{"id":"https://openalex.org/W4416024908","doi":"https://doi.org/10.4230/lipics.socg.2026.59","title":"Improved Bound for the k-Variate Elekes-R\u00f3nyai Theorem","display_name":"Improved Bound for the k-Variate Elekes-R\u00f3nyai Theorem","publication_year":2025,"publication_date":"2025-11-06","ids":{"openalex":"https://openalex.org/W4416024908","doi":"https://doi.org/10.4230/lipics.socg.2026.59"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2511.04233","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2511.04233","pdf_url":"https://arxiv.org/pdf/2511.04233","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2511.04233","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120304954","display_name":"Yaara Jahn","orcid":null},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Jahn, Yaara","raw_affiliation_strings":["Hebrew University of Jerusalem, Israel"],"raw_orcid":"https://orcid.org/0009-0009-1384-1136","affiliations":[{"raw_affiliation_string":"Hebrew University of Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040938021","display_name":"Orit E. Raz","orcid":"https://orcid.org/0000-0002-2910-436X"},"institutions":[{"id":"https://openalex.org/I124227911","display_name":"Ben-Gurion University of the Negev","ror":"https://ror.org/05tkyf982","country_code":"IL","type":"education","lineage":["https://openalex.org/I124227911"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Raz, Orit E.","raw_affiliation_strings":["Ben-Gurion University, Be'er-Sheva, Israel"],"raw_orcid":"https://orcid.org/0000-0002-2910-436X","affiliations":[{"raw_affiliation_string":"Ben-Gurion University, Be'er-Sheva, Israel","institution_ids":["https://openalex.org/I124227911"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.6317999958992004,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.6317999958992004,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied 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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.06239999830722809,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T12508","display_name":"Functional Equations Stability Results","score":0.029600000008940697,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5982999801635742},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.4426000118255615},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.4424000084400177},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.39259999990463257},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.38190001249313354},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.37869998812675476}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8578000068664551},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5982999801635742},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48980000615119934},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4625000059604645},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.4426000118255615},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.4424000084400177},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.39259999990463257},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.38190001249313354},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.37869998812675476},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2750000059604645},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.26489999890327454},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C194886279","wikidata":"https://www.wikidata.org/wiki/Q430271","display_name":"Fundamental theorem","level":3,"score":0.2563999891281128},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.25609999895095825},{"id":"https://openalex.org/C171636804","wikidata":"https://www.wikidata.org/wiki/Q1149458","display_name":"Compactness theorem","level":4,"score":0.2531999945640564}],"mesh":[],"locations_count":5,"locations":[{"id":"pmh:oai:arXiv.org:2511.04233","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2511.04233","pdf_url":"https://arxiv.org/pdf/2511.04233","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:drops-oai.dagstuhl.de:25866","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.59","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/LIPIcs.SoCG.2026.59/LIPIcs.SoCG.2026.59.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"InProceedings"},{"id":"pmh:doi:10.4230/lipics.socg.2026.59","is_oa":true,"landing_page_url":"https://books.google.co.il/books?id=8Hyp1SLis4IC","pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"ConferencePaper"},{"id":"doi:10.48550/arxiv.2511.04233","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2511.04233","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"Preprint"},{"id":"doi:10.4230/lipics.socg.2026.59","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.socg.2026.59","pdf_url":null,"source":null,"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":"pmh:oai:arXiv.org:2511.04233","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2511.04233","pdf_url":"https://arxiv.org/pdf/2511.04233","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4416024908.pdf"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Let":[0],"$f\\in":[1],"\\mathbb{R}[x_1,\\ldots,":[2],"x_k]$,":[3],"for":[4,71,124,145,148],"$k\\ge":[5],"2$.":[6],"For":[7],"any":[8],"finite":[9,75],"sets":[10],"$A_1,\\ldots,":[11,76],"A_k\\subset":[12,77],"\\mathbb{R}$,":[13,78],"consider":[14],"the":[15,28,46,60,100,128,141,164,176],"set":[16],"$$":[17,25],"f(A_1,\\ldots,":[18],"A_k):=\\{f(a_1,\\ldots,":[19],"a_k)\\mid":[20],"(a_1,\\cdots,a_k)\\in":[21],"A_1\\times\\cdots":[22],"\\times":[23],"A_k\\},":[24],"that":[26,70,120],"is,":[27],"image":[29],"of":[30,39,52,74,80,102,106,130,157,166,172],"$A_1\\times":[31],"\\cdots\\times":[32],"A_k$":[33],"under":[34],"$f$.":[35],"Extending":[36],"a":[37,50,103,107],"theorem":[38],"Elekes":[40],"and":[41,49,55,67,135],"R\u00f3nyai,":[42],"which":[43,149],"deals":[44],"with":[45,59],"case":[47,61],"$k=2$,":[48],"result":[51],"Raz,":[53],"Sharir,":[54],"De":[56],"Zeeuw,":[57],"dealing":[58],"$k=3$,":[62],"it":[63],"was":[64],"proved":[65],"Raz":[66],"Shem":[68],"Tov,":[69],"every":[72,125],"choice":[73],"each":[79],"size":[81],"$n$,":[82],"one":[83],"has":[84,90],"\\begin{equation}\\label{RSbound}":[85],"|f(A_1,\\ldots,A_k)|=\u03a9(n^{3/2}),":[86],"\\end{equation}":[87,123],"unless":[88],"$f$":[89,147],"some":[91],"degenerate":[92],"special":[93],"form.":[94],"In":[95],"this":[96],"paper,":[97],"we":[98,118],"introduce":[99],"notion":[101],"{\\it":[104],"rank}":[105],"$k$-variate":[108],"polynomial":[109],"$f$,":[110],"denoted":[111],"as":[112],"${\\rm":[113,137,150],"rank}(f)$.":[114],"Letting":[115],"$r={\\rm":[116],"rank}(f)$,":[117],"prove":[119],"\\begin{equation}":[121],"|f(A_1,\\ldots,A_k)|=\u03a9\\left(n^{\\frac{5r-4}{2r}-\\varepsilon}\\right),":[122],"$\\varepsilon>0$,":[126],"where":[127],"constant":[129],"proportionality":[131],"depends":[132],"on":[133,136,175],"$\\varepsilon$":[134],"deg}(f)$.":[138],"This":[139],"improves":[140],"previous":[142],"lower":[143,162],"bound,":[144],"polynomials":[146],"rank}(f)\\ge":[151],"3$.":[152],"We":[153],"present":[154],"an":[155],"application":[156],"our":[158],"main":[159],"result,":[160],"to":[161],"bound":[163],"number":[165],"distinct":[167],"$d$-volumes":[168],"spanned":[169],"by":[170],"$(d+1)$-tuples":[171],"points":[173],"lying":[174],"moment":[177],"curve":[178],"in":[179],"$\\mathbb{R}^d$.":[180]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-11-08T00:00:00"}