupta, Akash Kumar","raw_affiliation_strings":["Rutgers University, Piscataway, NJ, USA"],"raw_orcid":"https://orcid.org/0000-0002-0459-4411","affiliations":[{"raw_affiliation_string":"Rutgers University, Piscataway, NJ, USA","institution_ids":["https://openalex.org/I102322142"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139820100","display_name":"Nir Shalmon","orcid":"https://orcid.org/0009-0000-5804-8312"},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Shalmon, Nir","raw_affiliation_strings":["Tel Aviv University, Israel"],"raw_orcid":"https://orcid.org/0009-0000-5804-8312","affiliations":[{"raw_affiliation_string":"Tel Aviv University, Israel","institution_ids":["https://openalex.org/I16391192"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004182247","display_name":"Amir Shpilka","orcid":"https://orcid.org/0000-0003-2384-425X"},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Shpilka, Amir","raw_affiliation_strings":["Tel Aviv University, Israel"],"raw_orcid":"https://orcid.org/0000-0003-2384-425X","affiliations":[{"raw_affiliation_string":"Tel Aviv University, Israel","institution_ids":["https://openalex.org/I16391192"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":null,"topics":[],"keywords":[{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6802999973297119},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6629999876022339},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5787000060081482},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5353999733924866},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5091999769210815},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5062000155448914},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.4729999899864197},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.44999998807907104},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.43230000138282776}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8141999840736389},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6802999973297119},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6629999876022339},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5902000069618225},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5787000060081482},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5353999733924866},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5184999704360962},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5091999769210815},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5062000155448914},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4729999899864197},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.44999998807907104},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.43230000138282776},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4009999930858612},{"id":"https://openalex.org/C148607811","wikidata":"https://www.wikidata.org/wiki/Q746863","display_name":"Algebraic function","level":3,"score":0.35249999165534973},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.3402000069618225},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.32749998569488525},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.29589998722076416},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.2851000130176544},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.260699987411499},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2535000145435333},{"id":"https://openalex.org/C205979905","wikidata":"https://www.wikidata.org/wiki/Q215084","display_name":"Analytic function","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4230/lipics.ccc.2026.17","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.ccc.2026.17","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"ConferencePaper"}],"best_oa_location":{"id":"doi:10.4230/lipics.ccc.2026.17","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.ccc.2026.17","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"ConferencePaper"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"A":[0],"depth-4":[1],"algebraic":[2,228],"circuit":[3,14,58],"with":[4,172],"top":[5,136,167],"fan-in":[6,10,137],"k":[7],"and":[8,118,229,242],"bottom":[9],"2":[11],"is":[12,47,60,80,85,109,125,206,215,282],"a":[13,129,154,243,260],"\u03a6":[15,19,59,79],"of":[16,43,73,92,110,115,159,195,199,202,209,232,237,263,268,271,286,290],"the":[17,29,57,67,88,93,105,113,135,149,157,166,181,196,203,207,233,250,266,269,287,291],"form":[18],"=":[20,22,25],"\u2211_{i":[21],"1}^k":[23],"\u220f_{j":[24],"1}^{m_i}":[26],"Q_{ij},":[27],"where":[28],"polynomials":[30,94],"Q_{ij}":[31],"\u2208":[32,96,99],"\ud835\udd42[x\u2081,":[33],"\u2026,":[34],"x_n]":[35],"have":[36],"degree":[37],"at":[38,297],"most":[39],"2.":[40],"The":[41],"class":[42],"all":[44],"such":[45,160],"circuits":[46,78],"denoted":[48],"by":[49,128,274],"\u03a3^k":[50,74,120,189],"\u03a0":[51,75,121,190],"\u03a3":[52,76,122,191],"\u03a0\u00b2.":[53],"We":[54,101],"say":[55],"that":[56,162],"an":[61,210,227,283],"identity":[62,124],"if":[63],"it":[64],"formally":[65],"computes":[66],"zero":[68],"polynomial.":[69],"An":[70],"important":[71],"parameter":[72],"\u03a0\u00b2":[77,123,192],"their":[81],"(linear)":[82],"rank,":[83],"which":[84,131,214,293],"defined":[86],"as":[87,226],"vector":[89],"space":[90],"dimension":[91],"{Q_{ij}}_{i":[95],"[k],":[97],"j":[98],"[m_i]}.":[100],"prove":[102,259],"that,":[103],"when":[104,170],"base":[106],"field":[107],"\ud835\udd42":[108],"characteristic":[111],"zero,":[112],"rank":[114,158,204],"any":[116],"(simple":[117],"minimal)":[119],"upper":[126],"bounded":[127],"function":[130],"depends":[132,163],"only":[133,164],"on":[134,143,156,165],"k.":[138],"This":[139,221,280],"result":[140,179,222,236,248],"makes":[141],"progress":[142],"[Beecken":[144,173],"et":[145,174,239,254,277,299],"al.,":[146,175,240,255,278,300],"2013],":[147,176],"being":[148],"first":[150,182],"work":[151],"to":[152],"establish":[153],"bound":[155],"identities":[161],"fan-in.":[168],"Moreover,":[169],"combined":[171],"our":[177,200],"main":[178],"yields":[180],"deterministic,":[183],"polynomial":[184],"time":[185],"PIT":[186],"algorithm":[187],"for":[188],"circuits.":[193],"One":[194],"key":[197],"components":[198],"proof":[201],"bounds":[205],"derivation":[208],"approximate":[211,234,245],"Hansen-type":[212],"result,":[213],"interesting":[216],"in":[217,265],"its":[218],"own":[219],"right.":[220],"can":[223,294],"be":[224,295],"seen":[225],"higher-dimensional":[230],"analogue":[231],"Sylvester-Gallai":[235,247],"[Ai":[238],"2014],":[241],"distinct":[244],"fractional":[246],"than":[249],"one":[251],"from":[252],"[Garg":[253,298],"2023].":[256],"Additionally,":[257],"we":[258],"robust":[261],"version":[262,289],"it,":[264],"spirit":[267],"generalization":[270],"Hansen\u2019s":[272],"theorem":[273],"[Boaz":[275],"Barak":[276],"2013].":[279],"paper":[281],"extended":[284],"abstract":[285],"full":[288],"paper,":[292],"found":[296],"2026].":[301]},"counts_by_year":[],"updated_date":"2026-07-24T06:07:23.975819","created_date":"2026-07-24T00:00:00"}