On preperiodic points of rational functions defined over $\mathbb{F}_p(t)$

0.09 MB·

by  Jung Kyu Canci

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.
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JungKyuCanci,Universita¨tBasel,MathematischesInstitut,Rheinsprung21,CH-4051Basel,Switzerland
E-mailaddress:[e-mail]
LauraPaladino,Universita`diPisa,DipartimentodiMatematica,LargoBrunoPontecorvo5,56127Pisa,
Italy.
E-mailaddress:[e-mail]