A quantitative estimate for quasi-integral points in orbits
aa r X i v : . [ m a t h . N T ] O c t A QUANTITATIVE ESTIMATE FOR QUASI-INTEGRALPOINTS IN ORBITS
LIANG-CHUNG HSIA AND JOSEPH H. SILVERMAN
Abstract.
Let ϕ ( z ) ∈ K ( z ) be a rational function of degree d ≥ ϕ ( z ) is not apolynomial, and let α ∈ K . The second author previously provedthat the forward orbit O ϕ ( α ) contains only finitely many quasi- S -integral points. In this note we give an explicit upper bound forthe number of such points. Introduction
Let K/ Q be a number field, let S be a finite set of places of K , andlet 1 ≥ ε >
0. An element x ∈ K is said to be quasi- ( S, ε ) -integral if X v ∈ S [ K v : Q v ][ K : Q ] log + | x | v ≥ εh ( x ) . (1)We observe that x is in the ring of S -integers of K if and only if it isquasi-( S, ϕ ( z ) ∈ K ( z ) be a rational function of degree d ≥
2, let α ∈ K be a point, and let O ϕ ( α ) = (cid:8) α, ϕ ( α ) , ϕ ( α ) , . . . (cid:9) denote the forward orbit of α under iteration of ϕ . The second authorproved in [9] that if ϕ ( z ) is not a polynomial, then the orbit O ϕ ( α )contains only finitely many quasi-( S, ε )-integral points. More generally,if O ϕ ( α ) = ∞ and if β is not an exceptional point for ϕ , then there Date : November 7, 2018.1991
Mathematics Subject Classification.
Primary: 37P15 Secondary: 11B37,11G99, 14G99.
Key words and phrases.
Arithmetic dynamics, integral points.The fist author’s research is supported by NSC-97-2918-I-008-005 and NSC-96-2115-008-012-MY3. The second author’s research is supported by NSF DMS-0650017 and DMS-0854755. are only finitely many n ≥ ϕ n ( α ) − β is quasi-( S, ε )-integral. In this note we give an upper bound for thenumber of such n , making explicit the dependence on S , ϕ , α , and β .More precisely, we prove that the number of elements in the set (cid:8) n ≥ (cid:0) ϕ n ( α ) − β (cid:1) − is quasi-( S, ε )-integral (cid:9) (2)is smaller than 4 S γ + log + d h ( ϕ ) + ˆ h ϕ ( β )ˆ h ϕ ( α ) ! , (3)where γ depends only on d , ε , and [ K : Q ]. (See Section 2 for thedefinitions of the height h ( ϕ ) of the map ϕ and the canonical height ˆ h ϕ .)Our main result, Theorem 11 in Section 5, is a strengthened version ofthis statement.The specific form of the upper bound in (3) is interesting, especiallythe dependence on the wandering point α and the target point β . Forexample, if ˆ h ϕ ( α ) is sufficiently large (depending on β and ϕ ), thenthe bound is independent of α , β , and ϕ . It is also interesting to askwhether it is possible, for a given ϕ and α , to make the set (2) arbitrarilylarge by varying β . We discuss this question further in Remark 14.We briefly describe the organization of the paper. We start in Sec-tion 1 by setting notation and proving an elementary estimate for thechordal metric. Section 2 is devoted to height functions, both thecanonical height associated to a rational map and various results relat-ing heights and polynomials. In Section 3 we prove a uniform versionof the inverse function theorem for rational maps of degree d . Section 4states an estimate for the ramification of the iterate of a rational func-tion, taken from [9, 10], and a quantititative version of Roth’s theorem,taken from [8]. In Section 5 we combine the preliminary material toprove our main theorem. Finally, in Section 6, we use the main theoremto give an explicit upper bound for the number of S -integral points inan orbit. Remark . The original paper on finiteness of quasi- S -integral pointsin orbits [9] has been used by Patrick Ingram and the second author [5]to prove a dynamical version of the classical Bang–Zsigmondy theoremon primitive divisors [1, 13]. It has also been used by Felipe Voloch andthe second author [12] to prove a local–global criterion for dynamicson P . The quantitative results proven in the present paper shouldenable one to prove quantitative versions of both [5] and [12], but we NTEGRAL POINTS IN ORBITS 3 have not included these applications in this paper in order to keep itto a manageable length.
Remark . Quantitative estimates similar to those in this paper havebeen proven for the number of integral points on elliptic curves and oncertain other types of curves. See for example [3] and [8].
Acknowledgements.
The first author would like to thank his coauthorand the Department of Mathematics at Brown University for their hos-pitality during his visit when this work was initiated. The second au-thor would like to thank Microsoft Research New England for invitinghim to be a visiting researcher.1.
Preliminary Material and Notation
We set the following notation: K a number field; M K the set of places of K ; M ∞ K the set of archimedean (infinite) places of K ; M K the set of nonarchimedean (finite) places of K ;log + ( x ) the maximum of log( x ) and 0. We write log + d for log base d .For each v ∈ M K , we let | · | v denote the corresponding normalizedabsolute value on K whose restriction to Q gives the usual v -adic abso-lute value on Q . That is, if v ∈ M ∞ K , then | x | v is the usual archimedeanabsolute value, and if v ∈ M K , then | x | v = | x | p is the usual p -adic ab-solute value for a unique prime p . We also write K v for the completionof K with respect to | · | v , and we let C v denote the completion of analgebraic closure of K v .For each v ∈ M K , we let ρ v denote the chordal metric defined on P ( C v ), where we recall that for [ x , y ] , [ x , y ] ∈ P ( C v ), ρ v (cid:0) [ x , y ] , [ x , y ] (cid:1) = | x y − x y | v p | x | v + | y | v p | x | v + | y | v if v ∈ M ∞ K , | x y − x y | v max {| x | v , | y | v } max {| x | v , | y | v } if v ∈ M K .In this paper, we use the logarithmic version of the chordal metricto measure the distance between points in P ( C v ) . Definition.
The logarithmic chordal metric function λ v : P ( C v ) × P ( C v ) → R ∪ {∞} is defined by λ v (cid:0) [ x , y ] , [ x , y ] (cid:1) = − log ρ v (cid:0) [ x , y ] , [ x , y ] (cid:1) . LIANG-CHUNG HSIA AND JOSEPH H. SILVERMAN
Notice that λ v ( P, Q ) ≥ P, Q ∈ P ( C v ), and that two points P, Q ∈ P ( C v ) are close if and only if λ v ( P, Q ) is large. We also observethat λ v is a particular choice of an arithmetic distance function asdefined in [7, § λ P × P , ∆ , where ∆ isthe diagonal of P × P .The next lemma relates the logarithmic chordal metric λ v ( x, y ) tothe usual metric | x − y | v arising from the absolute value v . Lemma 3.
Let v ∈ M K and let λ v be the logarithmic chordal metricon P ( C v ) . Define ℓ v = 2 if v is archimedean, and ℓ v = 1 if v is non-archimedean. Then for x, y ∈ C v we have λ v ( x, y ) > λ v ( y, ∞ ) + log ℓ v = ⇒ λ v ( y, ∞ ) ≤ λ v ( x, y ) + log | x − y | v ≤ λ v ( y, ∞ ) + log ℓ v . Proof.
Notice that by the definition of chordal metric, λ v ( x, y ) = λ v ( x, ∞ ) + λ v ( y, ∞ ) − log | x − y | v . Therefore, λ v ( x, y ) + log | x − y | v = λ v ( x, ∞ ) + λ v ( y, ∞ ) ≥ λ v ( y, ∞ ) . This gives the lower bound for the sum λ v ( x, y ) + log | x − y | v . For the upper bound, if v is an archimedean place, then the assertionis the same as [10, Lemma 3.53]. We will not repeat the proof here.For the case where v is non-archimedean, notice that λ v satisfies thestrong triangle inequality, λ v ( x, y ) ≥ min ( λ v ( x, z ) , λ v ( y, z )) , and that this inequality is an equality if λ v ( x, z ) = λ v ( y, z ). Supposethat x and y satisfy the required condition in the statement of thelemma, i.e., λ v ( x, y ) > λ v ( y, ∞ ). (Notice that ℓ v = 1 in this case.)We claim that λ v ( x, ∞ ) ≤ λ v ( y, ∞ ). Assume to the contrary that λ v ( x, ∞ ) > λ v ( y, ∞ ). Then, by the strong triangle inequality for λ v wehave λ v ( x, y ) = min ( λ v ( x, ∞ ) , λ v ( y, ∞ )) = λ v ( y, ∞ ) . But this contradicts the assumption that λ v ( x, y ) > λ v ( y, ∞ ) hence theclaim. Now, λ v ( x, y ) + log | x − y | v = λ v ( x, ∞ ) + λ v ( y, ∞ ) ≤ λ v ( y, ∞ ) by the claim,which completes the proof of the lemma. (cid:3) NTEGRAL POINTS IN ORBITS 5 Dynamics and height functions
Let ϕ : P → P be a rational map on P of degree d ≥ K . We identify K ∪ {∞} ≃ P ( K ) by fixing an affinecoordinate z on P , so α ∈ K equals [ α, ∈ P ( K ), and the pointat infinity is [1 , ϕ : P → P with rational functions ϕ ( z ) ∈ K ( z ).Let P ∈ P . Then, the ( forward ) orbit of P under iteration of ϕ isthe set O ϕ ( P ) = (cid:8) ϕ n ( P ) : n = 0 , , , . . . (cid:9) . The point P is called a wandering point of ϕ if O ϕ ( P ) is an infinite set;otherwise, P is called a preperiodic point of ϕ . The set of preperiodicpoints of ϕ is denoted by PrePer( ϕ ). We say that a point A ∈ P is an exceptional point if it is preperiodic and ϕ − (cid:0) O ϕ ( A ) (cid:1) = O ϕ ( A ),which is equivalent to the assumption that the complete (forward andbackward) ϕ -orbit of A is a finite set. It is a standard fact that A is anexceptional point for ϕ if and only if A a totally ramified fixed pointof ϕ . (One direction is clear, and the other follows from that fact [10,Theorem 1.6] that if A is an exceptional point, then O ϕ ( A ) consists ofat most two points.)For a point P = [ x , x ] ∈ P ( K ), the height of P is h ( P ) = X v ∈ M K [ K v : Q v ][ K : Q ] log max (cid:0) | x | v , | x | v (cid:1) . Then the canonical height of P relative to the rational map ϕ is givenby the limit ˆ h ϕ ( P ) = lim n →∞ h ( ϕ n P ) d n . To simplify notation, we let d v = [ K v : Q v ][ K : Q ] . Using the definition of λ v , we see that h ( P ) = X v ∈ M K d v λ v ( P, ∞ ) + O (1) . More precisely, writing P = [ x , x ] and ∞ = [1 , h ( P ) = X v ∈ M K d v λ v ( P, ∞ ) + X v ∈ M ∞ K d v log max (cid:8) | x | v , | x | v (cid:9)p | x | v + | x | v ! . LIANG-CHUNG HSIA AND JOSEPH H. SILVERMAN
The quantity max { a, b } / √ a + b is between 1 / √ a, b ∈ R , so −
12 log 2 ≤ h ( P ) − X v ∈ M K d v λ v ( P, ∞ ) ≤ . For further material and basic properties of height functions, see forexample [10, §§ f = P a i z i ∈ K [ z ] and absolute value v ∈ M K , wedefine | f | v = max (cid:8) | a i | v (cid:9) and h ( f ) = h (cid:0) [ . . . , a i , . . . ] (cid:1) = X v ∈ M K d v log | f | v . We say that a rational function ϕ ( z ) = f ( z ) /g ( z ) ∈ K ( z ) of degree d is written in normalized form if f ( z ) = d X i =0 a i z i and g ( z ) = d X i =0 b i z i with a i , b i ∈ K, if a d and b d are not both zero, and if f and g are relatively primein K [ z ]. For v ∈ M K , we set | ϕ | v = max {| f | v , | g | v } , and then theheight of ϕ is defined by h ( ϕ ) = h (cid:0) [ a , . . . , a d , b , . . . , b d ] (cid:1) = X v ∈ M K d v log | ϕ | v . Directly from the definitions, we havemax (cid:0) h ( f ) , h ( g ) (cid:1) ≤ h ( ϕ ) . (4)The following basic properties of absolute values of polynomials willbe useful. Lemma 4.
Let v ∈ M K and let f, g ∈ K [ x ] be polynomials with coef-ficients in K . (a) | f + g | v ≤ ( | f | v + | g | v if v is archimedean, max {| f | v , | g | v } if v is nonarchimedean. (b) (Gauss’ Lemma) If v is nonarchimedean, then | f g | v = | f | v | g | v . (c) If v is archimedean and deg f + deg g < d , then d | f g | v ≤ | f | v | g | v ≤ d | f g | v Proof. (a) follows from the definition. For (b) and (c), see for example[6, Chapter 3, Propositions 2.1 and 2.3]. (cid:3)
NTEGRAL POINTS IN ORBITS 7
Proposition 5.
Let { f , . . . , f r } be a collection of polynomials in thering K [ x ] . (a) h ( f f · · · f r ) ≤ r X i =1 (cid:0) h ( f i ) + (deg f i + 1) log 2 (cid:1) ≤ r max ≤ i ≤ r (cid:8) h ( f i ) + (deg f i + 1) log 2 (cid:9) . (b) h ( f + f + · · · + f r ) ≤ r X i =1 h ( f i ) + log r. (c) Let ϕ ( z ) , ψ ( z ) ∈ K ( z ) be rational functions. Then h ( ϕ ◦ ψ ) ≤ h ( ϕ ) + (deg ϕ ) h ( ψ ) + (deg ϕ )(deg ψ ) log 8 . (d) Let ϕ ( z ) ∈ K ( z ) be a rational function of degree d ≥ .Then forall n ≥ we have h ( ϕ n ) ≤ (cid:18) d n − d − (cid:19) h ( ϕ ) + d (cid:18) d n − − d − (cid:19) log 8 . Proof.
The proofs of (a) and (b) can be found in [4, Proposition B.7.2],where the proposition is stated for multi-variable polynomials. As we’lluse the arguments in (a) for the proof of (c), we repeat the proofof (a) for the one-variable case. (Also, our situation is slightly differentfrom [4], since we are using a projective height, while [4] uses an affineheight.) Writing f i = P E a iE X E , we have f · · · f r = X E X e + ··· + e r = E a e · · · a re r ! X E , and hence for v ∈ M K , | f · · · f r | v = max E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X e + ··· + e r = E a e · · · a re r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v (5)and h ( f · · · f r ) = X v ∈ M K d v log | f · · · f r | v . If v is nonarchimedean, then by Gauss’ Lemma (Lemma 4(b)) we have | f · · · f r | v = r Y i =1 | f i | v . It remains to deal with archimedean place v . We note that thenumber of terms in the sum appearing in the right-hand side of (5) is LIANG-CHUNG HSIA AND JOSEPH H. SILVERMAN (cid:0) E + r − E (cid:1) . Hence | f · · · f r | v ≤ max E (cid:18)(cid:18) E + r − E (cid:19) max e + ··· + e r = E | a e · · · a re r | v (cid:19) ≤ max E (cid:18) E + r − max e + ··· + e r = E | a e · · · a re r | v (cid:19) . Further, if
E > deg( f . . . f r ), then the product a e · · · a re r is zero ,since in that case at least one of the a ij is zero. Hence | f · · · f r | v ≤ deg( f ··· f r )+ r − r Y i =1 | f i | v . (6)Let N v = 2 P i (deg f i +1) if v is archimedean, and N v = 1 if v is non-archimedean. Then we compute h ( f · · · f r ) = X v ∈ M K d v log | f · · · f r | v ≤ X v ∈ M K d v log N v + log r Y i =1 | f i | v ! ≤ r X i =1 ( h ( f i ) + (deg f i + 1) log 2) ≤ r max ≤ i ≤ r { h ( f i ) + (deg f i + 1) log 2 } , which completes the proof of (a).Next we give a proof of (c). Write ψ = ψ /ψ ∈ K ( z ) in normalizedform, so in particular ψ and ψ are relatively prime polynomials. Then( ϕ ◦ ψ )( z ) = P a i ψ i ψ d − i P b i ψ i ψ d − i , so by definition of the height of a rational function we have h ( ϕ ◦ ψ ) ≤ X v ∈ M K d v log max n(cid:12)(cid:12)(cid:12)X a i ψ i ψ d − i (cid:12)(cid:12)(cid:12) v , (cid:12)(cid:12)(cid:12)X b i ψ i ψ d − i (cid:12)(cid:12)(cid:12) v o . For the right hand side of the above inequality, if v is nonarchimedean,then by Gauss’ Lemma again we have (cid:12)(cid:12)(cid:12)X a i ψ i ψ d − i (cid:12)(cid:12)(cid:12) v ≤ max (cid:0) | f | v | ψ | iv | ψ | d − iv (cid:1) ≤ | ϕ | v | ψ | dv . Similarly, (cid:12)(cid:12)(cid:12)X b i ψ i ψ d − i (cid:12)(cid:12)(cid:12) v ≤ | ϕ | v | ψ | dv . NTEGRAL POINTS IN ORBITS 9
Hence for v nonarchimedean, | ϕ ◦ ψ | v ≤ | ϕ | v | ψ | dv . Next let v be an archimedean place of K . Then the triangle inequal-ity gives (cid:12)(cid:12)(cid:12)X a i ψ i ψ d − i (cid:12)(cid:12)(cid:12) v ≤ ( d + 1) | f | v max i (cid:8) | ψ i ψ d − i | v (cid:9) . Applying the estimate (6) to the product ψ i ψ d − i yields | ψ i ψ d − i | v ≤ d (deg ψ +1) | ψ | iv | ψ | d − iv ≤ d (deg ψ +1) | ψ | dv . Therefore, (cid:12)(cid:12)(cid:12)X a i ψ i ψ d − (cid:12)(cid:12)(cid:12) v ≤ ( d + 1)2 d (deg ψ +1) | f | v | ψ | dv ≤ ( d + 1)2 d (deg ψ +1) | ϕ | v | ψ | dv . Similarly, (cid:12)(cid:12)(cid:12)X b i ψ i ψ d − (cid:12)(cid:12)(cid:12) v ≤ ( d + 1)2 d (deg ψ +1) | ϕ | v | ψ | dv . We combine these estimates. To ease notation, we let N v = 1 for v nonarchimedean and N v = ( d + 1)2 d deg ψ = ( d + 1)4 deg ϕ deg ψ for v archimedean. Then h ( ϕ ◦ ψ ) ≤ X v ∈ M K d v log max n(cid:12)(cid:12)(cid:12)X a i ψ i ψ d − (cid:12)(cid:12)(cid:12) v , (cid:12)(cid:12)(cid:12)X b i ψ i ψ d − (cid:12)(cid:12)(cid:12) v o ≤ X v ∈ M K d v (cid:0) log | ϕ | v + d log | ψ | v + log N v (cid:1) ≤ h ( ϕ ) + dh ( ψ ) + (deg ϕ )(deg ψ ) log 4 + log( d + 1) ≤ h ( ϕ ) + dh ( ψ ) + (deg ϕ )(deg ψ ) log 8 , since d + 1 ≤ d ≤ d deg ψ . This completes the proof of (c).Finally, we prove (d) by induction on n . The stated inequality isclearly true for n = 1. Assume now it it true for n . Then h ( ϕ n +1 ) ≤ h ( ϕ n ) + d n h ( ϕ ) + d n +1 log 8 from (c) applied to ϕ n and ϕ , ≤ (cid:18) d n − d − h ( ϕ ) + d d n − − d − (cid:19) + d n h ( ϕ ) + d n +1 log 8from the induction hypothesis,= (cid:18) d n +1 − d − (cid:19) h ( ϕ ) + d (cid:18) d n − d − (cid:19) log 8 . This completes the proof of Proposition 5. (cid:3)
The following facts about height functions are well-known.
Proposition 6.
Let ϕ : P → P be a rational map of degree d ≥ defined over K . There are constants c , c , c , and c , depending onlyon d , such that the following estimates hold for all P ∈ P ( ¯ K ) . (a) (cid:12)(cid:12) h ( ϕ ( P )) − dh ( P ) (cid:12)(cid:12) ≤ c h ( ϕ ) + c . (b) (cid:12)(cid:12) ˆ h ϕ ( P ) − h ( P ) (cid:12)(cid:12) ≤ c h ( ϕ ) + c . (c) ˆ h ϕ ( ϕ ( P )) = d ˆ h ϕ ( P ) . (d) P ∈ PrePer( ϕ ) if and only if ˆ h ϕ ( P ) = 0 .Proof. See, for example, [4, §§ B.2,B.4] or [10, § (cid:3) A Distance Estimate
Our goal in this section is a version of the inverse function theoremthat gives explicit estimates for the dependence on the (local) heightsof both the points and the function. It is undoubtedly possible to givea direct, albeit long and messy, proof of the desired result. We insteadgive a proof using universal families of maps and arithmetic distancefunctions. Before stating our result, we set notation for the universalfamily of degree d rational maps on P .We write Rat d ⊂ P d +1 for the space of rational maps of degree d ,where we identify a rational map ϕ = f /g = P a i z i (cid:14) P b i z i with thepoint [ ϕ ] = [ f, g ] = [ a , . . . , a d , b , . . . , b d ] ∈ P d +1 . If ϕ ∈ Rat d ( ¯ Q ) is defined over ¯ Q , we define the height of ϕ as inSection 2 to be the height of the associated point in P d +1 ( ¯ Q ), h ( ϕ ) = h (cid:0) [ a , . . . , a d , b , . . . , b d ] (cid:1) . Over Rat d , there is a universal family of degree d maps, which wedenote byΨ : P × Rat d −→ P × Rat d , ( P, ψ ) (cid:0) ψ ( P ) , ψ (cid:1) . We note that Rat d is the complement in P d +1 of a hypersurface,which we denote by ∂ Rat d . (The set ∂ Rat d is given by the resul-tant Res( f, g ) = 0, so ∂ Rat d is a hypersurface of degree 2 d .) Since P is complete, we have ∂ ( P × Rat d ) = P × ∂ Rat d . The map Ψ is a finite map of degree d . Let R (Ψ) denote its ram-ification locus. Looking at the behavior of Ψ in a neighborhood ofa point ( P, ψ ), it is easy to see that the restriction of R (Ψ) to afiber P ψ = P × { ψ } is the ramification divisor of ψ , R (Ψ) (cid:12)(cid:12) P ψ = R ( ψ ) . NTEGRAL POINTS IN ORBITS 11
So the ramification indices of the universal map Ψ and a particularmap ψ are related by e ( P,ψ ) (Ψ) = e P ( ψ ) . (7) Proposition 7.
Let ψ ∈ K ( z ) be a nontrivial rational function, let S ⊂ M K be a finite set of absolute values on K , each extended in someway to ¯ K , and let A, P ∈ P ( K ) . Then X v ∈ S max A ′ ∈ ψ − ( A ) e A ′ ( ψ ) d v λ v ( P, A ′ ) ≥ X v ∈ S d v λ v (cid:0) ψ ( P ) , A (cid:1) + O (cid:0) h ( A ) + h ( ψ ) + 1 (cid:1) , where the implied constant depends only on the degree of the map ψ .Proof. The statement and proof of Proposition 7 use the machineryof arithmetic distance functions and local height functions on quasi-projective varieties as described in [7], to which we refer the readerfor definitions, notation, and basic properties. We begin with thedistribution relation for finite maps of smooth quasi-projective vari-eties [7, Proposition 6.2(b)]. Applying this relation to the map Ψ andpoints x, y ∈ P × Rat d yields δ (cid:0) Ψ( x ) , y ; v (cid:1) = X y ′ ∈ Ψ − ( y ) e y ′ (Ψ) δ ( x, y ′ ; v ) + O (cid:0) λ ∂ ( P × Rat d ) ( x, y ; v ) (cid:1) . (8)Here δ ( · , · ; v ) is a v -adic arithmetic distance function on P × Rat d and λ ∂ ( P × Rat d ) is a local height function for the indicated divisor. Inparticular, if we take x = ( P, ψ ) and y = ( A, ψ ), then the arithmeticdistance function δ and the chordal metric λ v defined in Section 1satisfy δ (cid:0) Ψ( x ) , y ; v (cid:1) = δ (cid:0) Ψ( P, ψ ) , ( A, ψ ); v (cid:1) = δ (cid:0) ( ψ ( P ) , ψ ) , ( A, ψ ); v (cid:1) = λ v (cid:0) ψ ( P ) , A (cid:1) . (9)Similarly, if y ′ = ( A ′ , ψ ) ∈ Ψ − ( y ), then δ ( x, y ′ ; v ) = δ (cid:0) ( P, ψ ) , ( A ′ , ψ ); v (cid:1) = λ v ( P, A ′ ) . Further, since ∂ ( P × Rat d ) = P × ∂ Rat d is the pull-back of a divisoron Rat d and ∂ ( P × Rat d ) = ( P × ∂ Rat d ) × ( P × Rat d )+( P × Rat d ) × ( P × ∂ Rat d ) , applying [7, Proposition 5.3 (a)] gives λ ∂ ( P × Rat d ) ( x, y ; v ) ≫≪ λ P × ∂ Rat d (cid:0) ( P, ψ ); v (cid:1) + λ P × ∂ Rat d (cid:0) ( A, ψ ); v (cid:1) ≫≪ λ ∂ Rat d ( ψ ; v ) . (10) Substituting (7), (9), and (10) into the distribution relation (8) yields λ v (cid:0) ψ ( P ) , A (cid:1) = X A ′ ∈ ψ − ( A ) e A ′ ( ψ ) λ v ( P, A ′ ) + O (cid:0) λ ∂ Rat d ( ψ ; v ) (cid:1) . (11)To ease notation, let A ′ v ∈ ψ − ( A ) be a point satisfying e A ′ v ( ψ ) λ v ( P, A ′ v ) = max A ′ ∈ ψ − ( A ) e A ′ λ v ( P, A ′ ) . Then for any A ′ ∈ ψ − ( A ) we have e A ′ ( ψ ) λ v ( P, A ′ ) = min (cid:8) e A ′ v ( ψ ) λ v ( P, A ′ v ) , e A ′ ( ψ ) λ v ( P, A ′ ) (cid:9) from choice of A ′ v , ≤ d min (cid:8) λ v ( P, A ′ v ) , λ v ( P, A ′ ) (cid:9) since ψ has degree d , ≤ dλ v ( A ′ v , A ′ ) + O (1) from the triangle inequality.(12)This is a nontrivial estimate for A ′ = A ′ v , so in (11) we pull off the A ′ v term and use (12) for the other terms to obtain λ v (cid:0) ψ ( P ) , A (cid:1) ≤ e A ′ v ( ψ ) λ v ( P, A ′ v ) + d X A ′ ∈ ψ − ( A ) A ′ = A ′ v λ v ( A ′ v , A ′ ) + O (cid:0) λ ∂ Rat d ( ψ ; v ) (cid:1) . (13)The next lemma gives an upper bound for λ v ( A ′ v , A ′ ). Lemma 8.
There is a constant C = C ( d ) such that the following holds.Let ψ ∈ Rat d ( ¯ Q ) , let A ∈ P ( ¯ Q ) , and let A ′ , A ′′ ∈ ψ − ( A ) be distinct points. Then X v ∈ M K d v λ v ( A ′ , A ′′ ) ≤ C (cid:0) h ( A ) + h ( ψ ) + 1 (cid:1) . Proof.
In the notation of [7], we have λ v ( A ′ , A ′′ ) = δ P × Rat d (cid:0) ( A ′ , ψ ) , ( A ′′ , ψ ); v (cid:1) = λ ( P × Rat d ) , ∆ (cid:0) ( A ′ , ψ ) , ( A ′′ , ψ ); v (cid:1) , where ∆ is the diagonal of ( P × Rat d ) . Summing over v gives heightfunctions X v ∈ M K λ v ( A ′ , A ′′ ) = h ( P × Rat d ) , ∆ (cid:0) ( A ′ , ψ ) , ( A ′′ , ψ ) (cid:1) + O (cid:0) h ∂ ( P × Rat d ) (cid:0) ( A ′ , ψ ) , ( A ′′ , ψ ) (cid:1) + 1 (cid:1) . Choosing an ample divisor H on P × Rat d , we use the fact thatheights with respect to a subscheme are dominated by ample heightsaway from the support of the subscheme [7, Proposition 4.2]. (This NTEGRAL POINTS IN ORBITS 13 is where we use the assumption that A ′ = A ′′ , which ensures that thepoint (cid:0) ( A ′ , ψ ) , ( A ′′ , ψ ) (cid:1) is not on the diagonal.) This yields X v ∈ M K λ v ( A ′ , A ′′ ) ≪ h P × Rat d ,H ( A ′ , ψ ) + h P × Rat d ,H ( A ′′ , ψ ) + 1 ≪ h ( A ′ ) + h ( A ′′ ) + h ( ψ ) + 1 . (14)We now use [11, Theorem 2], which says that there are positiveconstants C , C , C , depending only on the degree of ψ , such that h (cid:0) ψ ( P ) (cid:1) ≥ C h ( P ) − C h ( ψ ) − C . (15)(The paper [11] deals with general rational maps P n P n . In ourcase with n = 1, it would be a tedious, but not difficult, calculationto give explicit values for the C i , including of course C = deg ψ .)Applying (15) with P = A ′ and P = A ′′ , we substitute into (14) toobtain X v ∈ M K λ v ( A ′ , A ′′ ) ≪ h ( A ) + h ( ψ ) + 1 , which completes the proof of Lemma 8. (cid:3) We use Lemma 8 to bound the sum in the right-hand side of theinequality (13). We note that λ v ( A ′ , A ′′ ) ≥ P v ∈ S d v λ v ( A ′ , A ′′ ) ≪ h ( A ) + h ( ψ ) + 1for any set of places S . Further, the sum in (13) has at most d − X v ∈ S d v λ v (cid:0) ψ ( P ) , A (cid:1) ≤ X v ∈ S e A ′ v ( ψ ) d v λ v ( P, A ′ v ) + O (cid:0) h ( A ) + h ( ψ ) + 1 (cid:1) . Note that in this last inequality, the O (cid:0) h ( ψ ) (cid:1) term comes from twoplaces, Lemma 8 and X v ∈ S d v λ ∂ Rat d ( ψ ; v ) ≤ X v ∈ M K d v λ ∂ Rat d ( ψ ; v ) = h ∂ Rat d ( ψ ) = O (cid:0) h ( ψ ) + 1 (cid:1) , where the last equality comes from the fact that ∂ Rat d is a hypersurfaceof degree 2 d in P d +1 . This completes the proof of Proposition 7. (cid:3) A Ramification Estimate and a Quantitative Versionof Roth’s Theorem
In this section we state two known results that will be needed toprove our main theorem. The first says that away from exceptionalpoints, the ramification of ϕ m tends to spread out as m increases. Lemma 9.
Fix an integer d ≥ . There exist constants κ and κ < ,depending only on d , such that for all degree d rational maps ϕ : P → P , all points Q ∈ P that are not exceptional for ϕ , all integers m ≥ ,and all P ∈ ϕ − m ( Q ) , we have e P ( ϕ m ) ≤ κ ( κ d ) m . Proof.
This is [10, Lemma 3.52]; see in particular the last paragraphof the proof. It is not difficult to give explicit values for the con-stants. In particular, if Q is not preperiodic, then the stronger estimate e P ( ϕ m ) ≤ e d − is true for all m . (cid:3) The second result we need is the following quantitative version ofRoth’s Theorem.
Theorem 10.
Let S be a finite subset of M K that contains all infiniteplaces. We assume that each place in S is extended to ¯ K in somefashion. Set the following notation. s the cardinality of S . Υ a finite, G ¯ K/K -invariant subset of K . β a map S → Υ . µ > a constant. M ≥ a constant.There are constants r and r , depending only on [ K : Q ] , , and µ ,such that there are at most s r elements x ∈ K satisfying both of thefollowing conditions : X v ∈ S d v log + | x − β v | − v ≥ µh ( x ) − M. (16) h ( x ) ≥ r max v ∈ S { h ( β v ) , M, } . (17) Proof.
This is [8, Theorem 2.1], with a small change of notation. Forexplicit values of the constants, see [2]. (cid:3) A Bound for the Number of Quasi-Integral Points inan Orbit
In this section we prove our main result, which is an explicit upperbound for the number of iterates ϕ n ( P ) that are close to a given basepoint A in any one of a fixed finite number of v -adic topologies. Hereis the precise statement. Theorem 11.
Let ϕ ∈ K ( z ) be a rational map of degree d ≥ . Fixa point A ∈ P ( K ) which is not an exceptional point for ϕ , and let P ∈ P ( K ) be a wandering point for ϕ . For any finite set of places NTEGRAL POINTS IN ORBITS 15 S ⊂ M K and any constant ≥ ε > , define a set of non-negativeintegers Γ ϕ,S ( A, P, ε ) = ( n ≥ X v ∈ S d v λ v ( ϕ n P, A ) ≥ ε ˆ h ϕ ( ϕ n P ) ) . (a) There exist constants γ = γ (cid:0) d, ε, [ K : Q ] (cid:1) and γ = γ (cid:0) d, ε, [ K : Q ] (cid:1) such that ( n ∈ Γ ϕ,S ( A, P, ε ) : n > γ + log + d h ( ϕ ) + ˆ h ϕ ( A )ˆ h ϕ ( P ) !) ≤ S γ . (18)(b) In particular, there is a constant γ = γ (cid:0) d, ε, [ K : Q ] (cid:1) such that ϕ,S ( A, P, ε ) ≤ S γ + log + d h ( ϕ ) + ˆ h ϕ ( A )ˆ h ϕ ( P ) ! . (19)(c) There is a constant γ = γ (cid:0) K, S, ϕ, A, ǫ ) that is independent of P such that max Γ ϕ,S ( A, P, ε ) ≤ γ . Before giving the proof of Theorem 11, we make a number of remarks.
Remark . Note that as a consequence of Proposition 6(d), we haveˆ h ϕ ( P ) > P is wandering point for ϕ. Hence the right-hand side of(19) is well defined.
Remark . If we take ε = 1, then the set Γ ϕ,S ( A, P, ε ) more-or-lesscoincides with the set of points in the orbit O ϕ ( P ) that are S -integralwith respect to A . We say more-or-less because Γ ϕ,S ( A, P, ε ) is definedusing the canonical height of ϕ n ( P ), rather than the naive height. Butusing the inequality (cid:12)(cid:12) ˆ h ϕ ( P ) − h ( P ) (cid:12)(cid:12) ≪ h ( ϕ ) + 1 from Proposition 6and adjusting the constants, it is not hard to see that the estimate (19)remains true for the setΓ naive ϕ,S ( A, P, ε ) = ( n ≥ X v ∈ S d v λ v ( ϕ n P, A ) ≥ εh ( ϕ n P ) ) . (See the proof of Corollary 17.) For example, taking A = ∞ , theset Γ naive ϕ,S ( A, P, ε ) consists of the points ϕ n ( P ) such that z (cid:0) ϕ n ( P ) (cid:1) is( S, ε )-integral for some ε . This is the motivation for saying thatthe points in Γ ϕ,S ( A, P, ε ) are quasi-(
S, ε )-integral with respect to A ,where ε measures the degree of S -integrality. Remark . The dependence of the bounds (18) and (19) on h ( ϕ ),ˆ h ϕ ( A ), and ˆ h v ( P ) are quite interesting. A dynamical analogue of aconjecture of Lang asserts that the ratio h ( ϕ ) / ˆ h ϕ ( P ) is bounded, inde-pendently of ϕ and P , provided that ϕ is suitably minimal with respectto PGL ( K )-conjugation. See [10, Conjecture 4.98].On the other hand, there cannot be a uniform bound for the ratioˆ h ϕ ( A ) / ˆ h ϕ ( P ), since A and P may be chosen arbitrarily and indepen-dent of one another. This raises the interesting question of whetherthe bound for ϕ,S ( A, P, ε ) actually needs to depend on A . Even invery simple situations, it appears difficult to answer this question. Forexample, consider the map ϕ ( z ) = z , the initial point P = 2, and theset of primes S = {∞ , , } . As A ∈ Q ∗ varies, is it possible for theorbit O ϕ ( P ) to contain more and more points that are S -integral withrespect to A ? Writing A = x/y , we are asking ifsup x,y ∈ Z (cid:8) ( n, i, j ) ∈ N : y · n − x = 3 i j (cid:9) = ∞ . Remark . We observe that ϕ,S ( A, P, ε ) can grow as fast as log( ε − )as ε → + . For example, consider the map ϕ ( z ) = z d + z d − , thepoints A = 0 and P = p , and the set of primes S = { p } . Since ϕ n ( z ) = z ( d − n + h.o.t., we have (cid:12)(cid:12) ϕ n ( p ) (cid:12)(cid:12) p = p − ( d − n , so λ p ( ϕ n P, A ) = λ p ( ϕ n ( p ) ,
0) = − log (cid:12)(cid:12) ϕ n ( p ) (cid:12)(cid:12) p = ( d − n log p. Thus Γ ϕ,S ( A, P, ε ) consists of all n ≥ d − n log p ≥ ε ˆ h ϕ ( ϕ n P ) = εd n ˆ h ϕ ( P ) . Hence ϕ,S ( A, P, ε ) = $ log log pε ˆ h ϕ ( P ) !, log (cid:18) dd − (cid:19)% + 1= log( ε − )log( d/ ( d − o (log ε − ) as ǫ → + .In particular, if ε is small and d is large, so log( d/ ( d − ≈ / ( d − ϕ,S ( A, P, ε ) ≈ ( d −
1) log( ǫ − ) . Remark . See [3, 8] for a version of Theorem 11 for elliptic curves.These papers deal with points on an elliptic curve E that are quasi-( S, ǫ )-integral with respect to O , the zero point of E . It is also ofinterest to study points that are integral with respect to some otherpoint A , and in particular to see how the bound depends on A . The NTEGRAL POINTS IN ORBITS 17 distance function on E is translation invariant up to O (cid:0) h ( E ) (cid:1) , so wewant to estimate the size of the set (cid:26) P ∈ E ( K ) : X v ∈ S d v λ v ( P − A ) ≥ ε ˆ h E ( P ) (cid:27) . (20)Translating the points in (20) by A , we want to count points satisfying P d v λ v ( P ) ≥ ˆ h E ( P + A )+ O (cid:0) h ( E ) (cid:1) . The canonical height on an ellipticcurve is a quadratic form, so ˆ h E ( P + A ) ≤ h E ( P ) + 2ˆ h E ( A ). Usingthe results in [8], this leads to a bound for the set (20) in which thedependence on A appears as the ratio ˆ h E ( A ) / ˆ h E ( P min ), where P min isthe point of smallest nonzero height in E ( K ). This is analogous to thedependence on A in (19). Proof of Theorem 11.
To ease notation, we will write Γ S ( ε ) in place ofΓ ϕ,S ( A, P, ε ). For the given ε , we set m ≥ κ m ≤ ε κ , where κ and κ are the positive constants appearing in Lemma 9.Since κ <
1, there exists such an integer m . Notice that κ and κ depend only on d , and consequently m depends only on d and ε .More precisely, if we assume (without loss of generality) that ε < ,then m ≪ log( ε − ), where the implied constant depends only on d .Put e m = max A ′ ∈ ϕ − m ( A ) e A ′ ( ϕ m ) . Then Lemma 9 and our choice of m imply that e m ≤ κ ( κ d ) m ≤ ε d m . (21)Further, Proposition 7 says that for all Q ∈ P ( K ) we have e m X v ∈ S max A ′ ∈ ϕ − m ( A ) d v λ v ( Q, A ′ ) ≥ X v ∈ S d v λ v ( ϕ m Q, A ) − O (cid:0) h ( A ) + h ( ϕ m ) + 1 (cid:1) , (22)where the implied constant depends on deg( ϕ m ).Suppose first that n ≤ m for all n ∈ Γ S ( ε ). Then clearly S ( ε ) ≤ m , and from our choice of m we have S ( ε ) ≤ m ≤ log(5 κ ) + log( ε − )log( κ − ) + 1 . This upper bound has the desired form, since κ > > κ > d .We may thus assume that there exists an n ∈ Γ S ( ε ) such that n > m ,and we fix such an n ∈ Γ S ( ε ) . By the definition of Γ S ( ε ) we have ε ˆ h ϕ ( ϕ n P ) ≤ X v ∈ S d v λ v ( ϕ n P, A ) . Applying (22) to the point Q = ϕ n − m ( P ) yields ε ˆ h ϕ ( ϕ n P ) ≤ e m X v ∈ S d v max A ′ ∈ ϕ − m ( A ) λ v ( ϕ n − m P, A ′ )+ O (cid:0) h ( A ) + h ( ϕ m ) + 1 (cid:1) , (23)where the big- O constant depends on deg ϕ m = d m , so on d and ε .For each v ∈ S we choose an A ′ v ∈ ϕ − m ( A ) satisfying λ v ( ϕ n − m P, A ′ v ) = max A ′ ∈ ϕ − m A λ v ( ϕ n − m P, A ′ ) . (For ease of exposition, we will assume that z ( A ′ ) = ∞ for all A ′ ∈ ϕ − m A . If this is not the case, then we use z for some of the A ′ ’s, andwe use z − for the others.)Let S ′ ⊂ S be the set of places in S defined by S ′ = (cid:8) v ∈ S : λ v (cid:0) ϕ n − m ( P ) , A ′ v (cid:1) > λ v ( A ′ v , ∞ ) + log ℓ v (cid:9) , where we recall that ℓ v = 2 if v is archimedean and ℓ v = 1 otherwise.Set S ′′ = S r S ′ . Applying Lemma 3 to the places in S ′ and using thedefinition of S ′′ for the places in S ′′ , we find that ε ˆ h ϕ ( ϕ n ( P )) ≤ (cid:18)X v ∈ S ′ + X v ∈ S ′′ (cid:19) d v λ v ( ϕ n P, A ) since n ∈ Γ S ( A, P, ε ), ≤ e m (cid:18)X v ∈ S ′ + X v ∈ S ′′ (cid:19) d v λ v (cid:0) ϕ n − m ( P ) , A ′ v (cid:1) + O (cid:0) h ( A ) + h ( ϕ m ) + 1 (cid:1) from the definition of A ′ v and (23), ≤ e m X v ∈ S ′ d v (cid:0) λ v ( A ′ v , ∞ ) − log (cid:12)(cid:12) z (cid:0) ϕ n − m ( P ) (cid:1) − z ( A ′ v ) (cid:12)(cid:12) + log ℓ v (cid:1) + e m X v ∈ S ′′ d v (cid:0) λ v ( A ′ v , ∞ ) + log ℓ v (cid:1) + O (cid:0) h ( A ) + h ( ϕ m ) + 1 (cid:1) from Lemma 3, NTEGRAL POINTS IN ORBITS 19 ≤ e m X v ∈ S ′ d v log (cid:12)(cid:12) z (cid:0) ϕ n − m ( P ) (cid:1) − z ( A ′ v ) (cid:12)(cid:12) − + e m X v ∈ S d v (cid:0) λ v ( A ′ v , ∞ ) + log ℓ v (cid:1) + O (cid:0) h ( A ) + h ( ϕ m ) + 1 (cid:1) . We now use Proposition 6(b,c) to observe that X v ∈ S d v λ v ( A ′ v , ∞ ) ≤ X A ′ ∈ ϕ − m ( A ) X v ∈ S d v λ v ( A ′ , ∞ ) ≤ X A ′ ∈ ϕ − m ( A ) h ( A ′ ) ≤ X A ′ ∈ ϕ − m ( A ) (cid:0) ˆ h ϕ ( A ′ ) + O (cid:0) h ( ϕ ) + 1 (cid:1)(cid:1) ≤ ˆ h ϕ ( A ) + O (cid:0) h ( ϕ ) + 1 (cid:1)(cid:1) , Here the last line follows because there are at most d m terms in thesum, and ˆ h ϕ ( A ′ ) = d − m ˆ h ϕ ( A ). The constants depend only on m and d ,so on ε and d . Further, from the definition of ℓ v we have X v ∈ S d v log ℓ v ≤ log 2 . We also note from Proposition 5(d) that h ( ϕ m ) ≪ h ( ϕ ) + 1, with theimplied constant depending only on d and m . Hence ε ˆ h ϕ ( ϕ n ( P )) ≤ e m X v ∈ S ′ d v log + (cid:12)(cid:12) z (cid:0) ϕ n − m ( P ) (cid:1) − z ( A ′ v ) (cid:12)(cid:12) − + O (cid:0) ˆ h ϕ ( A ) + h ( ϕ ) + 1 (cid:1) . (24)We are going to apply Roth’s theorem (Theorem 10) to the setΥ = (cid:8) z ( A ′ ) : A ′ ∈ ϕ − m ( A ) (cid:9) ⊂ ¯ K, the map β : S ′ → Υ given by β ( v ) = A ′ v , and the points x = ϕ n − m ( P )for n ∈ Γ S ( ǫ ). We note that Υ is a G ¯ K/K -invariant set and that ≤ d m . We apply Theorem 10 to the set of places S ′ , taking M = 0and µ = . This gives constants r and r , depending only on [ K : Q ], d ,and ε , such that the set of n ∈ Γ S ( ǫ ) with n > m can be written as aunion of three sets, (cid:8) n ∈ Γ S ( ǫ ) : n > m (cid:9) = T ∪ T ∪ T , characterized as follows: T ≤ S ′ r ,T = ( n > m : X v ∈ S ′ d v log + (cid:12)(cid:12) z (cid:0) ϕ n − m ( P ) (cid:1) − z ( A ′ v ) (cid:12)(cid:12) − ≤ h (cid:0) ϕ n − m ( P ) (cid:1)) ,T = (cid:26) n > m : h (cid:0) ϕ n − m ( P ) (cid:1) ≤ r max v ∈ S ′ (cid:8) h ( A ′ v ) , (cid:9)(cid:27) . We already have a bound for the size of T , so we look at T and T .We start with T and use Proposition 6(b,c) to estimate h ( A ′ v ) ≤ ˆ h ϕ ( A ′ ) + c h ( ϕ ) + c = d − m ˆ h ϕ ( A ) + c h ( ϕ ) + c ,h (cid:0) ϕ n − m ( P ) (cid:1) ≥ ˆ h ϕ (cid:0) ϕ n − m ( P ) (cid:1) − c h ( ϕ ) − c = d n − m ˆ h ϕ ( P ) − c h ( ϕ ) − c . Hence T ⊂ (cid:8) n > m : d n − m ˆ h ϕ ( P ) ≤ c ˆ h ϕ ( A ) + c h ( ϕ ) + c (cid:9) , so every n ∈ T satisfies n ≤ m + log + d c ˆ h ϕ ( A ) + c h ( ϕ ) + c ˆ h ϕ ( P ) ! ≤ c + log + d ˆ h ϕ ( A ) + h ( ϕ )ˆ h ϕ ( P ) ! . (25)Finally, we consider the set T . Again using Proposition 6(b,c) torelate h (cid:0) ϕ n − m ( P ) (cid:1) to d n − m ˆ h ϕ ( P ), we find that every n ∈ T satisfies X v ∈ S ′ d v log + (cid:12)(cid:12) z (cid:0) ϕ n − m ( P ) (cid:1) − z ( A ′ v ) (cid:12)(cid:12) − ≤ d n − m ˆ h ϕ ( P ) + c h ( ϕ ) + c . We substitute this estimate into (24) to obtain ε ˆ h ϕ (cid:0) ϕ n ( P ) (cid:1) ≤ e m d n − m ˆ h ϕ ( P ) + c (cid:0) ˆ h ϕ ( A ) + h ( ϕ ) + 1 (cid:1) . We know from (21) that e m ≤ εd m /
5, and also ˆ h ϕ (cid:0) ϕ n ( P ) (cid:1) = d n ˆ h ϕ ( P ),which yields εd n ˆ h ϕ ( P ) ≤ (cid:16) ε d m (cid:17) d n − m ˆ h ϕ ( P ) + c (cid:0) ˆ h ϕ ( A ) + h ( ϕ ) + 1 (cid:1) . A little bit of algebra gives the inequality n ≤ log d c ˆ h ϕ ( A ) + h ( ϕ ) + 1 ε ˆ h ϕ ( P ) ! ≤ c + log + d ˆ h ϕ ( A ) + h ( ϕ )ˆ h ϕ ( P ) ! . (26)Combining the estimate for T with the bounds (25) and (26) for thelargest elements in T and T completes the proof of (a).We note that (b) follows immediately from (a). NTEGRAL POINTS IN ORBITS 21
Finally, we prove (c). Our first observation is that the set Υ = z (cid:0) ϕ − m ( A ) (cid:1) used in the application of Roth’s theorem does not dependon the point P . So the largest element in the finite set T is boundedindependently of P . (Of course, since Roth’s theorem is not effective,we do not have an explicit bound for max Υ in terms K , S , ε , ϕ and A ,but that is not relevant.)Our second observation is to note that the quantityˆ h min ϕ,K def = inf (cid:8) ˆ h ϕ ( P ) : P ∈ P ( K ) wandering for ϕ (cid:9) is strictly positive. To see this, let P ∈ P ( K ) be any ϕ -wanderingpoint. Thenˆ h min ϕ,K = inf (cid:8) ˆ h ϕ ( P ) : P ∈ P ( K ) and 0 < ˆ h ϕ ( P ) ≤ ˆ h ϕ ( P ) (cid:9) . This last set is finite, so the infimum is over a finite set of positivenumbers, hence is strictly positive. Therefore in the upper bounds (25)and (26) for max T and max T , we may replace ˆ h ϕ ( P ) with ˆ h min ϕ,K toobtain upper bounds that are independent of P . This proves thatmax( T ∪ T ∪ T ) may be bounded independently of P , which com-pletes the proof of (c). (cid:3) A Bound for the Number of Integral Points in anOrbit
In this section, we use Theorem 11 to give a uniform upper boundfor the number of S -integral points in an orbit. Corollary 17.
Let K be a number field, let S ⊂ M K be a finite setof places that includes all archimedean places, let R S be the ring of S -integers of K , and let d ≥ . There is a constant γ = γ (cid:0) d, [ K : Q ] (cid:1) suchthat for all rational maps ϕ ∈ K ( z ) of degree d satisfying ϕ ( z ) / ∈ K [ z ] and all ϕ -wandering points P ∈ P ( K ) , the number of S -integral pointsin the orbit of P is bounded by (cid:8) n ≥ z (cid:0) ϕ n ( P ) (cid:1) ∈ R S (cid:9) ≤ S γ + log + d h ( ϕ )ˆ h ϕ ( P ) ! . Proof.
By definition, an element α ∈ K is in R S if and only if | α | v ≤ v / ∈ S , or equivalently, if and only if h ( α ) = X v ∈ S d v log max (cid:8) | α | v , (cid:9) . We note that for v ∈ M K we have λ v ( α, ∞ ) = λ v (cid:0) [ α, , [1 , (cid:1) = log max (cid:8) | α | v , (cid:9) . The formula for λ v when v is archimedean is slightly different, but thetrivial inequality max { t, } ≤ √ t + 1 shows that for v ∈ M ∞ K we havelog max (cid:8) | α | v , (cid:9) ≤ λ v ( α, ∞ ) . Hence α ∈ R S = ⇒ h ( α ) ≤ X v ∈ S d v λ v ( α, ∞ ) . Let n ≥ z (cid:0) ϕ n ( P ) (cid:1) ∈ R S . Then h (cid:0) ϕ n ( P ) (cid:1) ≤ X v ∈ S d v λ v (cid:0) ϕ n ( P ) , ∞ (cid:1) . (27)Proposition 6 tells us that h (cid:0) ϕ n ( P ) (cid:1) ≥ ˆ h ϕ (cid:0) ϕ n ( P ) (cid:1) − c h ( ϕ ) − c = d n ˆ h ϕ ( P ) − c h ( ϕ ) − c , (28)where c and c depend only on d . Combining (27) and (28) gives X v ∈ S d v λ v (cid:0) ϕ n ( P ) , ∞ (cid:1) ≥ d n ˆ h ϕ ( P ) − c h ( ϕ ) − c . (29)We consider two cases. First, if d n ˆ h ϕ ( P ) ≤ c h ( ϕ ) + 2 c , then the number of possible values of n is at mostlog + d c h ( ϕ ) + 2 c ˆ h ϕ ( P ) ! , which has the desired form. Second, if d n ˆ h ϕ ( P ) ≥ c h ( ϕ ) + 2 c , then (29) implies that X v ∈ S d v λ v (cid:0) ϕ n ( P ) , ∞ (cid:1) ≥ d n ˆ h ϕ ( P ) = 12 ˆ h ϕ (cid:0) ϕ n ( P ) (cid:1) . (30)Now Theorem 11(b) with ε = and A = ∞ tells us that the numberof n satisfying (30) is at most4 S γ + log + d h ( ϕ ) + ˆ h ϕ ( ∞ )ˆ h ϕ ( P ) ! , (31)where γ depends only on [ K : Q ] and d . (Note that our assumptionthat ϕ ( z ) is not a polynomial is equivalent to the assertion that ∞ is not an exceptional point for ϕ . This is needed in order to applyTheorem 11.) It only remains to observe thatˆ h ϕ ( ∞ ) ≤ h ( ∞ ) + c h ( ϕ ) + c and h ( ∞ ) = h (cid:0) [0 , (cid:1) = 0 NTEGRAL POINTS IN ORBITS 23 to see that the bound (31) has the desired form. (cid:3)
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Department of Mathematics, National Central University, Chung-Li, 32054 Taiwan, R. O. C.
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