{"id":"https://openalex.org/W4412376432","doi":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","title":"Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets","display_name":"Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets","publication_year":2015,"publication_date":"2015-07-31","ids":{"openalex":"https://openalex.org/W4412376432","doi":"https://doi.org/10.55016/ojs/cdm.v10i1.62182"},"language":"en","primary_location":{"id":"doi:10.55016/ojs/cdm.v10i1.62182","doi":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","is_oa":true,"landing_page_url":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","pdf_url":"https://cdm.ucalgary.ca/article/download/62182/46756","source":{"id":"https://openalex.org/S2498033248","display_name":"Contributions to Discrete Mathematics","issn_l":"1715-0868","issn":["1715-0868"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","norway-1"],"host_organization":"https://openalex.org/P4310312896","host_organization_name":"University of Calgary Press","host_organization_lineage":["https://openalex.org/P4310312896"],"host_organization_lineage_names":["University of Calgary Press"],"type":"journal"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Contributions to Discrete Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://cdm.ucalgary.ca/article/download/62182/46756","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087560321","display_name":"Terence Tao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Terence Tao","raw_affiliation_strings":["UCLA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"UCLA","institution_ids":["https://openalex.org/I161318765"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5087560321"],"corresponding_institution_ids":["https://openalex.org/I161318765"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.35662109,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"10","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980000257492065,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.0003000000142492354,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.0003000000142492354,"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/asymmetric-expansion","display_name":"asymmetric expansion","score":0.6079999804496765},{"id":"https://openalex.org/keywords/definable-sets","display_name":"definable sets","score":0.8550000190734863},{"id":"https://openalex.org/keywords/szemeredi-regularity-lemma","display_name":"Szemer\u00e9di regularity lemma","score":0.8980000019073486},{"id":"https://openalex.org/keywords/dense-graphs","display_name":"dense graphs","score":0.6539999842643738},{"id":"https://openalex.org/keywords/etale-fundamental-group","display_name":"\u00e9tale fundamental group","score":0.8569999933242798},{"id":"https://openalex.org/keywords/algebraic-varieties","display_name":"algebraic varieties","score":0.6990000009536743}],"concepts":[{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.8560483455657959},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8473026752471924},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5759062170982361},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.47337764501571655},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34681886434555054},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34327036142349243},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.55016/ojs/cdm.v10i1.62182","doi":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","is_oa":true,"landing_page_url":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","pdf_url":"https://cdm.ucalgary.ca/article/download/62182/46756","source":{"id":"https://openalex.org/S2498033248","display_name":"Contributions to Discrete Mathematics","issn_l":"1715-0868","issn":["1715-0868"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","norway-1"],"host_organization":"https://openalex.org/P4310312896","host_organization_name":"University of Calgary Press","host_organization_lineage":["https://openalex.org/P4310312896"],"host_organization_lineage_names":["University of Calgary Press"],"type":"journal"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Contributions to Discrete Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.55016/ojs/cdm.v10i1.62182","doi":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","is_oa":true,"landing_page_url":"https://doi.org/10.55016/ojs/cdm.v10i1.62182","pdf_url":"https://cdm.ucalgary.ca/article/download/62182/46756","source":{"id":"https://openalex.org/S2498033248","display_name":"Contributions to Discrete Mathematics","issn_l":"1715-0868","issn":["1715-0868"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","norway-1"],"host_organization":"https://openalex.org/P4310312896","host_organization_name":"University of Calgary Press","host_organization_lineage":["https://openalex.org/P4310312896"],"host_organization_lineage_names":["University of Calgary Press"],"type":"journal"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Contributions to Discrete Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G7201218755","display_name":"Global Behaviour of Critical Nonlinear PDE","funder_award_id":"0649473","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412376432.pdf","grobid_xml":"https://content.openalex.org/works/W4412376432.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W6605240630","https://openalex.org/W6606270696","https://openalex.org/W6631016710","https://openalex.org/W6685362491","https://openalex.org/W6820391742","https://openalex.org/W6823816502"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W3006682748","https://openalex.org/W1097487277","https://openalex.org/W2949964783","https://openalex.org/W2768610039","https://openalex.org/W2735923117","https://openalex.org/W2159703794","https://openalex.org/W2089427484","https://openalex.org/W2115913661","https://openalex.org/W2964302506"],"abstract_inverted_index":{"Let":[0],"$P:":[1],"\\F":[2,4],"\\times":[3],"\\to":[5],"\\F$":[6,47],"be":[7,228],"a":[8,14,32,57,81,99,119,194,197],"polynomial":[9],"of":[10,18,85,87,163,173,200,203,219,229,242],"bounded":[11,198],"degree":[12],"over":[13,170],"finite":[15,171],"field":[16],"$\\F$":[17],"large":[19,59,174],"characteristic.":[20,175],"In":[21],"this":[22],"paper":[23],"we":[24],"establish":[25,151],"the":[26,37,65,161,179,184,190,213,220,238],"following":[27],"dichotomy:":[28],"either":[29],"$P$":[30,63],"is":[31,80,154],"\\emph{moderate":[33],"asymmetric":[34,126],"expander}":[35],"in":[36,183,187],"sense":[38],"that":[39,50,90,159,188,217],"$|P(A,B)|":[40,130],"\\gg":[41,110],"|\\F|$":[42,133],"whenever":[43,113],"$A,":[44],"B":[45],"\\subset":[46],"are":[48,205,222],"such":[49],"$|A|":[51,114],"|B|":[52,136],"\\geq":[53,115,137],"C":[54,116],"|\\F|^{2-1/8}$":[55],"for":[56,75,102,123,139,207],"sufficiently":[58],"$C$,":[60],"or":[61,70,95],"else":[62],"takes":[64],"form":[66],"$P(x,y)":[67,71],"=":[68,72,131],"Q(F(x)+G(y))$":[69],"Q(F(x)":[73],"G(y))$":[74],"some":[76,140,208,234],"polynomials":[77,86],"$Q,F,G$.":[78],"This":[79,176,224],"reasonably":[82],"satisfactory":[83,121],"classification":[84,101,122],"two":[88],"variables":[89],"moderately":[91],"expand":[92],"(either":[93],"symmetrically":[94],"asymmetrically).":[96],"We":[97],"obtain":[98],"similar":[100],"weak":[103],"expansion":[104,127],"(in":[105,128],"which":[106,129,204,226],"one":[107],"has":[108],"$|P(A,A)|":[109],"|A|^{1/2}":[111],"|\\F|^{1/2}$":[112],"|\\F|^{1-1/16}$),":[117],"and":[118],"partially":[120],"almost":[124],"strong":[125],"(1-O(|\\F|^{-c}))":[132],"when":[134],"$|A|,":[135],"|\\F|^{1-c}$":[138],"small":[141,209],"absolute":[142],"constant":[143],"$c>0$).":[144],"The":[145],"main":[146],"new":[147],"tool":[148],"used":[149],"to":[150],"these":[152],"results":[153],"an":[155,243],"\\emph{algebraic":[156],"regularity":[157,181],"lemma}":[158],"describes":[160],"structure":[162],"dense":[164],"graphs":[165],"generated":[166],"by":[167],"definable":[168],"subsets":[169],"fields":[172],"lemma":[177,182,192,215],"strengthens":[178],"Sz\\'emeredi":[180],"algebraic":[185,244],"case,":[186],"while":[189],"latter":[191],"decomposes":[193],"graph":[195],"into":[196],"number":[199],"components,":[201],"most":[202],"$\\eps$-regular":[206],"but":[210],"fixed":[211],"$\\epsilon$,":[212],"former":[214],"ensures":[216],"all":[218],"components":[221],"$O(|\\F|^{-1/4})$-regular.":[223],"lemma,":[225],"may":[227],"independent":[230],"interest,":[231],"relies":[232],"on":[233],"basic":[235],"facts":[236],"about":[237],"\\'etale":[239],"fundamental":[240],"group":[241],"variety.":[245]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":2}],"updated_date":"2026-08-24T07:32:12.397491","created_date":"2025-10-10T00:00:00"}