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ON PREPERIODIC POINTS OF RATIONAL FUNCTIONS DEFINED OVER F (t) p 6 1 JUNGKYUCANCIANDLAURAPALADINO 0 2 n Abstract. LetP∈P1(Q)beaperiodicpointforamonicpolynomialwithcoefficientsin Z. Withelementary techniques oneseesthattheminimalperiodicity of Pisatmost2. a J Recentlyweprovedageneralizationofthisfacttothesetofallrationalfunctionsdefined overQwithgoodreductioneverywhere(i.e. atanyfiniteplaceofQ). Thesetofmonic 7 polynomialswithcoefficientsinZcanbecharacterized,uptoconjugationbyelementsin 2 PGL2(Z),asthesetofallrationalfunctionsdefinedoverQwithatotallyramifiedfixed ] pointinQandwithgoodreductioneverywhere. Let pbeaprimenumberandletFpbe T thefieldwithpelements.Inthepresentpaperweconsiderrationalfunctionsdefinedover N therationalglobalfunctionfieldFp(t)withgoodreductionateveryfiniteplace.Weprove someboundsforthecardinalityoforbitsinFp(t)∪{∞}forperiodicandpreperiodicpoints.. . h t a m Keywords.preperiodicpoints,functionfields. [ 1 1. Introduction v Inarithmeticdynamicthereisagreatinterestaboutperiodicandpreperiodicpointsof 3 9 a rationalfunction φ: P1 → P1. A point P is said to be periodic for φ if there exists an 2 integer n > 0 such that φn(P) = P. The minimal number n with the above properties 7 is called minimal or primitive period. We say that P is a preperiodic point for φ if its 0 (forward)orbit O (P) = {φn(P) | n ∈ N} contains a periodic point. In other words P is . φ 1 preperiodicifitsorbitO (P)isfinite. Inthiscontextanorbitisalsocalledacycleandits φ 0 sizeiscalledthelengthofthecycle. 6 Let p be a prime and, as usual, let F be the field with p elements. We denote by 1 p K a global field, i. e. a finite extension of the field of rational numbers Q or a finite : v extension of the field F (t). Let D be the degree of K over the base field (respectively p i X Q in characteristic 0 andF (t) in positivecharacteristic). We denotebyPrePer(φ,K)the p r setof K–rationalpreperiodicpointsforφ. By consideringthe notionof height, onesees a that the set PrePer(φ,K) is finite for any rational map φ: P → P defined over K (see 1 1 forexample[13]or[5]). ThefinitenessofthesetPrePer(f,K)followsfrom[5,Theorem B.2.5,p.179]and[5,TheoremB.2.3,p.177](eveniftheselasttheoremsareformulatedin thecaseofnumberfields,theyhaveasimilarstatementinthefunctionfieldcase).Anyway, thebounddeducedbythoseresultsdependsstrictly onthecoefficientsofthemapφ (see also [13, Exercise 3.26 p.99]). So, during the last twenty years, many dynamists have searched for bounds that do not depend on the coefficients of φ. In 1994 Morton and Silverman stated a conjecture known with the name ”Uniform Boundedness Conjecture for Dynamical Systems”: for any number field K, the number of K-preperiodic points 2010MathematicsSubjectClassification. 37P05,37P35. L.PaladinoispartiallysupportedbyIstitutoNazionalediAltaMatematica,grantresearchAssegnodiricerca Ing. G.Schirillo, and partially supported bythe European Commission and byCalabria Region through the EuropeanSocialFund. 1 2 JUNGKYUCANCIANDLAURAPALADINO of a morphism φ: P → P of degree d ≥ 2, defined over K, is bounded by a number N N dependingonlyontheintegersd,N andD =[K :Q]. Thisconjectureisreallyinteresting evenforpossibleapplicationontorsionpointsofabelianvarieties. Infact,byconsidering the Lattès map associated to the multiplication by two map [2] over an elliptic curve E, onesees that the UniformBoundednessConjeturefor N = 1 and d = 4 implies Merel’s Theoremon torsion pointsof elliptic curves(see [6]). The Lattès map has degree 4 and itspreperiodicpointsareinone-to-onecorrespondencewiththetorsionpointsof E/{±1} (see[11]).SoaproofoftheconjectureforeveryN,couldprovideananalogousofMerel’s Theoremforallabelianvarieties.Anyway,itseemsveryhardtosolvethisconjecture,even forN =1. LetR bethe ringof algebraicintegersof K. Roughlyspeaking: we saythatanendo- morphismφofP has(simple)goodreductionata placepifφcanbewrittenintheform 1 φ([x : y]) = [F(x,y),G(x,y)],where F(x,y)andG(x,y)arehomogeneouspolynomialof the same degree with coefficients in the local ring R at p and such that their resultant p Res(F,G)isap–unit.InSection3wepresentmorecarefullythenotionofgoodreduction. ThefirstauthorstudiedsomeproblemslinkedtoUniformBoundednessConjecture. In particular,whenN = 1,K isanumberfieldandφ :P →P isanendomorphismdefined 1 1 over K, he provedin [3, Theorem 1] that the lenght of a cycle of a preperiodicpoint of φ is boundedby a number dependingonly on the cardinality of the set of places of bad reductionofφ. Asimilarresultinthefunctionfieldcasewasrecentlyprovedin[4]. Furthermoreinthe samepaperthereisaboundprovedfornumberfields,thatisslightlybetterthantheonein [3]. Theorem1.1(Theorem1,[4]). LetKbeaglobalfield.LetS beafinitesetofplacesofK, containingallthearchimedeanones,withcardinality|S| ≥ 1. Let pbethecharacteristic of K. Let D = [K : F (t)] when p > 0, or D = [K : Q] when p = 0. Then there p exists a numberη(p,D,|S|),dependingonlyon p, D and|S|, such thatif P ∈ P (K) is a 1 preperiodicpointforanendomorphismφofP definedoverKwithgoodreductionoutside 1 S,then|O (P)|≤η(p,D,|S|).Wecanchoose φ η(0,D,|S|)=max (216|S|−8+3) 12|S|log(5|S|) D, 12(|S|+2)log(5|S|+5) 4D n (cid:2) (cid:3) (cid:2) (cid:3) o inzerocharacteristicand (1) η(p,D,|S|)=(p|S|)4Dmax (p|S|)2D,p4|S|−2 . (cid:8) (cid:9) inpositivecharacteristic. ObservethattheboundsinTheorem1.1donotdependonthedegreedofφ. Asaconse- quenceof that result, we could givethe followingboundforthe cardinalityof the set of K-rationalpreperiodicpointsforanendomorphismφofP definedoverK. 1 Corollary1.1.1(Corollary1.1,[4]). LetKbeaglobalfield.LetS beafinitesetofplaces ofKofcardinality|S|containingallthearchimedeanones.LetpbethecharacteristicofK. LetDbethedegreeofKovertherationalfunctionfieldF (t),inthepositivecharacteristic, p andoverQ,inthezerocharacteristic. Letd ≥2beaninteger. Thenthereexistsanumber C =C(p,D,d,|S|),dependingonlyon p,D,dand|S|,suchthatforanyendomorphismφ ofP ofdegreed,definedoverK andwithgoodreductionoutsideS,wehave 1 #PrePer(φ,P (K))≤C(p,D,d,|S|). 1 PREPERIODICPOINTSOVERFp(t) 3 Theorem 1.1 extends to global fields and to preperiodic points the result proved by MortonandSilvermanin[7, CorollaryB]. Thecondition|S| ≥ 1 in itsstatementis only atechnicalone. Inthecaseofnumberfields,werequirethatS containsthearchimedean places (i.e. the ones at infinity), then it is clear that the cardinality of S is not zero. In the function field case any place is non archimedean. Recall that the place at infinity in thecase K = F (t) istheoneassociatedtothevaluationgivenbytheprimeelement1/t. p When K is a finite extensionof F (t), the placesatinfinityof K arethe onesthatextend p theplaceofF (t)associatedto1/t. TheargumentsusedintheproofofTheorem1.1and p Corollary1.1.1workalsowhenS doesnotcontainalltheplacesatinfinity. Anyway,the mostimportantsituationiswhenalltheonesatinfinityareinS. Forexample,inorderto havethatanypolynomialinF (t)isanS–integer,wehavetoputinS allthoseplaces.Note p thatinthenumberfieldcasethequantity|S|dependsalsoonthedegreeDoftheextension KofQ,becauseS containsallarchimedeanplaces(whoseamountdependsonD). EvenwhenthecardinalityofS issmall, theboundsinTheorem1.1isquitebig. This isaconsequenceofoursearchingforsomeuniformbounds(dependingonlyon p,D,|S|). TheboundC(p,D,d,|S|)inCorollary1.1.1canbeeffectivelygiven,butinthiscasetoothe boundisbig,evenforsmallvaluesoftheparametersp,D,d,|S|.Foramuchsmallerbound seeforinstancetheoneprovedbyBenedettoin[1]forthecasewhereφisapolynomial. Inthemoregeneralcasewhenφisarationalfunctionwithgoodreductionoutsideafinite S,theboundinTheorem1.1canbesignificantlyimprovedforsomeparticularsetsS. For example if K = Q and S contains only the place at infinity, then we have the following bounds(see[4]): • IfP∈P (Q)isaperiodicpointforφwithminimalperiodn,thenn≤3. 1 • IfP∈P (Q)isapreperiodicpointforφ,then|O (P)|≤12. 1 φ HereweprovesomeanalogousboundswhenK =F (t). p Theorem1.2. Letφ: P → P ofdegreed ≥ 2definedoverF (t)withgoodreductionat 1 1 p everyfiniteplace.IfP∈P (F (t))isaperiodicpointforφwithminimalperiodn,then 1 p • n≤3 if p=2; • n≤72 if p=3 • n≤(p2−1)p if p≥5. MoregenerallyifP∈P (F (t))isapreperiodicpointforφwehave 1 p • |O (P)|≤9 if p=2; φ • |O (P)|≤288 if p=3; φ • |O (P)|≤(p+1)(p2−1)p if p≥5. φ Observe that the bounds do not depend on the degree of φ but they depend only on the characteristic p. In the proof we will use some ideas already written in [2], [3] and [4]. Theoriginalidea ofusingS–unittheoremsin the contextof thearithmeticof dynamical systemsisduetoNarkiewicz[9]. 2. Valuations,S-integersandS-units Weadoptthepresentnotation:letKbeaglobalfieldandv avaluationonKassociated p toanonarchimedeanplacep. LetR ={x∈ K |v (x)≥1}bethelocalringofKatp. p p Recall that we can associate an absolute value to any valuation v , or more precisely p a place p that is a class of absolute values (see [5] and [12] for a reference about this topic). With K = F (t), all places are exactly the onesassociated either to a monic irre- p duciblepolynomialinF [t]ortotheplaceatinfinitygivenbythevaluationv (f(x)/g(x)= p ∞ deg(g(x))−deg(f(x)),thatisthevaluationassociatedto1/t. 4 JUNGKYUCANCIANDLAURAPALADINO InanarbitraryfiniteextensionKofF (t)thevaluationsofKaretheonesthatextendthe p valuationsofF (t). Weshallcallplacesatinfinitytheonesthatextendtheabovevaluation p v on F (t). The other ones will be called finite places. The situation is similar to the ∞ p one in number fields. The non archimedean places in Q are the ones associated to the valuationsatanyprime pofZ. Butthereisalsoaplacethatisnotnon–archimedean,the oneassociatedtotheusualabsolutevalueonQ. WithanarbitrarynumberfieldK wecall archimedeanplacesalltheonesthatextendto K theplacegivenbytheabsolutevalueon Q. FromnowonS willbeafinitefixedsetofplacesofK. Weshalldenoteby R ≔{x∈K |v (x)≥0foreveryprime p<S} S p theringofS-integersandby R∗ ≔{x∈ K∗ |v (x)=0foreveryprime p<S} S p thegroupofS-units. AsusualletF bethealgebraicclosureofF . ThecasewhenS = ∅istrivialbecause p p if so, then the ring of S–integersis already finite; more precisely R = R∗ = K∗ ∩F . S S p ThereforeinwhatfollowsweconsiderS ,∅. InanycasewehavethatK∗∩F iscontainedinR∗. RecallthatthegroupR∗/K∗∩F p S S p hasfiniterankequalto|S|−1(e.g. see[10,Proposition14.2p.243]). Thus,sinceK∩F p isafinitefield,wehavethatR∗ hasrankequalto|S|. S 3. Goodreduction Weshallstatethenotionofgoodreductionfollowingthepresentationgivenin[11]and in[4]. Definition3.0.1. LetΦ:P →P bearationalmapdefinedoverK,oftheform 1 1 Φ([X :Y])=[F(X,Y):G(X,Y)] whereF,G ∈ K[X,Y]arecoprimehomogeneouspolynomialsofthesamedegree. Wesay thatΦisinp–reducedformifthecoefficientsof F andG areinR [X,Y]andatleastone p ofthemisap-unit(i.e. aunitinR ). p Recall that R is a principal local ring. Hence, up to multiplying the polynomials F p andGbyasuitablenon-zeroelementofK,wecanalwaysfindap–reducedformforeach rationalmap.Wemaynowgivethefollowingdefinition. Definition 3.0.2. Let Φ : P → P be a rational map defined over K. Suppose that the 1 1 morphism Φ([X : Y]) = [F(X,Y) : G(X,Y)] is written in p-reduced form. The reduced map Φ : P → P is defined by [F (X,Y) : G (X,Y)], where F andG are the p 1,k(p) 1,k(p) p p p p polynomialsobtainedfromF andGbyreducingtheircoefficientsmodulop. Withtheabovedefinitionswegivethefollowingone: Definition3.0.3. ArationalmapΦ: P → P ,definedoverK,hasgoodreductionatpif 1 1 degΦ = degΦ . Otherwisewesaythatithasbadreductionatp. GivenasetS ofplaces p ofKcontainingallthearchimedeanones.WesaythatΦhasgoodreductionoutsideS ifit hasgoodreductionatanyplacep<S. Notethattheabovedefinitionofgoodreductionisequivalentto askthatthehomoge- neousresultantofthepolynomialF andGisinvertibleinR ,whereweareassumingthat p Φ([X :Y])=[F(X,Y):G(X,Y)]iswritteninp-reducedform. PREPERIODICPOINTSOVERFp(t) 5 4. Divisibilityarguments We define the p-adic logarithmic distance as follows (see also [8]). The definition is independentofthechoiceofthehomogeneouscoordinates. Definition4.0.4. Let P = x :y ,P = x :y betwo distinctpointsinP (K). We 1 1 1 2 2 2 1 (cid:2) (cid:3) (cid:2) (cid:3) denoteby (2) δ (P ,P )=v (x y −x y )−min{v (x ),v (y )}−min{v (x ),v (y )} p 1 2 p 1 2 2 1 p 1 p 1 p 2 p 2 thep-adiclogarithmicdistance. The divisibility arguments,thatwe shalluse to producethe S–unitequationusefulto proveourbounds,areobtainedstartingfromthefollowingtwofacts: Proposition4.0.1. [8,Proposition5.1] δ (P ,P )≥min{δ (P ,P ),δ (P ,P )} p 1 3 p 1 2 p 2 3 forallP ,P ,P ∈P (K). 1 2 3 1 Proposition 4.0.2. [8, Proposition 5.2] Let φ: P → P be a morphism defined over K 1 1 withgoodreductionataplacep. ThenforanyP,Q∈P(K)wehave δ (φ(P),φ(Q))≥δ (P,Q). p p Asadirectapplicationofthepreviouspropositionswehavethefollowingone. Proposition4.0.3. [8,Proposition6.1]Letφ: P →P beamorphismdefinedoverKwith 1 1 goodreductionataplacep. LetP∈P(K)beaperiodicpointforφwithminimalperiodn. Then • δ (φi(P),φj(P))=δ (φi+k(P),φj+k(P)) foreveryi, j,k ∈Z. p p • Leti, j∈Nsuchthatgcd(i− j,n)=1. Thenδ (φi(P),φj(P))=δ (φ(P),P). p p 5. ProofofTheorem1.2 Wefirstrecallthefollowingstatements. Theorem5.1(MortonandSilverman[8], Zieve[14]). Let K,p,pbeasabove. LetΦbe anendomorphismofP ofdegreeatleasttwodefinedoverKwithgoodreductionatp. Let 1 P ∈ P (K)beaperiodicpointforΦwithminimalperiodn. Letmbetheprimitiveperiod 1 ofthereductionofPmodulopandr themultiplicativeperiodof(Φm)′(P)ink(p)∗. Then oneofthefollowingthreeconditionsholds (i) n=m; (ii) n=mr; (iii) n= pemr,forsomee≥1. InthenotationofTheorem5.1,if(Φm)′(P) = 0modulop,thenwesetr = ∞. Thus,if Pisaperiodicpoint,thenthecases(ii)and(iii)arenotpossiblewithr=∞. Proposition 5.1.1. [8, Proposition 5.2] Let φ: P → P be a morphism defined over K 1 1 withgoodreductionataplacep. ThenforanyP,Q∈P(K)wehave δ (φ(P),φ(Q))≥δ (P,Q). p p Lemma5.1.1. Let (3) P= P 7→P 7→...7→P 7→P =[0:1]7→[0:1]. −m+1 −m+2 −1 0 beanorbitforanendomorphismφdefinedoverKwithgoodreductionoutsideS. Forany a,bintegerssuchthat0<a<b≤m−1andp<S,itholds 6 JUNGKYUCANCIANDLAURAPALADINO a) δ (P ,P )≤δ (P ,P ); p −b 0 p −a 0 b) δ (P ,P )=δ (P ,P ). p −b −a p −b 0 Proofa)ItfollowsdirectlyfromProposition5.1.1. b)ByProposition4.0.1andparta)wehave δ (P ,P )≥min{δ (P ,P ),δ (P ,P )}=δ (P ,P ). p −b −a p −b 0 p −a 0 p −b 0 Letrbethelargestpositiveintegersuchthat−b+r(b−a)<0. Then δ (P ,P )≥min{δ (P ,P ),δ (P ,P ),...,δ (P ,P )} p −b 0 p −b −a p −a b−2a p −b+r(b−a) 0 =δ (P ,P ). p −b −a TheinequalityisobtainedbyapplyingProposition4.0.1severaltimes. (cid:3) Lemma5.1.2(Lemma3.2[4]). LetK beafunctionfieldofdegreeDoverF (t)andS a p nonemptyfinitesetofplacesofK. LetP ∈P (K)withi∈{0,...n−1}bendistinctpoints i 1 suchthat (4) δ (P ,P )=δ (P,P ), foreachdistinct0≤i, j≤n−1andforeachp<S. p 0 1 p i j Thenn≤(p|S|)2D. Since F (t) is a principal ideal domain, every point in P (F (t)) can be written in S– p 1 p coprimecoordinates,i. e.,foreachP∈ P (F (t))wemaywriteP =[a:b]witha,b∈R 1 p S andmin{v (a),v (b)} = 0,foreachp < S. Wesaythat[a : b]areS–coprimecoordinates p p forP. ProofofTheorem1.2WeusethesamenotationofTheorem5.1. AssumethatS contains onlytheplaceatinfinity.Case p=2. LetP∈P (F (t))beaperiodicpointforφ. Without 1 p lossof generalitywe can supposethat P = [0 : 1]. Observethatm is boundedby 3 and r = 1. ByTheorem5.1,wehaven = m·2e,witheanonnegativeintegralnumber. Upto consideringthem–thiterateofφ,wemayassumethattheminimalperiodicityofPis2e. Sonowsupposethatn=2e,withe≥2. Considerthefollowing4pointsofthecycle: [0:1]7→[x :y ]7→[x :y ]7→[x :y ]... 1 1 2 2 3 3 wherethepoints[x :y]arewrittenS–coprimeintegralcoordinatesforalli∈{1,...,n−1}. i i By applying Proposition 4.0.3, we have δ ([0 : 1],P ) = δ ([0 : 1],P ), i. e. x = x , p 1 p 3 3 1 becauseofR∗ ={1}. Fromδ ([0:1],P )=δ (P ,P )wededuce S p 1 p 1 2 x (5) y = 2y +1. 2 1 x 1 Furthermore, by Proposition 4.0.3 we have δ ([0 : 1],P ) = δ (P ,P ). Since x = x , p 1 p 2 3 3 1 then (6) y x −x y = x . 3 2 3 2 1 This last equality combined with (5) provides y = y , implying [x : y ] = [x : y ]. 3 1 1 1 3 3 Thuse ≤ 1 and n ∈ {1,2,3,6}. The nextstep is to provethatn , 6. If n = 6, with few calculationsoneseesthatthecyclehasthefollowingform. (7) [0:1]7→[x :y ]7→[A x :y ]7→[A x :y ]7→[A x :y ]7→[x :y ]7→[0:1], 1 1 2 1 2 3 1 3 2 1 4 1 5 PREPERIODICPOINTSOVERFp(t) 7 where A ,A ∈ R and everythingis writtenin S-coprimeintegralcoordinates. We may 2 3 S applyProposition4.0.3,thenbyconsideringthep–adicdistancesδ (P ,P)forallindexes p 1 i 2≤i≤5foreveryplacep,weobtainthatthereexistssomeS–unitsu suchthat i (8) y = A y +u ; y = A y +A u ; y = A y +A u ; y =y +A u . 2 2 1 2 3 3 1 2 3 4 2 1 3 4 5 1 2 5 SinceR∗ ={1},wehavethattheidentitiesin(8)become S y = A y +1; y = A y +A ; y = A y +A ; y =y +A 2 2 1 3 3 1 2 4 2 1 3 5 1 2 whereA ,A arenonzeroelementsinF [t]. Byconsideringthep–adicdistanceδ (P ,P ) 2 3 p p 2 4 foreachfiniteplacep,fromProposition4.0.3weobtainthat v (A x )=δ (P ,P )=v (A x (A y +A )−A x (A y +1))=v (A A x −A x ), p 2 1 p 2 4 p 2 1 2 1 3 2 1 2 1 p 2 3 1 2 1 i. e. A x = A A x −A x (becauseR∗ = {1}). ThenA A x = 0thatcontradictsn = 6. 2 1 2 3 1 2 1 S 2 3 1 Thusn≤3. SupposenowthatPisapreperiodicpoint. Withoutlossofgeneralitieswecanassume thattheorbitofPhasthefollowingshape: (9) P= P 7→P 7→...7→P 7→P =[0:1]7→[0:1]. −m+1 −m+2 −1 0 Indeeditissufficienttotakeinconsiderationasuitableiterateφn (withn ≥ 3),sothatthe orbitof the point P, with respect the iterate φn, containsa fixed point P . By a suitable 0 conjugationbyanelementofPGL (R ),wemayassumethatP =[0:1]. 2 S 0 Forall1 ≤ j ≤ m−1,letP = [x :y ]bewritteninS–coprimeintegralcoordinates. −j j j ByLemma5.1.1,forevery1 ≤ i < j ≤ m−1thereexistsT ∈ R suchthat x = T x . i,j S i i,j j Considerthep–adicdistancebetweenthepointsP andP . AgainbyLemma5.1.1,we −1 −j have δ (P ,P )=v (x y −x y /T )=v (x /T ), p −1 −j p 1 j 1 1 1,j p 1 1,j for all p < S. Then, there exists u ∈ R∗ such that y = y +u /T , for all p < S. j S j (cid:0) 1 j(cid:1) 1,j Thus, there exists u ∈ R∗ such that [x ,y ] = [x ,y + u ]. Since R∗ = {1}, then j S −j −j 1 1 j S P = [x : y +1]. Sothelengthoftheorbit(9)isatmost3. Wegetthebound9forthe −j 1 1 cardinalityoftheorbitofP. Case p>2. Since D = 1 and |S| = 1, then the bound for the number of consecutive points as in Lemma5.1.2canbechosenequalto p2. ByTheorem5.1theminimalperiodicitynfora periodicpointP∈P (Q)forthemapφisoftheformn=mrpewherem≤ p+1,r ≤ p−1 1 andeisanonnegativeinteger. Letusassumethate≥2.Sincep>2,byProposition4.0.3,foreveryk ∈{0,1,2,...,pe−2} and i ∈ {2,...,p − 1}, we have that δ (P ,P ) = δ (P ,P ), for any p < S. Then p 0 1 p 0 k·p+i P = [x ,y ]. Furthermoreδ (P ,P ) = δ (P ,P ) implying that there exists a k·p+i 1 k·p+i p 0 1 p 0 k·p+i elementu ∈R∗ suchthat k·p+i S (10) P =[x :y +u ]. k·p+i 1 1 k·p+i SinceR∗ hasp−1elementsandthereare(pe−2+1)(p−2)numbersoftheshapek·p+i S asabove,wehave(pe−2+1)(p−2)≤ p−1. Thuse=2and p=3. Then n ≤ 72 if p = 3 and n ≤ (p2 −1)p if p ≥ 5. For the more generalcase when P is preperiodic, consider the same arguments used in the case when p = 2, showing [x ,y ] = [x ,y +u ], with u ∈ R∗. Thus, the orbitof a point P ∈ P (Q) containing −j −j 1 1 j j S 1 P ∈ P (Q),asin(9), haslengthatmost|R∗|+2 = p+1. Theboundinthepreperiodic 0 1 S caseisthen288for p=3and(p+1)(p2−1)pfor p≥5. ✷ 8 JUNGKYUCANCIANDLAURAPALADINO Withsimilarproofs,wecangetanalogousboundsforeveryfiniteextensionK ofF (t). p Theboundsof Theorem1.2, with K = F (t), areespeciallyinteresting,fortheyarevery p small. References [1] R.Benedetto,Preperiodicpointsofpolynomialsoverglobalfields,J.ReineAngew.Math.608(2007), 123-153. [2] J.K.Canci,Cyclesforrationalmapsofgoodreductionoutsideaprescribedset,Monatsh.Math.,149 (2006),265-287. [3] J.K.Canci,Finiteorbitsforrationalfunctions,Indag.Math.N.S.,18(2)(2007),203-214. [4] J.K.Canci,L.Paladino,Preperiodicpointsforrationalfunctionsdefinedoveraglobalfieldinterms ofgoodreduction,http://arxiv.org/pdf/1403.2293.pdf. [5] M.HindryandJ.H.Silverman,DiophantineGeometryAnIntroduction,Springer-Verlag,GTM201, 2000. 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JungKyuCanci,Universita¨tBasel,MathematischesInstitut,Rheinsprung21,CH-4051Basel,Switzerland E-mailaddress:[email protected] LauraPaladino,Universita`diPisa,DipartimentodiMatematica,LargoBrunoPontecorvo5,56127Pisa, Italy. E-mailaddress:[email protected]