Rational values of transcendental functions and arithmetic dynamics

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain a lower bound of the form $cD^{n/4 - \varepsilon}$ on the degree of the splitting field of $P^{\circ n}(z)=P^{\circ n}(\alpha)$, where $P$ is a polynomial of degree $D\geq 2$ over a number field, $P^{\circ n}$ is its $n$-th iterate and $c$ depends effectively on $P, \alpha$ and $\varepsilon$. Our $c$ is positive for each algebraic $\alpha$ for which the set $\{P^{\circ n}(\alpha):n\in\mathbb{N}\}$ is infinite.


Introduction
We will give improvements, in certain cases, of the Bombieri-Pila Theorem on rational values of transcendental analytic functions. Our improvements apply to certain classes of analytic functions on the disk, by which we shall always mean the open unit disk in the complex plane. For z in the disk, we let φ(z) = 1/(1 − |z|). We consider classes of functions defined by growth conditions involving φ. For our first result we take functions decaying exponentially with respect to φ. More precisely, we suppose that f is analytic on the unit disk and that there is some subset S of the unit disk containing (0, 1) such that, for some positive real numbers a, b with b > 1, we have |f (z)| ≤ a b φ(z) for z ∈ S. Theorem 1.1. Suppose that f is analytic on the unit disk with |f (z)| ≤ 1 there and that there are positive real numbers a, b with b > 1 such that |f (z)| < ab −φ(x) for z in S. There is a positive real constant c a,b , depending only on a, b and effectively computable from them, with the following property. For d ≥ 2 and H ≥ e the number of algebraic α ∈ S such that f (α) is algebraic and non-zero with [Q(α, f (α)) : Q] ≤ d and H(α), H(f (α)) ≤ H is at most c a,b d 9 (log d) 2 (log H) 9 .
Date: August 24, 2018. 1 Moreover if, for some 0 < r < 1, we only consider α such that |α| ≤ r then the bound can be improved to c a,b,r d 4 (log d) 2 (log H) 4 , where c a,b,r is effectively computable from a, b and r.
Here H(α) is the multiplicative height of α. See for instance Chapter 3 of Waldschmidt's book [39] for a discussion of heights. Note that the rapid decay of f rules out the possibility that f is algebraic, unless f = 0, and indeed such a bound may fail for algebraic functions.
There have been many results of this type since the fundamental work of Bombieri and Pila [6]. Building on this work, Pila proved [27] a bound of the form c ε H ε for a transcendental analytic function f on [0, 1]. This bound is essentially optimal, as Surroca [36,37] and Bombieri and Pila [28] showed. Surroca also showed that, in the situation of Pila's result, a bound of the form c(log H) 2 holds at some unbounded sequence of heights. But to have bounds of this form holding at all heights we must make further assumptions on f . One possibility is to assume that f satisfies some nice form of differential equation. In this direction, see work by Pila [29,31], Thomas and the second author [18,19], Binyamini and Novikov [4] and Comte and Miller [12]. In a different direction, Masser [24] proved a very precise zero estimate for the Riemann zeta function and using it showed that a c(log H) 2 (log log H) −2 bound holds for the restriction of the zeta function to (2,3). Besson [3] established a similar bound for the restriction of the gamma function to a bounded interval (and also computed an explicit constant for Masser's result [2]). Independently, the first two authors proved [7] bounds of the form c(log H) 3+ε for gamma on the interval (0, ∞), and for suitable restrictions of entire functions satisfying certain growth conditions, and further results for entire functions satisfying fairly weak growth conditions in [8].
We will prove a similar result to Theorem 1.1 for functions which grow exponentially along S, but before discussing that we mention an application of Theorem 1.1 to arithmetic dynamics. For this, suppose that P is a polynomial in one variable, of degree D at least 2, with coefficients in a number field K. We write P •n for the n-th iterate of P , so P •(n+1) = P • P •n and P •0 (z) = z. Fix some α in K. For n ≥ 1 consider the set S α,n = {β : P •n (β) = P •n (α)}.
The field generated by this set is a Galois extension of K since it is the splitting field of a polynomial. So we have a sequence of Galois extensions K n = K(S α,n )/K. Applying Theorem 1.1 with a strategy of the third author from [33], and using p-adic methods of Ingram and DeMarco et al [17,14], we will prove the following. Theorem 1.2. Suppose that P , K and α are as above. Let ε > 0. Then there are c > 0, depending only on P and ε, and c ′ > 0 depending only on P such that Here h(α) is the logarithmic height of α, so h(α) = log H(α), and the functionĥ P is the canonical height attached to P (see [35,Theorem 3.20]). Note that this height vanishes at α if and only if α is preperiodic for P (i.e. the forward orbit of α under P is finite). The quantity [Q(α) : Q]h(α) in the theorem could be replaced by any upper bound for log p where p > 1 is an upper bound for the residue characteristics for which |α| v > 1. Our theorem is related to a long-standing open problem in arithmetic dynamics (see for instance Conjecture 1.2 in [20]), in which in place of the equation P •n (β) = P •n (α) we instead consider P •n (β) = α. Their conjecture predicts a stronger growth of the order D n with a constant depending in an unspecified manner on α. Ingram has obtained results towards this conjecture using the p-adic Böttcher map [17], however we are not aware of any bounds that hold for all α that are not pre-periodic. Although the problem we consider is different, we are able to obtain a nontrivial result for all non-preperiodic α. These seem to be the first such results in this generality. Moreover our results are effective.
Our sequence K n from above is reminiscent of the fields obtained by considering the splitting field of equations such as 2 n z = α where z, α lie in a commutative algebraic group and 2 n is multiplication by 2 n . In fact if P = z 2 and α = 1 we are reduced to considering 2 n z = α in the multiplicative group G m . In that particular situation we get cyclotomic extensions with abelian Galois group. For general polynomials P however the extensions K n are not abelian. Another example is considering the same equation in elliptic curves. Both of these examples correspond to adjoining torsion points to a number field. For this problem in the elliptic case see [23] where alternatively a bound can also be deduced from Serre's open image theorem if the curve does not have CM. The archimedian part of the proof of Theorem 1.2 uses a strategy developed by the third author in [33] to provide alternative proofs in the elliptic case.
In our situation there is usually no commutative group present and dynamical systems of rational maps provide a rich arithmetic structure that is often very different from the one provided by commutative groups. We are considering the one-variable case which corresponds to algebraic groups of dimension one. Here Lattés maps correspond to elliptic curves and the polynomial z 2 or Tschebycheff polynomials to the multiplicative group G m but the space of rational maps is much larger and these examples are usually considered very special cases in dynamics.
The main new ingredient in our proof of Theorem 1.2 lies in the following. Theorem 1.3. Given P as above, there is an effectively computable R > 0 depending only on P with the following property. For each ε > 0 there exists an effectively computable positive constant c ε depending only on ε and P such that if α is algebraic with |α| ≥ R then Note that since |α| is large, α cannot be preperiodic.
Although it might be desirable to have such a lower bound for the degree of each element of S α,n , this cannot hold as we could have for instance α ∈ Q and β = α. Instead one could ask about 'primitive' solutions β which do not satisfy P •m (β) = P •m (α) for m < n. We don't prove a bound on these, but we are able to show that the proportion of numbers in S α,n of low degree decays exponentially. We fix R as in Theorem 1.3 and α algebraic with |α| ≥ R. For δ > 0 let p δ,α,n = #{β ∈ S α,n : [Q(β) : Q] ≤ D δn } D n .
Then we have the following.
Theorem 1.4. There exists an effectively computable constant c ε depending only on ε, P and δ such that To prove Theorem 1.3 we use the theory of Böttcher maps in combination with the counting result, Theorem 1.1. It will be clear from the proof that the same result holds for any rational function with a superattracting fixed point at infinity of order D ≥ 2. This strategy of combining the Böttcher map with counting results also opens up the possibility of applying the Pila-Zannier strategy (see for instance [40]) to certain problems in arithmetic dynamics. We will pursue this elsewhere.
The proof of Theorem 1.1 develops the work of the first two authors in [7], but the proof is more involved as there is less room for large disks on the unit disk. The methods are quite flexible and by combining them with an idea from [8] we are able to give a similar result for functions which grow exponentially with respect to φ. We prove Theorem 1.5. Suppose that f is analytic in the unit disk and that there are positive reals a, b, c 0 with b > 1 such that |f (z)| ≤ φ(z) c 0 φ(z) for |z| ≥ 1/2 and |f (x)| ≥ ab φ(x) for x in (0, 1). Then there is a positive real number c with the following property. For H ≥ e the number of rationals q in (0, 1) such that f (q) is also rational and both q and f (q) have multiplicative height at most H is at most c(log H) 18 .
Here, c depends only on a, b and c 0 and can be computed from them.
In fact we give a more precise form for points of bounded degree, with polynomial dependence on the degree, and with growth on a possibly larger set S containing (0, 1).
In common with almost all counting results of Bombieri-Pila type, our proof requires the construction of a polynomial vanishing at the relevant points, and satisfying various other properties. This construction is carried out in the next section. Sections 3 and 4 then give the proofs of Theorems 1.1 and 1.5 respectively. In Section 5 we prove Theorems 1.3 and 1.4 and then in Section 6 we combine Theorem 1.3 with the p-adic methods mentioned above to prove Theorem 1.2. In a short final section we give some further examples of functions to which our counting results apply.
We are grateful to Patrick Ingram for pointing out the Böttcher map to us and to Tom Tucker for his suggestion of combining nonarchimedean and archimedean methods.
All three authors thank the Engineering and Physical Sciences Research Council for support under grant EP/N007956/1.

Polynomials for functions on the disk
Recall that for z in the unit disk we let φ(z) = 1 1−|z| . In this section we use Masser's polynomial construction [24, Proposition 2, page 2039] to capture algebraic points of bounded height and degree on algebraic curves. First we recall the height we will be using, so fix an algebraic number α with minimal polynomial P over the integers, and suppose that P has positive leading coefficient, a say. The height of α is defined to be H(α) = a max{1, |z|} 1 deg P with the product taken over the roots z of P . Below we will also be using H as a bound on the heights of the points considered. This shouldn't lead to confusion. Here is the result of Masser's that we need. Proposition 2.1. For any integer d ≥ 1 and real A > 0, Z > 0, M > 0, H ≥ 1, T ≥ √ 8d, let f 1 , f 2 be analytic functions on an open set containing the closed disk of radius 2Z around the origin and suppose |f 1 (z)|, |f 2 (z)| ≤ M on this disk. Let Z be a finite set of complex numbers and suppose that, for all z, z ′ ∈ Z, then there is a nonzero polynomial P of degree at most T such that P (f 1 (z), f 2 (z)) = 0 for all z in Z. Moreover P can be taken to have integer coefficients of modulus at most 2(T + 1) 2 H T .
Proof. Without the 'moreover' statement, this is Proposition 2 on page 2039 of Masser's paper [24] (except that there T was assumed integral, but this plays no role in the proof). The 'moreover' statement follows from Masser's proof, see the discussion on page 1145 of [7].
We will need the following simple lemma as in [32].
Lemma 2.2. Let D R be a closed disk of radius R. Then D R can be covered by k closed disks of radius r > 0, where We now suppose that f is analytic on the disk and that there is a c 0 > 0 such that Given positive reals a, b, with a ≥ b > 1, we let l a,b = log a log b . For the relevance of this number see Lemma 3.2.
There is a constant c 1 > 0 depending only on c 0 and effectively computable from it with the following property. If a and b are positive reals with a ≥ b e > 1 and a ≥ e then there is a nonzero polynomial P of degree at most Proof. We write c for various positive constants, all effectively computable from c 0 . Let D be the closed disk centred at 0 and of radius 1− with c absolute. Let a 1 , . . . , a N be the centres of these disks. We may assume a 1 , . . . , a N ∈ D. For n = 1, . . . , N we define the translates f 1,n (z) = z + a n , f 2,n (z) = f (z + a n ). We will apply Proposition 2.1 to each of these pairs of functions.
A . Let Z n be the set of α − a n such that: We now apply Proposition 2.1 with Z = Z n and f 1 = f 1,n , f 2 = f 2,n . Clearly we have requirements (c) and (d). And (a) and (b) are fine, by our choices of A, Z. For |z| ≤ 2Z we have So by (2) we can take To ensure that (1) holds, we can take Then the proposition gives a nonzero polynomial P n of degree at most T ′ with integer coefficients of modulus at most 2(T ′ + 1) 2 H T ′ such that P n (f 1,n (z), f 2,n (z)) = 0 for z in Z n . Let P be the product of all these P n . Then since N < c (l a,b d log H) 2 the polynomial P has degree at most T = cl 3 a,b log l a,b d 4 log d(log H) 3 log log H and if α satisfies (i) and (ii) in the statement of the lemma then P (α, f (α)) = 0. To complete the proof, we need to estimate the coefficients of P . To this end, we write And then for any α, β with α + β ≤ T we have with the summation taken over all sequences and For these n we certainly have i (n) , j (n) ≤ T ′ and so there are at most T ′ + 1 choices for each of i (n) , j (n) and then there are at most (T ′ + 1) 2N summands in (4). Each of these summands is a product of N integers each bounded in modulus by 2(T ′ + 1) 2 H T ′ and so each summand is bounded by 2 N (T ′ + 1) 2N H T ′ N and we can conclude that After perhaps rechoosing the constant in T , we have If we in fact have |f (z)| ≤ c 0 rather than just (2) then we proceed exactly as above, but take M = max{1, c 0 } in place of (3) above. We then find that for (1) we can take T ′ = cd 2 log H and then the proof continues as above, and leads to T = cl 2 a,b d 4 (log H) 3 .

Functions on the disk with decay
Recall that for z in the unit disk, φ(z) = 1 1−|z| . Suppose that f is analytic on the disk and that there are positive real a and b > 1 such that We assume (as we clearly can) that a ≥ b e > 1 and a ≥ e. Further that |f | ≤ 1. In this setting, we prove the following.
There is an absolute effectively computable constant c > 0 such that the number of algebraic α ∈ S such that f (α) is algebraic and non-zero with [Q(α, f (α)) : Q] ≤ d and This clearly implies Theorem 1.1 (apart from the "moreover" part which we shall deduce at the end of this section). We begin with a bound on the size of the α in the theorem. It is here that it is crucial Proof. By a fundamental property of the height we have Thus and so As in the earlier work of the first two authors [7,8] we use the following result to count zeros of functions. Let θ(z) = 1+z 1−z , then θ −1 (z) = z−1 z+1 . We need some properties of this transformation for the proof of the next theorem.
A Moebius transformation maps orthogonal circles to orthogonal circles. In particular, since θ maps the interval [−1, 1] to the real interval [0, +∞], it maps each circle of radius 0 < r < 1 centred at 0 to a circle C r that intersects the real line orthogonally in the points θ(−r) < 1, θ(r) > 1. Note that θ sends the unit circle to the imaginary line. Denote by U r the closed disk that has C r as its boundary and note that C r ′ is contained in U r for r ′ ≤ r. Thus U r is the image of the disk of radius r centred at 0 under θ. We need the following to continue.
Proof. We note thatθ(z) = θ(z)/s is again a Moebius transformation that sends [−1, 1] to [0, +∞]. It is enough to show thatθ sends the circle of radius r around 0 into U r ′ . For this it is enough to note that the image underθ of this circle intersects the real line orthogonally at the points θ(−r)/s ≥ θ(−r ′ ) and θ(r)/s ≤ θ(r ′ ).
We can now prove Theorem 3.1. For this we assume |f (z)| ≤ 1 on the unit disk. All the constants can be effectively computed. We will write c for various positive constants. Recall that d ≥ 2 and H ≥ e e . By Lemma 3.2 the α we wish to count satisfy |α| ≤ 1 − 1/2l a,b d log H. So we can apply Lemma 2.3 to find an absolute constant c 1 > 0 such that there is a nonzero polynomial P (X, Y ) of degree at most T = c 1 l 2 a,b d 4 (log H) 3 with integer coefficients satisfying |P | ≤ 2(T + 1) 2 H T (where |P | is the maximum modulus of the coefficients of P ) and such that P (α, f (α)) = 0 for all the α's we're counting. Fix such a P .
Note that for x > 1 we have φ(θ −1 (x)) = 1 2 (x + 1) and so for these x we have (6) |f (θ −1 (x))| ≤ a b 1 2 (x+1) . Let R(X) = P (X, 0) and Q = P − R. We can assume that R is nonzero since if R is the zero polynomial then P is divisible by Y and we can divide by Y until the corresponding R is non-zero, and this doesn't affect us since we're only counting α's for which f (α) = 0. Note that Q is divisible by Y . We now seek a point x 0 such that |P (θ −1 (x 0 ), f (θ −1 (x 0 )))| is not too small (and is, in particular, nonzero). We consider Q and R separately, and first find a large interval on which |Q(θ −1 (x), f (θ −1 (x)))| is very small. More precisely, we want an interval of length at least T + 1 on which Since Y divides Q there is a polynomial Q ′ of degree at most T and with |Q ′ | ≤ |Q| such that Q = Y Q ′ . We have (6) for the second inequality. We know that |Q| ≤ |P | ≤ 2(T + 1) 2 H T and so log |Q| ≤ cT log H.
Combining the last two estimates shows that for (7) it suffices to have log 2 + T log(x + 1) + log a + 2 log(T + 1) For this it is sufficient to ensure that l −1 a,b (x + 1) ≥ max{cT log(x + 1), cT log H}. So we can choose x around cl a,b T log T log H/ log log H and so we can take [cl a,b T log T log H/ log log H, 2cl a,b T log T log H/ log log H] as our interval.
We now consider R. The function is a polynomial in X of degree at most T with integer coefficients (we write ⌊T ⌋ for the integer part of T ). Since our interval has length greater than T + 1 there is an integer m inside it such that (m + 1) ⌊T ⌋ R(θ −1 (m)) is not zero. As R has integer coefficients this number is also an integer and we have But m is in the interval on which (7) holds and so we find that Now suppose that α in S is algebraic and such that f (α) is also algebraic and non-zero, and [Q(α, f (α)) : Q] ≤ d and H(α), H(f (α)) ≤ H. By Lemma 3.2 we have (9) |α| ≤ 1 − 1 2l a,b d log H .
Then θ −1 (θ(α)/m) is a zero of g and we want to bound the absolute value of θ −1 (θ(α)/m). To this end we find 0 ≤ r < 1 such that r ≥ |α| and θ(−r) ≤ θ(−|α|)/m. By (9) and the fact that it is sufficient to find an r such that 2 and, by Lemma 3.4, |θ −1 (θ(α)/m)| ≤ r. So it suffices to count the zeros of g in a disk of radius r. Then let Now we can use some standard estimates to deduce that For example as exp(t) ≤ 1 + 2t for 0 ≤ t ≤ 1 we deduce that We also bound − log |g(0)|. From (8) 9 .
In an application that we give later, we only need count points (α, f (α)) for α in a compact subset of the disk. In this case, we can improve the bound.
To begin, we need only take T = cd 2 log H, where c now depends on the compact subset. We proceed as above, but take with c again depending on the compact set. We then find a final bound of cl a,b log l a,b d 4 (log d) 2 (log H) 4 .

Functions with growth
We now prove Theorem 1.5. Large parts of the proof are very similar to the previous proof and we leave these parts for the reader. Suppose that f is analytic on the unit disk and that there are positive real numbers a, c 0 and b > 1 such that (2) holds for |z| ≥ 1/2 and such that for x ∈ S |f (x)| ≥ ab φ(x) .
As before, S is a subset of the unit disk containing the interval (0, 1) and l a,b = log a log b . We assume a ≥ max{b e , e}. (In the case where this extra assumption is not satisfied, one could obtain the conclusion of Theorem 1.5 by applying the following result to a function obtained from f by scaling.)  Then We can now start the proof of Theorem 4.1. We write c for various positive constants which are effectively computable from c 0 . Applying Lemma 2.3 gives us a nonzero polynomial P of degree at most T = cl 3 a,b log l a,b d 4 log d(log H) 3 log log H with integer coefficients and with |P | ≤ 2(T + 1) 2 H T such that P (α, f (α)) = 0 for the α's we're interested in. Let θ and θ −1 be as in the previous proof. For real x > 1 we have φ(θ −1 (x)) = 1 2 (x + 1) and so (10) |f (θ −1 (x))| ≥ ab 1 2 (x+1) . Let L ≤ T be the degree of P in Y , and write where P 0 , . . . , P L are polynomials in X with integer coefficients satisfying |P i | ≤ |P |. Let By (10) f (θ −1 (x)) is nonzero for x > 1 and so .
We will use this to find an x which isn't too large and is such that P (θ −1 (x), f (θ −1 (x))) isn't too small, much as we did in the previous proof. Let R(X) = P ′ (X, 0)(= P L (X)) and Q(X, Y ) = P ′ (X, Y ) − R(X). Note that R(X) is not the zero polynomial, by the definition of L. For now, suppose that Q is also not the zero polynomial. Since Y divides Q we can argue exactly as we did after (7) to show that We can then proceed as before to find an m in this interval such that Then by (11) we also have In fact, |P (θ −1 (m), f (θ −1 (m)))| is a bit larger than this, but we don't seem to be able to exploit this. Anyway, we can now proceed as in the previous proof. We see that it suffices to count the zeros of g(z) = P (θ −1 (mθ(z)), f (θ −1 (mθ(z)))) in a disk of radius We estimate this as before, except that now the bound on M is larger as our f is now growing. Let Using reasoning from the previous section we get 1 log(R/r) ≤ cl 2 a,b dT log T (log H) 2 / log log H.
Before estimating M = max{|g(z)| : |z| ≤ R} we first estimate M ′ = max{|f (θ −1 (mθ(z)))| : |z| ≤ R}. For |z| ≤ R we have and so by (2) we have But what if Q is the zero polynomial? Then our original P has the form P (X, Y ) = P L (X)Y L . The first factor has at most T zeros since it is a polynomial of degree at most T . The second factor is non-zero at all the points we care about. So we end up with a much better bound in this case. This completes the proof.

Dynamics over C
In this section we work in C and assume that we have fixed an embedding of Q in C. We consider the dynamical system associated to a polynomial P ∈ K[z] of degree D ≥ 2 where K is a number field. In what follows we may and will assume that P is monic. This is because we can make P monic by passing to a conjugate γ −1 P (γX) for γ = 0 lying in an extension of degree at most D − 1 over a field of definition for P . The filled Julia set of P is the set of z ∈ C such that P •n (z) ∞. For each such polynomial P (in fact over C) there exists a neighbourhood U ∞ of ∞ contained in the complemement of the filled Julia set and a bi-holomophism is an open disk of radius r centred at 0, that satisfies and Φ P (P (z)) = Φ P (z) D .
We will assume that 0 is not contained in U ∞ . This is a theorem of Böttcher [25,Theorem 6.7]. For example if P = z D then Φ P = 1/z and we can pick U ∞ = {z : |z| > 1}. In fact if all critical points of P lie in its filled Julia set then we can always take U ∞ to be the complement of the filled Julia set of P and r = 1. In general we can't continue Φ P as a holomorphic function to the whole complement of the filled Julia set [25, p.6-7].
In what follows we pick U ∞ such that Φ −1 P can be extended to D(0, r 1/D ) and such that r ≤ e −π/6 . This choice can be made effectively and makes the computations more transparent. For any α ∈ D(0, r) we can now define the function This f * has the property that with S α,n as defined on page 2, and further 0 < |f * (τ )| < c exp(−2πℑ(τ )) (14) for some effective positive c depending only on P . We pull back f * to the unit disk via µ(z) = i(1 + z)/(1 − z) and by (14) this pullback f = f * • µ and the set satisfy the conditions of Theorem 1.1 (with effectively computable a, b).
Recall that to each polynomial (or even just rational) map P we can associate a canonical dynamical heightĥ P : with the propertyĥ P (P (z)) = Dĥ P (z) and for a constant c P,h depending (effectively) only on P (in particular h P = 0) [35,Theorem 3.20].
Proof of Theorems 1.3 and 1.4. Sinceĥ P (β) =ĥ P (α) for β ∈ S α,n we deduce from the above that H( where d is a bound for the degree of f (k/D N + i/24) over Q and c is effective. Theorem 1.3 now follows from (13). Now pick a set S δ ⊂ {k/D N + i/24; k = ⌊− D n −1 2 ⌋, . . . , ⌊ D n −1 2 ⌋} such that f (µ −1 (τ )) has degree at most D δn for τ ∈ S δ . Plugging this bound in Theorem 1.1 for our f we deduce Theorem 1.4. Now we want to investigate how the involved constants behave if we vary P in a family of polynomials.
Let V be a quasi-affine variety over a number field K with coordinate functions T = (T 1 , . . . , T m ) on A m and let P ∈ K[X], where K = K(T ), of degree deg(P ) = D ≥ 2. (Here we work with X instead of z to distinguish it from the complex variable.) For each c ∈ V (Q) such that the coefficients of P are defined at c we can specialize to P c ∈ Q[X] and associate the canonical dynamical heightĥ c =ĥ Pc to P c . We define the Weil-height on V in the usual way for (t 1 , . . . , t m ) = T (c) where K is a number field containing t 1 , . . . , t m , M K is the set of places of K, suitably normalized such that |p| v = p −1 for a prime p satisfying v|p, and n v is the local degree [K v : Q v ]. We shrink V if necessary such that P c is always defined and such that deg P c = deg P for all c ∈ V (Q).
for δ 1 , δ 2 effectively computable positive real constants depending only on P .
Proof. We set bP = a 0 X D +· · ·+a D with a 0 , . . . , where O K is the ring of integers of K, a 0 , . . . , a D , b are co-prime and we set D V to be the maximum of the degrees of b, a i , i = 0, . . . , D.
It is more convenient to work withP = P (1/X) = A/(bX D ) with A ∈ O K [T, X]. Note that (A, bX D ) = 1 and that this holds for every specialization P c that we consider. By the theory of resultants there exist polynomials . Now we can follow the proof in [16, section 5.1] but we repeat some of the arguments for the reader's convenience. Let (t 1 , . . . , t m ) = T (c) be the coordinates of the specialized point, r = R(c) and x be an algebraic specialization of X. At a nonarchimedian place | · | we have that |r| and |rx 2D−1 | and therefore also |r| max{1, |x| 2D−1 } are bounded above by max{1, |t 1 |, . . . , |t m |} D V (2D−1) max{1, |x|} D−1 max{|A|, |bx D |}.
At an archimedean place we get |r| max{1, |x| 2D−1 } bounded above by Taking the product over all places the factor r cancels out and we obtain Proof. For the proof we just have to follow the proof in [25, Theorem 6.7] and we repeat some of the arguments. We can write and since the coefficients of P (z) are uniformly bounded on B, the O(1/|z|) term is bounded uniformly for |z| > r > 0. We set Z such that z = exp(Z) and set F (Z) = log P (exp(Z)). With the right lifting of F we get F (Z) = DZ + O(exp(−ℜ(Z))) and we can choose σ so large (and r so big) such that for ℜ(Z) > σ, |F (Z) − DZ| < 1. Now the sequence of functions L k = F •k (Z)/D k converges uniformly on this domain to a function L that satisfies L • F = DL. Moreover it satisfies L(Z + 2πi) = L(Z) + 2πi. Thus the function Φ(z) = exp(−L(log z)) is well defined on D(∞, r) and satisfies Φ(P ) = Φ D (Note that |P (z)| > r for |z| > r.) Moreover as the convergence is uniform in B the statement about holomorphy follows. Moreover we can bound the points of low degree in S α,n,c . For every ε > 0 and δ > 0 there exists a constant c ε such that the number of points in S α,n,c of degree at most D δn divided by D n is bounded above by Proof. For the proof, given Lemma 5.1, we only have to make sure that the constant in Theorem 1.1 can be chosen uniformly for c ∈ B. From the construction of the Böttcher map it follows that we only need a uniform bound on the coefficients of P , which we have.
We note that if we have an effective bound for the length of P c as c varies over B all the constants in Theorem 5.3 are effective.

General Galois-bounds
We now prove Theorem 1.2.
Theorem 6.1. Suppose that K is a number field and that P ∈ K[X] has degree D at least 2. Let ε > 0. There exist c > 0, depending only on P and ε, and c ′ > 0 depending only on P such that if α ∈ K and As before, we may assume P is monic. Now fix a number field K. In what follows, given a prime p we write | · | v for the extension of a p-adic valuation on Q to K with the standard normalization and K v for the completion of K with respect to v. We write C v for the completion of the algebraic closure of K v . And we let D v (0, r) and D v (∞, r) denote the sets of points in C v such that |z| v < r and |z| v > r, respectively. Suppose that P (z) = z D + a 1 z D−1 + · · · + a D is a polynomial over K. For primes p not dividing D we define And for primes p that do divide D we set We will use the following result, due to De Marco et al ([14, Theorem 6.5]) which extends work of Ingram ([17]).
Theorem 6.2 (Ingram [17], De Marco et al [14]). In the setting described above, there exists an injective analytic function Moreover, Φ v has the property that if z lies in a finite extension of K v then this extension also contains Φ v (z).
Before we proceed with the proof we need more information about cyclotomic extensions of p-adic fields. First some (basic) group theory. Lemma 6.3. For a prime p let a be an integer coprime to p. If p = 2 let e be the order of a in (Z/pZ) * and m be maximal with the property a e = 1 mod p m . If p = 2 let e be the order of a in (Z/4Z) * and m be maximal such that a e = 1 mod 2 m . For n ≥ m the order of a in (Z/p n Z) * is ep n−m .
Proof. We prove this by induction on n ≥ m, and prove simultaneously that n is maximal such that a ep n−m = 1 mod p n . (Where we mean maximal on the right side of the equality for fixed n − m). For n = m this is our assumption, so assume that the result holds for some n ≥ m. So a ep n−m = 1 mod p n and so a ep n−m = 1 + bp n for some b. If p|b then a ep n−m = 1 mod p n+1 , a contradiction. So (b, p) = 1. We have If p > 2 then we have n ≥ 2, and if p = 2 then n ≥ 3. Either way, the sum on the right of this expression will be divisible by p n+2 , and so a ep n+1−m = 1 +bp n+1 for someb with (b, p) = 1. Thus n+1 is maximal such that a ep n+1−m = 1 mod p n+1 .
The order, f say, of a in (Z/p n+1 Z) * , divides ep n+1−m . On the other hand e|f , since a f = 1 mod p. So N = ep k for some k ≤ n + 1 − m. If k < n + 1 − m then k = l − m for some l < n + 1, and then a ep l−m = 1 mod p n+1 , contradicting our inductive assumption on the maximality of l. So f = ep n+1−m , as required. Now we can deduce the following. Lemma 6.4. Let a and D be coprime positive integers. For a prime q dividing D let m q be as m in Lemma 6.3. Let m be maximal among these m q . Then for n at least m, the order of a in (Z/D n Z) * is at least D n−m .
Proof. Let D = q|D q nq be the prime decomposition of D. Suppose that a f = 1 mod D n . Then a f = 1 mod q nnq for all q. By the previous lemma, q nnq−mq divides f , and so q|D q nnq−mq divides f . And we're done since q|D q nnq−mq ≥ D n−m .
For a positive integer l we denote by ζ l a primitive l-th root of unity. Proof. By [26,Proposition 7.12] the extension K is unramified over Q p . In particular p stays prime in K. Hence we can apply the Eisenstein criterion to the cyclotomic polynomial φ p k in K just as in the proof of [26,Proposition 7.13].
Combining the previous lemmas we obtain the following. Proof. We write D = p npD where n p ,D are positive integers such that p does not divideD. Since Q p (ζ D n ) = Q p (ζD n , ζ p np ) we obtain the corollary from Lemmas 6.5 and 6.6.
Before we prove Theorem 6.1 we prove the p-adic version of the Galois bounds. These turn out to have a stronger asymptotic than the complex ones but in general with a worse dependence on the point α. Proof. For z ∈ D(∞, δ v ) we have |P •n (z)| v = |z| D n v . From this we can deduce that |β| v = |α| v for β ∈ S α,n . Moreover, the pre-images of a critical point of P lie outside of D(∞, δ v ) for all places v. Thus P •n (X) − P •n (α) has non-vanishing discriminant and so D n distinct solutions. Hence as Φ v is injective there exists β ∈ S α,n such that Φ v (β) = Φ v (α)ζ D n . By the properties of Φ v , So by our choice of β we have ζ D n ∈ Q p (β, α) and so by Lemma  From this point on we also allow v to be archimedean. For the archimedean places v we set δ v = R as in Theorem 1.3. In order to prove Theorem 6.1, we first observe that the set of α such that neither Theorem 1.3 nor Theorem 6.8 apply is a set of bounded height. In the following result K is, as before, a number field over which P is defined. However this time we do not require α ∈ K. For any number field K, we denote by M K the set of places on K normalized in the usual way. Proof of Theorem 6.1. By Lemma 6.9 the set of α ∈ Q such that |α| v ≤ δ v for all places v has bounded height, say by a constant c 1 . Recall that |ĥ P − h| ≤ c 2 where c 2 only depends on P . If h(α) > c 1 and so |α| v > δ v for some v, we can apply Theorems 6.8 or 1.3. If h(α) ≤ c 1 andĥ P (α) > 0 then we pick a positive integer k such thatĥ P (P •k (α)) = D kĥ P (α) > c 1 + c 2 . It is clear that [Q(α) : Q(P •k (α))] ≤ D k and, since P •n (α) = P •n (β) implies P •n (P •k (α)) = P •n (P •k (β)), we can again apply Theorem 1.3 or Theorem 6.8 to P •k (α). If we pick k minimal with the described property we get (1 + 2 ∞ n=1 q n 2 ) 4 be the modular λ-function, where q = exp(πiτ ). See for instance chapter 7, section 7 of [10]. By (8.1) on page 117 of [10] λ is bounded on the half plane ℑτ ≥ 1 and there is some c > 0 such that (15) 0 < |λ(τ )| < ce −πℑτ for τ with sufficiently large imaginary part, and real part in [−1, 1] say. As in section 5 we construct a related function on the disk. In order to obtain a bounded function on the disk, we don't work with the whole upper-half plane, but the half-plane given by ℑτ > 1. The transformation z → 2i 1−z takes the unit disk to this half plane and so the function λ * (z) = λ 2i 1 − z is bounded on the unit disk. Let S be the union of the interval (0, 1) with the inverse image under the Moebius transformation above of the set S ′ of τ with real part in [−1, 1] and imaginary part large enough so that (15) holds. Bounds on the number of algebraic points of bounded height and degree on the graph of the function λ * and on the graph of λ restricted to ℑτ > 1 are clearly equivalent, with suitable changes in constants. And by (15) above the function λ * satisfies the hypotheses of our Theorem 1.1. So we get a bound c ′ d 9 (log d) 2 (log H) 9 for algebraic points of degree at most d and height at most H (with d ≥ 2 and H ≥ e) on the graph of λ restricted to the set S ′ . Here c ′ is absolute, and could in principle be computed. This is weak compared to Schneider's theorem [1, Theorem 6.3, page 56], which implies that λ(τ ) will be transcendental when τ is algebraic and not quadratic. But exactly the same argument applies to the derivatives λ ′ , λ ′′ of λ, and leads to results which appear to be new, although they could perhaps also be obtained using Binyamini's result in [5]. (For what is known about transcendence here see [15]).
Similarly, our result applies to the discriminant function ∆(τ ) = (2π) 12 q n≥1 (1 − q n ) 24 where we now use q = exp(2πiτ ) and to other cusp forms (see for instance Theorem 8.1 on page 62 and Proposition 7.4 on page 59 of [34]). Indeed it applies to any modular form n≥0 c n q n with algebraic c 0 , by applying the above method to n≥1 c n q n .