{"id":"https://openalex.org/W4385515481","doi":"https://doi.org/10.1080/10586458.2023.2239265","title":"Orbits on K3 Surfaces of Markoff Type","display_name":"Orbits on K3 Surfaces of Markoff Type","publication_year":2023,"publication_date":"2023-08-03","ids":{"openalex":"https://openalex.org/W4385515481","doi":"https://doi.org/10.1080/10586458.2023.2239265"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2023.2239265","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2023.2239265","pdf_url":null,"source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","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/A5108329825","display_name":"Elena Fuchs","orcid":null},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Elena Fuchs","raw_affiliation_strings":["Department of Mathematics, University of California Davis, Davis, CA, USA"],"raw_orcid":"https://orcid.org/0000-0002-0978-5137","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091166307","display_name":"Matthew Litman","orcid":"https://orcid.org/0000-0002-0908-9369"},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Matthew Litman","raw_affiliation_strings":["Department of Mathematics, University of California Davis, Davis, CA, USA"],"raw_orcid":"https://orcid.org/0000-0002-0908-9369","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058156788","display_name":"Joseph H. Silverman","orcid":"https://orcid.org/0000-0003-3887-3248"},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Joseph H. Silverman","raw_affiliation_strings":["Department of Mathematics, Brown University, Providence, RI, USA"],"raw_orcid":"https://orcid.org/0000-0003-3887-3248","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Brown University, Providence, RI, USA","institution_ids":["https://openalex.org/I27804330"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035058492","display_name":"Austin Tran","orcid":null},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Austin Tran","raw_affiliation_strings":["Department of Mathematics, University of California Davis, Davis, CA, USA"],"raw_orcid":"https://orcid.org/0000-0003-3725-7822","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5058156788"],"corresponding_institution_ids":["https://openalex.org/I27804330"],"apc_list":null,"apc_paid":null,"fwci":2.7116,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.90292534,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"33","issue":"4","first_page":"663","last_page":"700"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9984999895095825,"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"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9984999895095825,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10849","display_name":"Finite Group Theory Research","score":0.9976000189781189,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7870895862579346},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.6072997450828552},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5769135355949402},{"id":"https://openalex.org/keywords/automorphism","display_name":"Automorphism","score":0.558070182800293},{"id":"https://openalex.org/keywords/orbit","display_name":"Orbit (dynamics)","score":0.5156663656234741},{"id":"https://openalex.org/keywords/genus","display_name":"Genus","score":0.49820590019226074},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.49718955159187317},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.44869565963745117},{"id":"https://openalex.org/keywords/surface","display_name":"Surface (topology)","score":0.4344331622123718},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4293258786201477},{"id":"https://openalex.org/keywords/analogy","display_name":"Analogy","score":0.4177272319793701},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.41552940011024475},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3955538868904114},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.26057183742523193},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2377851903438568},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.17327046394348145},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08528468012809753},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06281322240829468}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7870895862579346},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6072997450828552},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5769135355949402},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.558070182800293},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.5156663656234741},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.49820590019226074},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.49718955159187317},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.44869565963745117},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.4344331622123718},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4293258786201477},{"id":"https://openalex.org/C521332185","wikidata":"https://www.wikidata.org/wiki/Q185816","display_name":"Analogy","level":2,"score":0.4177272319793701},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.41552940011024475},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3955538868904114},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26057183742523193},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2377851903438568},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.17327046394348145},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08528468012809753},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06281322240829468},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C59822182","wikidata":"https://www.wikidata.org/wiki/Q441","display_name":"Botany","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/10586458.2023.2239265","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2023.2239265","pdf_url":null,"source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1495270607","https://openalex.org/W1976677460","https://openalex.org/W1992162705","https://openalex.org/W2000693464","https://openalex.org/W2002372149","https://openalex.org/W2003165831","https://openalex.org/W2039731427","https://openalex.org/W2048858343","https://openalex.org/W2062868118","https://openalex.org/W2063201429","https://openalex.org/W2089175547","https://openalex.org/W2513851937","https://openalex.org/W2962713058","https://openalex.org/W2963364721","https://openalex.org/W2964274337","https://openalex.org/W2964276714","https://openalex.org/W2982348275","https://openalex.org/W2982435174","https://openalex.org/W3017109513","https://openalex.org/W4241489498","https://openalex.org/W4248269519","https://openalex.org/W4311037871","https://openalex.org/W6644637806"],"related_works":["https://openalex.org/W2392206215","https://openalex.org/W2365201483","https://openalex.org/W2355561779","https://openalex.org/W2352407775","https://openalex.org/W2186567693","https://openalex.org/W108701362","https://openalex.org/W2469799552","https://openalex.org/W2768582344","https://openalex.org/W2615704157","https://openalex.org/W2972371517"],"abstract_inverted_index":{"AbstractLet":[0],"W\u2282P1\u00d7P1\u00d7P1":[1],"be":[2,500,611],"a":[3,10,67,125,151,161,170,261,266,283,327,396,409,477,507,563,574,591,597,693],"surface":[4,65,325,332],"given":[5,256],"by":[6,73,169,257,788],"the":[7,21,36,74,87,105,120,135,216,230,248,252,258,278,294,300,304,349,353,449,483,489,492,526,570,580,634,641,672,680,688,715,723,743,749,753,782],"vanishing":[8,259],"of":[9,45,69,90,107,113,129,139,163,165,172,251,254,260,348,380,436,467,485,525,658,752],"(2,":[11,262,564],"2,":[12,263,565],"2)-form.":[13],"These":[14],"surfaces":[15,131,255,365,382,528],"admit":[16],"three":[17,22,55,75,295],"involutions":[18,296,682],"coming":[19],"from":[20,502],"projections":[23],"W\u2192P1\u00d7P1,":[24],"so":[25,356,656,703,777],"we":[26,40,85,100,133,146,343,371,442,469,511,548,649,677,704,761],"call":[27],"them":[28],"tri-involutive":[29],"K3":[30,175,267,324,387],"(TIK3)":[31],"surfaces.":[32,96],"By":[33],"analogy":[34],"with":[35,333,346,736],"classical":[37,121,279],"Markoff":[38,46,122,280],"equation,":[39],"say":[41],"that":[42,227,246,273,293,321,363,376,418,441,452,546,593,624,646,697,720,763],"W":[43,515,522,595],"is":[44,51,265,317,326,395,408,448,456,523,596,620,626,712],"type":[47],"(MK3)":[48],"if":[49,422,425,521,594],"it":[50,357,497,560],"symmetric":[52],"in":[53,158,233,373,383,412,430,465,488,530,557,579,654,684,701,714,765],"its":[54],"coordinates":[56,553],"and":[57,79,94,104,117,145,200,207,223,241,314,337,423,541,583,666,708,733,779],"invariant":[58],"under":[59],"double":[60],"sign":[61,80,542,667],"changes.":[62,81],"An":[63],"MK3":[64,95,130,527],"admits":[66],"group":[68,635],"automorphisms":[70],"G":[71,460],"generated":[72],"involutions,":[76],"coordinate":[77,539,664,751],"permutations,":[78],"In":[82,340,389,568],"this":[83,148,234,341,711,759],"paper":[84,342],"study":[86,101],"G-orbit":[88,137],"structure":[89,138],"points":[91,305,464,691],"on":[92,660],"TIK3":[93,509,599],"Over":[97],"finite":[98,156,180,421],"fields,":[99],"fibral":[102,493],"connectivity":[103],"existence":[106],"large":[108],"orbits,":[109,313],"analogous":[110],"to":[111,153,192,399,461,472,554,628,747],"work":[112,344],"Bourgain,":[114],"Gamburd,":[115],"Sarnak":[116,209],"others":[118],"for":[119,141,203,210,218,274,307,360,369,491,506,519,616,631,647,758,781],"equation.":[123],"For":[124,276,299,504,774],"particular":[126],"1-parameter":[127],"family":[128,162,253],"Wk,":[132],"compute":[134],"full":[136,450],"Wk(Fp)":[140],"all":[142],"primes":[143],"p\u2264113,":[144],"use":[147,445,742],"data":[149],"as":[150,640,652,669,671],"guide":[152],"find":[154,762],"many":[155,224],"G-orbits":[157],"Wk(C),":[159],"including":[160],"orbits":[164],"size":[166],"288":[167],"parameterized":[168],"curve":[171],"genus":[173],"9.KEYWORDS:":[174],"surfacearithmetic":[176],"dynamicsfinite":[177],"orbitsorbits":[178],"over":[179],"fields2010":[181],"AMS":[182],"SUBJECT":[183],"CLASSIFICATION:":[184],"Primary":[185],"37P55Secondary":[186],"14J2837F8037P2537P35":[187],"AcknowledgmentsThe":[188],"authors":[189],"would":[190],"like":[191],"thank":[193,215],"Philip":[194],"Boalch,":[195],"Wei":[196],"Ho,":[197],"Ram":[198],"Murty,":[199],"Igor":[201],"Shparlinski":[202],"their":[204,219,315,426],"helpful":[205,225],"advice":[206],"Peter":[208],"his":[211],"encouragement.":[212],"We":[213,244,319,416,479,544,622,644,718,741,767],"also":[214,536,678],"referees":[217],"very":[220],"careful":[221],"reading":[222],"suggestions":[226],"greatly":[228],"improved":[229],"paper.":[231],"Calculations":[232],"article":[235],"were":[236],"done":[237],"using":[238,459,551],"Magma":[239],"[Citation8]":[240],"GP-PARI":[242],"[Citation31].Notes1":[243],"remark":[245,623],"although":[247,547,607],"generic":[249,381,508],"member":[250],"2)-form":[264],"surface,":[268,510,600],"there":[269],"are":[270,297,366,386,420,428,434,605],"special":[271],"members":[272],"not.":[275],"example,":[277,505,775],"equation":[281],"defines":[282],"rational":[284],"surface.2":[285],"See":[286,613],"Definition":[287,586],"3.1,":[288],"but":[289,630],"briefly,":[290],"non-degeneracy":[291,354],"means":[292],"well-defined.3":[298],"Cayley":[301],"cubic":[302],"M1,4,":[303],"(2,t,t)":[306],"positive":[308],"integers":[309],"t":[310],"generate":[311],"distinct":[312],"union":[316],"M1,4(Z).4":[318],"recall":[320,545],"an":[322,391,404,734],"algebraic":[323,603],"smooth":[328],"projective":[329],"geometrically":[330],"connected":[331],"trivial":[334],"canonical":[335],"bundle":[336],"irregularity":[338],"zero.":[339],"directly":[345],"equations":[347],"form":[350,578,692],"(6)":[351],"satisfying":[352,414],"condition,":[355],"not":[358,444,481,610],"important":[359],"our":[361,364,384,558,632],"purposes":[362],"K3.":[367],"However,":[368],"completeness,":[370],"show":[372],"Section":[374,531],"B":[375],"minimal":[377],"regular":[378],"models":[379],"families":[385],"surfaces.5":[388],"general,":[390],"(a,":[392,405],"b,":[393,406],"c)-form":[394,407],"global":[397],"section":[398],"O(P1)3(a,b,c),":[400],"or":[401],"more":[402,638,770],"prosaically,":[403],"polynomial":[410,571],"f":[411],"K[X1,X2;Y1,Y2;Z1,Z2]":[413],"f(uX1,uX2;vY1,vY2;wZ1,Zw2)=uavbwcf(X1,X2;Y1,Y2;Z1,Z2).6":[415],"note":[417,645,719],"\u03c012,\u03c013,\u03c023":[419],"only":[424],"fibers":[427,466,659],"0-dimensional,":[429],"which":[431,447],"case":[432],"they":[433,608],"maps":[435],"degree":[437,575],"2.7":[438],"The":[439],"reason":[440],"do":[443,480],"{\u03c6\u2208G:\u03c6(F)=F},":[446],"subgroup":[451],"leaves":[453],"F":[454,550],"invariant,":[455],"because":[457],"when":[458],"move":[462],"around":[463],"W,":[468],"will":[470,498,535],"want":[471],"apply":[473],"one":[474,524],"generator":[475],"at":[476],"time.8":[478],"include":[482,537],"set":[484,706],"generators":[486,771],"Gen(G)":[487,534,651],"notation":[490,745],"automorphism":[494],"groups,":[495],"since":[496],"generally":[499],"clear":[501],"context.":[503],"take":[512,650],"Gen(G)={\u03c31,\u03c32,\u03c33}.":[513],"If":[514,676],"has":[516],"extra":[517],"symmetries,":[518],"example":[520,760],"described":[529,653,683],"6,":[532],"then":[533,601,687],"some":[538],"permutations":[540,665],"shifts.9":[543],"write":[549],"affine":[552],"ease":[555],"notation,":[556],"calculations":[559],"always":[561],"represents":[562],"2)":[566],"form.":[567],"particular,":[569],"F(x0,y0,z)":[572],"denotes":[573],"2":[576],"homogeneous":[577],"variables":[581],"Z1":[582],"Z2;":[584],"cf.":[585],"3.2.10":[587],"Lemma":[588],"5.6(a)":[589],"contains":[590],"proof":[592],"non-degenerate":[598],"these":[602],"sets":[604],"one-dimensional,":[606],"need":[609],"irreducible.11":[612],"Remark":[614,685],"5.7":[615],"examples":[617],"where":[618],"Cy0,z0(1)":[619],"reducible.12":[621],"(\u03bc23)1\u22caS3":[625],"isomorphic":[627],"S4,":[629],"applications":[633],"G\u00b0":[636],"appears":[637],"naturally":[639],"semi-direct":[642],"product.13":[643],"MK3-surfaces,":[648],"(16),":[655],"\u0393-connectivity":[657],"MK3-surfaces":[661],"may":[662],"employ":[663],"changes,":[668],"well":[670],"usual":[673],"\u03c3i":[674],"automorphisms.14":[675],"allow":[679],"\u03b4-inversion":[681],"8.7,":[686],"4":[689,732],"singular":[690],"single":[694],"orbit.15":[695],"Note":[696],"we're":[698],"really":[699],"working":[700],"P1,":[702],"formally":[705],"0\u22121=\u221e":[707],"\u221e\u22121=0.16":[709],"Indeed,":[710],"true":[713],"ring":[716],"Z[2\u22121,x,y,z,k].17":[717],"\u03b2=0":[721],"gives":[722],"contradiction":[724],"1=0,":[725],"while":[726],"\u03b2=1":[727],"yields":[728],"k":[729],"=":[730],"\u2013":[731],"orbit":[735],"fewer":[737],"than":[738,772],"64":[739],"elements.18":[740],"convenient":[744],"v[j]":[746],"denote":[748],"jth":[750],"vector":[754],"v.19":[755],"Somewhat":[756],"surprisingly,":[757],"G\u03c3\u00b7P1=G\u00b7P1=\u011c\u00b0G\u03c3\u00b7P1":[764],"W13(F71).20":[766],"have":[768],"listed":[769],"needed.":[773],"\u03c33=\u03c423\u00b0\u03c32\u00b0\u03c423,":[776],"Aut(Wx0(1))=\u3008\u03c32,\u03c423,\u03f523\u3009,":[778],"similarly":[780],"other":[783],"fibers.Additional":[784],"informationFundingSilverman's":[785],"research":[786],"supported":[787],"Simons":[789],"Collaboration":[790],"Grant":[791],"#712332.":[792]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}