On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document}H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document}H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in \{0,d\}$$\end{document}k∈{0,d} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (k,d) = 1$$\end{document}gcd(k,d)=1. We therefore study the behaviour in the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< k < d$$\end{document}0 1$$\end{document}gcd(k,d)>1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.


Introduction
LetQ denote the algebraic closure of Q in C. For an algebraic number α ∈Q, let H(α) denote the (absolute multiplicative Weil) height of α, as defined in Section 1.5 of [3]. We have H(α) ∈Q ∩ [1, ∞) for all α ∈Q. By a well-known theorem of Northcott [18] (see also Theorem 1.6.8 in [3]), there are at most finitely many algebraic numbers of bounded degree and bounded height. This article seeks to answer the question: "How many α ∈Q are there of fixed degree d and fixed height H?" In particular, we are interested in whether the height assumes many values, but each single value is assumed only rarely, or whether there are only few values that are however assumed very often. For fundamental properties of the height, we refer to Section 1.5 of [3].
Much is known about counting algebraic numbers or more generally points in P n (Q) of fixed degree (over Q or over any fixed number field) and bounded height: Schanuel first proved, in [21], an asymptotic for the number of algebraic points of bounded height that are defined over a fixed number field. Further results, including the asymptotic for the number of quadratic points (over Q) of bounded height, were obtained by Schmidt in [22] and [23]. If n is larger than the degree of the point (over Q), then Gao found and proved the correct asymptotic in [12]. He also determined the correct order of magnitude for any n and any degree (over Q). Masser and Vaaler then counted algebraic numbers of fixed degree and bounded height in [17] (over Q) and [16] (over any fixed number field). If the degree of the point (over any fixed number field) is at most slightly less than 2n 5 , then Widmer obtained the correct asymptotic in [28]. More recently, Guignard [14] counted quadratic points (over any fixed number field) if n ≥ 3; he also counted points whose degree (over any fixed number field) is an odd prime less than or equal to n − 2. However, Guignard uses a slightly different height, corresponding to another choice of norm at the infinite places.
The same problem has also been studied for integral points, i.e. elements ofQ n whose coordinates are algebraic integers: In Theorem 5.2 in Chapter 3 of [15], Lang gives an asymptotic for the number of algebraic integers of bounded height that lie in a fixed number field (with an unspecified constant in the main term). The work [7] of Chern and Vaaler, which was also used crucially in [17], yields an asymptotic for the number of algebraic integers of fixed degree over Q and bounded height. In [1], Barroero extended the results of Lang and Chern and Vaaler to count algebraic integers of fixed degree over any fixed number field and bounded height. Widmer counted, in [29], integral points of fixed degree (over any fixed number field) and bounded height under the assumption that the degree of the point is either 1 or at most slightly less than n. In [13], Grizzard and Gunther counted (among other things) algebraic integers of fixed degree (over Q), fixed norm, and bounded height. This last result is somewhat related to our work in that the d-th power of the height of an algebraic integer of degree d (over Q) with no conjugate inside the open unit disk is equal to the absolute value of its norm.
We emphasize that all these results give much more precise asymptotics than the ones obtained in this article. However, already when counting rational numbers of fixed height, Euler's phi function appears, so it is clear that such precise asymptotics cannot be obtained in general when counting algebraic numbers of fixed degree and fixed height. Instead, we strive to obtain an asymptotic for the logarithm of the associated counting function.
In order to state our results, we have to introduce some notation: The conjugates (over Q) of an algebraic number are the complex zeroes of its minimal polynomial over Q. While there is no nice asymptotic for the logarithm of the counting function associated to our question from the beginning, we have managed to obtain such an asymptotic in many cases if the number of conjugates that lie inside the open unit disk is also prescribed. if these limits exist. It will follow from Lemma 3.1 that A(k, d) is an infinite set. Thus, B(k, d) contains arbitrarily large elements and at least the limit superior and inferior corresponding to a(k, d) certainly exist. We remark that it is not clear if the conjugates inside the open unit disk are the right thing to take into account here. The Galois group of the normal closure of Q(α), the degree [Q H(α) d : Q], and the normal closure of Q H(α) d also seem to play an important role as will become apparent. Of course, these objects are not independent of one another (e.g. k ∈ {0, d} is equivalent to [Q H(α) d : Q] = 1).
The main results of this article can be summarized as follows: Demanding that the action of the Galois group of the normal closure of Q(α) on the conjugates of α ∈ A(k, d, H) is sufficiently generic implies that there are few such α if 0 < k < d. The following is our strongest result in this direction: We also show that the height function together with the degree is in some sense "almost injective" if the degree is at least 2: If d = 4 and k = 2, then we obtain finer results than those given by In the construction in the proof of Theorem 6.2, the field Q(H 4 ) is made to vary in an infinite set unless κ = 4. This suggests that in general fixing the field Q(H d ) might lead to more uniform growth behaviour. The following is a simplified version of Theorem 7.1: Theorem 1.5 Let δ ∈ (0, 1) and > 0 and let K ⊂Q be a fixed Galois extension of Q. Let k, d ∈ N such that 0 < k < d and let H ∈Q ∩ [1, ∞) such that the normal closure of Q(H d ) is equal to K .
Suppose that α ∈ A(k, d, H). Set L = Q(α) ∩ K and l = d[L : Q] −1 and let β ∈ L be the Q(α)/L-norm of α. There exists a constant C = C(k, d, K, δ, ) > 0 such that if for every field embedding σ : Theorem 7.1 is then used together with upper bounds for |A(k, d, H)| from Theorem 5.2 for the determination of the limit superior corresponding to a(k, d) in Theorem 7.5. We also give examples that show the necessity of the dependence of C on K and δ.
In Sect. 8, we count polynomials with integer coefficients of fixed degree d and fixed Mahler measure M as defined in Section 1.6.4 of [3]. Among these polynomials, those that are irreducible in Z[t] are in a 1-to-d correspondence with the algebraic numbers of degree d and height M 1 d . However, we also count the polynomials that are reducible in Z[t] and this leads to somewhat simpler results although even fewer of the considered limits exist. We obtain Theorem 8.1, an analogue of Theorem 1.1 in this context.
Following a suggestion of Norbert A'Campo, we study the dynamical behaviour of the height function in Sect. 9. The dynamical behaviour of the Mahler measure has been studied initially by Dubickas in [9] and [10] and subsequently by Zhang in [31] as well as by Fili, Pottmeyer, and Zhang in [11]. We obtain the following result: Theorem 1.6 (= Theorem 9.3). Let α ∈Q and define inductively α 0 = α, α n = H(α n−1 ) (n ∈ N). Then either there exist N, a ∈ N and b ∈ Q, b > 0, such that α n = a b for all n ≥ N or lim n→∞ α n = 1.
In particular, the periodic points of H are precisely the a b for a ∈ N and b ∈ Q, b > 0. Our proofs are mostly elementary. Our constructions of many algebraic numbers of a given height rely on point counting results for lattices by  (in the proofs of Theorem 2.1 and Lemma 3.1) and Widmer in [30] (in the proof of Theorem 7.1). The first of these results generalizes a theorem of Davenport in [8] while the second one generalizes a theorem of Skriganov in [25].
The main result of [2] is formulated in an arbitrary o-minimal structure; we will however apply it only in the structure of semialgebraic sets, where a subset of R n (n ∈ N) is called semialgebraic or definable (in the structure of semialgebraic sets) if it is a finite union of sets defined by a finite number of polynomial equations and inequalities with real coefficients. By the Seidenberg-Tarski theorem, the structure of semialgebraic sets is o-minimal, which implies that besides polynomial equations and inequalities with real coefficients, we can also use existential and universal quantifiers to define semialgebraic sets. For a general introduction to o-minimal structures, see [26].
For a real number ξ , we denote by [ξ ] the largest integer which does not exceed ξ . We use φ to denote Euler's phi function and μ to denote the Möbius function. For a finite field extension L/K , we denote the corresponding field norm by N L/K . If K is a number field, then we denote its ring of integers by O K . The norm of an ideal I of O K is denoted by N (I). The imaginary unit in C is denoted by √ −1 and the real and imaginary part of a complex number are denoted by Re and Im respectively. For a real-valued function f on S ⊂ R n , we write O(f ) for any function g : S → R such that there exists a constant C = C(f, g) ≥ 0 with |g(s)| ≤ Cf (s) for all s ∈ S. If n = 1, S is unbounded, and f (s) > 0 for |s| large enough, we say that a function g : If α is an algebraic number of degree d, a minimal polynomial of α in Z[t] is an irreducible element of Z[t] that has α as a zero. There are two choices for a minimal polynomial of α in Z[t] as (Z[t]) * = {±1}. The following simple observation will be used at different places throughout this article: If a is the leading coefficient of a minimal polynomial of α in Z[t] and α 1 , . . . , α d−k are the conjugates of α that lie outside the open unit disk, then H(α) d = |a||α 1 | · · · |α d−k | = ±aα 1 · · · α d−k (see Propositions 1.6.5 and 1.6.6 in [3]). We can write ±α i instead of |α i | (i = 1, . . . , d − k) since the non-real conjugates appear in complex conjugate pairs and the real conjugates are equal to their absolute value up to sign.

The case k ∈ {0, d}
In this section, we treat the case where k ∈ {0, d}, which is the easiest one to resolve. Theorem 2.1 Let d ∈ N. The following hold: (ii) Let us define and g j (x 1 , y 1 , . . . , x d , y d ) = The set Z is definable in the o-minimal structure of all semialgebraic subsets of R n (n ∈ N). Let π : R d × R → R d be the canonical projection. For T ∈ R, T = 0, the set Z T = π(Z ∩(R d ×{T })) parametrizes polynomials w 0 t d +· · ·+w d−1 t +T of degree d with real coefficients and positive leading coefficient that have no complex zeroes inside the open unit disk and whose constant coefficient is equal to T . Note that Z T = |T | · Z T /|T | (T = 0) and that the coordinates of a point in Z T can all be bounded by some constant multiple of |T |, depending on d. It follows that the volume of Z T is |T | d times the volume of Z T /|T | (T = 0) and that the volume of any orthogonal projection of Z T on some j-dimensional coordinate subspace of R d has j-dimensional volume at most a constant multiple of |T | j , depending on d.
We then deduce from Theorem 1.3 in [2] that Here and in the rest of this proof, the implicit constants in the O notation depend only on d. The volume V u is positive for ; a > 0, gcd(a, . . . , ±T ) = 1, all complex zeroes of P are at least 1 in absolute value}|, then we have N d (T ) = S|TÑ d (S). Using Möbius inversion together with an elementary bound for the divisor function, we deduce that Here S|T μ(S) In fact, for d ≥ 2, the product is at least ∞ k=2 1 − 1 k 2 = 1 2 , so bounded from below uniformly. What we really want iŝ and all complex zeroes of P are at least 1 in absolute value}| Hence, we obtain that  If [Q(α) : Q] = d ≥ 2 and H(α) = 1 for some α ∈Q, then α is a root of unity by Kronecker's theorem (Theorem 1.5.9 in [3]), so α ∈ A(0, d, 1) and d = φ(n) for some n ∈ N. On the other hand, if d = φ(n) for some n ∈ N, then any primitive n-th root of unity belongs to A(0, d, 1). It follows that 1 never belongs to B(d, d) if d > 1 and that 1 belongs to B(0, d) if and only if d ∈ φ(N).
Let now N be a natural number that is greater than or equal to 2. We want to show that the positive real d-th root N 1 d of N belongs to B(0, d) ∩ B (d, d). For this, we define a natural number p as follows: If N = 2, we set p = 1. If N ≥ 3, then we let p ∈ N be a prime number such that p < N and p does not divide N . Such a prime number always exists: If N = 3, we set p = 2. If N ≥ 4 and no such prime number existed, then N would be divisible by the product of all prime numbers that are smaller than N . Now − 1 ≥ 5 must have a prime factor and this prime factor must be greater than or equal to N . It follows that ≤ N ≤ − 1, a contradiction. The

Some useful lemmas
In this section, we collect some simple but useful lemmas. The first one shows that specifying the number of conjugates inside the open unit disk does not change the growth rate obtained by Masser and Vaaler in [17].  We can again apply Theorem 1.3 from [2] to the following definable family of semialgebraic sets: where and for j = 1, . . . , d and the σ j are again the elementary symmetric polynomials in d variables.
If againZ T = π(Z ∩ (R d+1 × {T })) for the projection π : R d+1 × R → R d+1 and T ≥ 1, then it is easy to see that all coordinates of a point inZ T are bounded by some constant multiple of T , depending on d, thatZ T = T ·Z 1 , and thatZ 1 has non-empty interior. Similarly as above, N d,k (T ) := |Z T ∩Z d+1 | counts the number of polynomials P(t) ∈ Z[t] of degree d with positive leading coefficient and precisely k complex zeroes inside the open unit disk (counted with multiplicities) such that the product of the leading coefficient and the absolute values of the complex zeroes outside the open unit disk, each absolute value raised to the power of the respective zero's multiplicity, is at most T . IfÑ d,k (T ) denotes the number of such polynomials with coprime coefficients, then we have that

Using another Möbius inversion and Theorem 1.3 from [2], we deduce that
for some constant C > 0, where C as well as the implicit constant in the O notation depend only on d and k. The proof of Lemma 2 in [17] shows that the number of reducible polynomials that we count in this way is of lower growth order. We can therefore deduce the lemma by setting T = H d .
The next lemma follows straightforwardly from Lemma 3.1.

Lemma 3.2 Let d ∈ N and k ∈ {0, . . . , d}. If the limits a(k, d) and b(k, d) both exist, then a(k, d)
Proof If they added up to some smaller number, we would immediately obtain a contradiction with Lemma 3.1 for H big enough, so suppose they add up to some bigger number. If b(k, d) = 0, then a(k, d) > d(d + 1) and we immediately get a contradiction with Lemma 3.1 for H big enough. So we can assume that b(k, d) > 0.
We can find some and δ < , the right-hand side grows asymptotically faster than The next lemma is the first and weakest in a series of results saying that for k ∈ {1, . . . , d− 1}, there cannot exist too many α ∈ A(k, d, H) whose Galois group is "large". Furthermore, we have and assume either that the Galois group of the normal closure of Q(α) acts transitively on the k-element subsets of the set of conjugates of α and α 1 , . . . , α k are the conjugates of α that do not lie inside the open unit disk. By assumption, we have 0 < k < d. We can assume without loss of generality that α = α 1 since α 1 determines α up to d possibilities. Now note that where α k+1 is a conjugate of α, distinct from the α j (j = 1, . . . , k) (here we use that k < d).
The numerator and denominator of the right-hand side are products of conjugates of ±H d by our assumption on either the Galois group of the normal closure of Q(α) or the degree of H d . So aα k is determined by H up to finitely many possibilities (bounded in terms of only d and k), so it can be assumed fixed. The same holds for aα k j for all j = 1, . . . , d by conjugating. And aα k together with a determines α up to k possibilities (here we need that k > 0), so it remains to bound the number of possibilities for a.
But a d−k |b| k = d j=1 a|α j | k is already determined up to finitely many possibilities , and a has to divide this natural number as k < d. Since | d j=1 aα k j | ≤ H d 2 , it follows from well-known bounds for the divisor function that there are at most C (d, )H possibilities for a.
The next two lemmas contain general facts from algebraic number theory that will be useful at several places in this article. Proof Let v be a finite place of Q(α 1 , . . . , α d ) and | · | v an associated absolute value. We have From the Gauss lemma (Lemma 1.6.3 in [3]) and the definition of a, we deduce that As v was arbitrary, the lemma follows.
There exists a constant C = C(D, ) such that Lemma 3.5 essentially follows from the proof of Proposition 2.5 in [5]. For the reader's convenience, we reproduce the proof here.
Proof We can assume without loss of generality that 0 < |N | ≤ H D since otherwise the set whose cardinality we wish to bound has at most one element.
Let U K denote the group of algebraic units in K . We call two elements of K \{0} associate if their quotient belongs to U K . It follows from [4], pp. 219-220, our bound for |N | in terms of H, and elementary bounds for the divisor function that the number of pairwise non-associate elements of Hence we can assume that α = α 0 ξ for some ξ ∈ U K and fixed α 0 ∈ O K with N K /Q (α) = N K /Q (α 0 ) = N and max{H(α), H(α 0 )} ≤ H. It follows that H(ξ ) ≤ H 2 . We want to bound the number of possibilities for ξ .
Let σ i : K → C denote the distinct embeddings of K in C (i = 1, . . . , D) and set for η ∈ U K . Then ν is a group homomorphism from the multiplicative group U K to the additive group R D and its image ν( This cube can be covered by at most C (D, )H 2 translates of the unit cube [0, 1] D . Since ν is a group homomorphism, it therefore suffices to show that the number of then the absolute value of each coefficient of P η is bounded by exp(D(1 + log 2)). This completes the proof of the lemma.

The case k ∈ {1, d − 1} or d prime
In this section, we completely resolve the cases where k ∈ {1, d − 1} or d is prime. We also determine b(k, d) for all k and d. Although many of the results in this section will be superseded by Theorem 5.2, we have included them because they can be proved in a different, somewhat easier way.
(In particular, all these limits exist.) Proof (i) This follows from Lemma 3.3 as the Galois group of the normal closure of Q(α) always acts transitively on the 1-element and the (d − 1)-element subsets of the set of conjugates of α.
(ii) It follows from Lemma 3.
with Galois group isomorphic to the full symmetric group S d is bounded from above by CH 2 for some constant C = C(k, d, ). Furthermore, the number of α ∈ H ≤H A(k, d, H ) with Galois group not isomorphic to the full symmetric group is of growth order o H d(d+1) (see [27]). But by Lemma 3.1, the number of α of degree d with precisely k conjugates inside the open unit disk and height at most H grows asymptotically like some constant positive multiple of H d(d+1) , which yields a contradiction for H large enough.
(iii) We follow a similar strategy as in the proof of Lemma 3.3. Let d be a prime number, 0 < k < d, H ∈ [1, ∞), and α ∈ A(k, d, H). The Galois group of the normal closure of Q(α) must contain an element of order d since d is prime and the Galois group acts transitively on the d-element set of conjugates of α. Since d is prime, such an element of order d must act as a d-cycle on the conjugates of α. If these conjugates are α 1 , . . . , α d , we can assume without loss of generality that this d-cycle acts on them by acting on the indices as ( We aim to write some l-th power of aα d−k 1 as a quotient of products of conjugates of ±H d , where l ∈ N and the number of conjugates that appear are bounded in terms of k and d only. Once this is achieved, we can conclude as in the proof of Lemma 3.3. To , where A is a permutation matrix corresponding to the cycle (12 · · · d).
If finite (which we will later prove it to be), the index [Z d : ] can be bounded by (d − k) d 2 through an application of Hadamard's determinant inequality.
Assuming for the moment that [Z d : ] < ∞, we deduce that (n, 0, . . . , 0) ∈ for some natural number n ≤ (d − k) where the number of conjugates that appears is bounded in terms of k and d only as we wanted.
It remains to prove that [Z d : ] < ∞. Equivalently, we can show that the vector By adapting the proof of Theorem 4.1(iii), we can now strengthen Lemma 3.3. . The Galois group of the normal closure of Q(α) can be identified with a subgroup G of the symmetric group S d . To a (formal) product d i=1 α e i i with e i ∈ Z we again associate a vector (e 1 , . . . , e d ) ∈ Z d . The group G then acts on Q d by permuting the coordinates. We will denote the vector associated to i∈I α i by v. As we have seen in the proof of Theorem 4.1(iii), it suffices to show that the vector space V generated over Q by the gv for g ∈ G must be Q d in order to prove the lemma.
Certainly, this vector space is G-invariant. Since G acts 2-transitively, we know that there are only 4 G-invariant vector subspaces of Q d , i.e. {0}, Q(1, 1, 1, . . . , 1), and Q d (see [24], Exercise 2.6). We can immediately exclude the first two since neither of them contains the vector v. Furthermore, the vector g∈G gv is non-zero and lies in Q (1, 1, 1, . . . , 1), so we can also exclude the third one. It follows that V = Q d and we are done.
The next theorem shows that the height function together with the degree is in some sense "almost injective" if the degree is at least 2. By Theorem 4.1(i) each summand here is bounded by CH 2 for some C = C(d, ) and we are done.

The case gcd(k, d) = 1
In this section, we prove Theorem 5.2, which will give a useful unconditional upper bound for |A(k, d, H)|. Theorem 5.2 also provides a further strengthening of Lemmas 3.3 and 4.2. We first prove an auxiliary lemma that will also be useful later.
We deduce that the coefficients of the polynomial d−k i=1 (t − α i ) belong to Q(H d ). In order to prove the first part of the lemma, we will make use of the following simple facts: If K 2 /K 1 is a finite Galois extension of fields of characteristic 0 within a fixed algebraic closure K 1 and ξ ∈ K 1 , then [K 2 (ξ ) : We In particular, a(k, d) = 0 if gcd(k, d) = 1.
We will see later that the exponent in the bound for |A(k, d, H)| is indeed sharp for every choice of (k, d). Let us also note at this stage that one might hope a priori to prove that a(k, d) = 0 for all d ∈ N and k ∈ {1, . . . , d − 1} with gcd(k, d) = 1 by showing the following: For any transitive subgroup G of the symmetric group S d and any vector v ∈ Q d with exactly k entries equal to 1 and d − k entries equal to 0, the set Gv generates Q d . Unfortunately, this statement is wrong. One can construct a counterexample with G equal to the subgroup generated by the d-cycle (12 · · · d) from any counterexample to the following statement: Any sum of k distinct d-th roots of unity is non-zero. If we denote e 2π √ −1 n by ζ n for n ∈ N, then a construction by Rédei (see [19], Satz 9) yields counterexamples like where the right-hand side is a sum of 13 distinct 42-nd roots of unity. If G is a 2-transitive subgroup of S d , then it follows from the proof of Lemma 4.2 that the statement is correct.
We see that Theorem 5.2(i) yields an upper bound with exponent as soon as the normal closure of Q(H d ) contains α. If we restrict ourselves to α such that Q(α) is Galois over Q, we can for example obtain such a bound as soon as [Q(H d ) : Q] = d. In Theorem 6.1, we will see another case where Theorem 5.2(i) can be applied with l = 1. A(k, d, H). Let K be the normal closure of Q(H d ) and set l = [K (α) : K ]. By Lemma 5.1, l divides gcd(k, d). Thus, part (ii) of the theorem directly follows from part (i), after adjusting the constant C. Since l ≤ gcd(k, d) < d, we must have l = 1 if Gal(Q/Q) acts primitively on the set of conjugates of α, so part (iii) also follows from part (i).

Proof of Theorem 5.2 Let H ∈Q ∩ [1, ∞) and let α ∈
We now fix l and prove part (i): Let α 1 , . . . , α d be the conjugates of α, numbered so that Let a be the (non-zero) leading coefficient of a minimal polynomial of α in Z[t], chosen such that H(α) d = aα 1 · · · α d−k . It follows from Lemma 5.1 that So the number of possibilities for a is bounded by C 1 H 3 for some constant C 1 , depending only on d and . Hence we can assume that a ∈ Z\{0} is fixed.
All the γ j lie in the fixed number field K that is determined uniquely by H and d. We The algebraic integers γ j (j = 1, . . . , l) lie in the given number field K of degree at most d! and their height is bounded by C 4 H d , where C 4 depends only on d and k. It therefore follows from Lemma 3.5 that the number of possibilities for each of them, if their K /Qnorm is fixed, is bounded by C 5 H 3l , where C 5 depends only on d, k, and . Part (i) of the theorem now follows since aα l + l j=1 (−1) j γ j α l−j = 0 and so α is determined up to Galois conjugation by l, a, and the γ j (j = 1, . . . , l).

The case (k, d) = (2, 4)
One might be tempted to conjecture that a(k, d) = 0 for all d ≥ 2 and 0 < k < d, but this is not true. We begin our investigations by studying the simplest non-trivial case, namely (k, d) = (2, 4). In this case, there are three possibilities for [Q(H 4 ) : Q], namely 2, 4, or 6. In the last case, we can apply Lemma 3.3 to obtain that |A(2, 4, H)| grows more slowly than H for every > 0. We now show in the next theorem that the same holds in the middle case, where [Q(H 4 ) : Q] = 4. In the first case, i.e. if the normal closure of Q(α) is Q(α), K coincides with the normal closure of Q(α) as both are equal to Q(α).
In the second case, i.e. if the normal closure of Q(α) is a number field of degree 8, the Galois group of the normal closure of Q(α) is isomorphic to the dihedral group D 4 and Q(H 4 ) is a quartic subfield of that normal closure. If K is not equal to the normal closure of Q(α), then the extension Q(H 4 )/Q is Galois. Suppose now that the conjugates of α are the α i (i = 1, . . . , 4) and that the Galois group is generated by field automorphisms acting on the conjugates α i by acting on their indices as the cycle (1234) and the transposition (13). Since [Q(H 4 ) : Q] = 4, we can assume after a cyclic renumbering that H 4 = ±aα 1 α 2 , where a ∈ N is the leading coefficient of a minimal polynomial of α in Z[t]. The only subfield of the normal closure of Q(α) of degree 4 that is Galois over Q corresponds under the Galois correspondence to the cyclic normal subgroup of D 4 generated by (13) (24). But this element does not fix H 4 since |α 1 α 2 | ≥ 1 > |α 3 α 4 |. So Q(H 4 ) cannot be Galois over Q and it follows also in this case that K is equal to the normal closure of Q(α).
The theorem now follows from Theorem 5.2(i) with l = 1.
In the case where [Q(H 4 ) : Q] = 2, it follows from Theorem 5.2 that we have |A(2, 4, H)| ≤ C( )H 4+ for all such H. However, the next theorem shows that one cannot always expect this growth and in fact one cannot obtain a uniform growth rate in H even after partitioning A(2, 4) into an arbitrary finite number of subsets. In Sect. 7, we will prove that |A(2, Proof Let κ ∈ [0, 4]. We fix m ∈ N prime with m ≡ 1 mod 4 and denote its positive square root by √ m. We define u 1 + u 2 √ m = u 1 − u 2 √ m (u 1 , u 2 ∈ Q). If κ < 4, we apply a theorem of Chebyshev [6] (Bertrand's postulate) to find a prime number b 2 ∈ N such that which preserves (6.1), we can assume that 0 < β < 1.
Let F be the fixed field of the stabilizer H of {α 1 , α 2 } in the Galois group G of the normal closure of Q(α). By Lemma 5.1, every σ ∈ G which fixes β must lie in H. Since the converse implication holds trivially, it follows that F = Q(β) and β ∈ {±aα 3 α 4 }. We deduce from Lemma 3.4 that a divides a 2 4 j=1 α j = ββ in Z. Since |β| < 1, it follows from well-known bounds for the divisor function that the number of possibilities for a is bounded by C 1 H 4 for a certain constant C 1 that depends only on .
From now on, we assume that a is fixed and count the number of possibilities for α. We have aα 2 1 − γ α 1 ± β = 0, where γ = a(α 1 + α 2 ). For a given α 1 , there are exactly four possible α. From now on, we assume that α = α 1 . It then suffices to bound the number of possibilities for γ . Now γ lies in F , so γ ∈ Q(β). Furthermore, we have that γ = a(α 1 + α 2 ) ∈ Q(β) is an algebraic integer by Lemma 3.4. Since m ≡ 1 mod 4, we have γ ∈ Z + Z √ m, so c 2 , a),c 1 =ã −1 c 1 , andc 2 =ã −1 c 2 . By the usual bound for the divisor function, the number of possibilities forã is bounded from above by C 2 a 16 ≤ C 2 H 4 with a constant C 2 that depends only on . In the following, we assume thatã is fixed.
Recall thatc 2 can only be 0 if a = 1, in which case gcd(a,ã 2 )ã −1 = aã −1 ≤ a 1 2 . Thanks to (6.5) and the above, the number of possibilities for the pair (c 1 ,c 2 ) is then bounded from above by 24 H κ a gcd(a,ã 2 )ã −1 + a If κ < 2, we have to study more closely the case thatc 2 = 0. We use that a is the leading coefficient of a minimal polynomial of α in Z[t]. Ifc 2 = 0 and γ = c 1 , we can therefore conclude that a divides all coefficients of the polynomial Here, the sign of β is the same as that of β. In particular, a divides c 1 (β + β) = 2b 1 c 1 as well as . It follows from (6.1) that 0 < |b 1 | ≤ |b 2 | √ m + 1 ≤ 2m 4 2(4−κ) + 1. As κ < 2, this implies together with (6.3) that 0 < |b 1 | < m for H ≥ H 1 = H 1 (κ). We assume from now on that H ≥ H 1 . Since m is prime, it then follows that δ = gcd(b 1 , b 2 2 ). But b 2 is prime and does not divide b 1 , so δ = 1. Since any common divisor of a and b 1 must also divide δ, it follows that gcd(a, b 1 ) = 1.
Putting everything together, we obtain that the number of possibilities for α is bounded by  (2,4). Since H( √ β) = H, the lower bound holds with c = 1. We now assume that κ > 0. We choose γ = c 1 + c 2 √ m with 2 4 − β > 0 and an arbitrary complex square root otherwise. It follows that α is an algebraic integer of degree dividing 4. Note that α = 0 and γ = β+α 2 α is uniquely determined by α.
We begin by controlling the cases where [Q(α) : Q] < 4. If α were a rational integer, then α would be a common divisor of b 1 and b 2 . As b 1 and b 2 are coprime by construction, it would follow that α = ±1 and therefore This contradicts (6.6). So α cannot be a rational integer.  √ m] + 1 − c 2 √ m < 1 and 0 < β < 1 together with (6.6) and fundamental properties of the height, we can bound the height of γ ± δ from above by If we write η = ηζ u l , where η and η are two possible values for γ + δ, then it follows that h Hence there are at most C 5 log |β| possibilities for the unit and hence for γ +δ, where C 5 is an absolute constant. Now γ +δ determines γ −δ since (γ +δ)(γ −δ) = 4β and β is fixed. And γ + δ together with γ − δ determines γ , so there are at most C 5 log |β| possibilities for γ as well. It follows that α is quadratic for at most C 6 |β| κ 8 = C 6 H κ 2 choices of γ , where C 6 = C 6 (κ) depends only on κ.
Summarizing, we find that α has degree < 4 for at most C 6 H κ 2 choices of γ .
If β > 0 and γ 2 4 < |β|, then If β > 0 and γ 2 4 ≥ |β|, we have since γ < |β| + 1 and γ ≥ 2|β| Recall that β > 0. If |γ | < 2β 1 2 , then γ 2 4 − β is purely imaginary and Otherwise, we have So in any case, α has two conjugates inside and two conjugates outside the open unit disk. Finally, we can compute that so H(α) = |β| 1 4 = H. Thanks to (6.2), the number of choices for γ can be estimated as We can then use (6.4) to deduce that the number of choices for γ is equal to at least H κ 4 if κ < 4. If κ = 4, we get that the number of choices for γ is equal to at least H κ

The case gcd(k, d) > 1
In fact, the situation is even worse than Theorem 6. Before the proof, we make some remarks on this theorem: The number of possibilities for L given K is bounded in terms of d and k. As β is a product of l conjugates of α, we can bound its height by H l . Since K /Q is a Galois extension, we have that [K (α) : K ] = [Q(α) : L] = l for any α ∈ A L,β (k, d, H). An upper bound for |A L,β (k, d, H)| of the same growth order as (7.2) (up to H 2 ) is therefore provided by Theorem 5.2(i). However, the following examples show that it is not possible in general to prove the lower bound (7.2) with C depending on k, d, δ, and , but not on K , or with C depending on k, d, K , and , but not on δ. For reasons of space, we grudgingly leave it to the reader to work out the details in the examples.
The necessity of the dependence on K is shown by the following example: The necessity of the dependence on δ is shown by the following example: We deduce that a = a. This implies that α satisfies an equation we must have γ ∈ I. Let γ denote the image of γ under the non-trivial field automorphism of K . Then one can show that max{|γ |, |γ |} ≤ Applying Theorem 2.1 in [30] with S = ((1, 1), (1, 1)) and C = {(0, 0)} to the image of I under a Minkowski embedding (cf. the proof of Lemma 7.4 below and note that the K /Q-norm of every element of I is divisible by a) shows that the number of such γ is bounded independently of (a, b), but H → ∞ as b → ∞.
We now prove Theorem 7.1.
It follows from (2) that the σ (P γ ) for σ : L → C are pairwise distinct. Together with (5) and the fact that K /Q is Galois, this implies that Q γ is irreducible in Q[t] and therefore in Z[t].
Let α γ be a complex zero of P γ . It follows that  (5), we must have L = Q(α γ ) ∩ K . We also have that N Q(α γ )/L (α γ ) = β so that α γ ∈ A L,β (k, d, H). Since γ is uniquely determined by α γ , a, and L, we have reduced the proof of (7.2) to proving the following Lemma 7.4: Proof We will use c 1 , c 2 , . . . for positive constants that depend only on k, d, K , δ, and .
Recall that we have assumed that l ≥ 2. We can assume without loss of generality that < 1 2 . We want to use Theorem 2.1 in [30]. Let r and s denote the number of real embeddings and pairs of complex conjugate embeddings of L respectively. For each pair of complex conjugate embeddings of L, we choose one element of the pair. Furthermore, we order both the real embeddings of L and the pairs of complex conjugate embeddings of L in fixed ways each. Let denote the image of I inside R r × C s under the thus obtained Minkowski embedding. We identify C with R 2 by identifying (v, w) ∈ R 2 with v + w √ −1 ∈ C and we identify each γ i with its image in (i = 1, . . . , l − 1). In the following, we use the notation of [30]: We set n = r + s, N = [L : Q], C = {0} ⊂ R N , m j = β j = 1 for 1 ≤ j ≤ r, and m j = β j = 2 for r + 1 ≤ j ≤ r + s. For each j ∈ {1, . . . , n}, let σ j : L → C denote the associated embedding used to define the Minkowski embedding.
Thanks to ( [30] then yields a main term which is greater than or equal to c 2 H d for the number of γ i satisfying (3) and (4), for fixed i. Choosing B = Q max and using our lower bound for μ( , B), we find that the corresponding error term is bounded from above by We turn to (1). The ideal I = aO L + aβO L is contained in I and [I : I ] divides [I : (1) is not satisfied, then γ 1 is contained in IP for some prime ideal P such that IP divides I . Note that N (P) then divides a. Applying Theorem 2.1 in [30] as above to each ideal IQ instead of I with Q a product of pairwise distinct such P and then using the inclusion-exclusion principle, we see that imposing (1) means that the main term gets multiplied by a factor while the error term gets multiplied by 2 u , where u is the number of possibilities for P.
As a ≤ H d , the factor in the main term can be bounded from below by c 4 H − while u is bounded from above by log H + c 5 thanks to Théorème 11 in [20]. We next consider (2). If (2) is not satisfied, then σ (γ 1 ) = σ (γ 1 ) for two distinct embeddings σ , σ of L in C and so γ 1 lies in a lower-dimensional linear subspace of R N , obtained by equating two coordinates or setting a coordinate equal to 0. The intersection of such a subspace with Z Q is a bounded convex set of volume 0 that is contained in Z Q and so Theorem 2.1 in [30] shows that the number of such γ 1 can be absorbed into the error term.
It remains to be shown that the number of γ which satisfy conditions (1) to (4), but not (5) is of lower growth order than H d(l−1)− . LetP be some monic irreducible factor of P γ in K [t]. Setl = degP. We definep 1 , . . . ,p˜l ∈ K by aP(t) = at˜l +p 1 t˜l −1 + · · · +p˜l and set K = Q(p 1 , . . . ,p˜l). SinceP is irreducible in K [t],P divides Q γ ∈ Z[t], and K /Q is Galois, LetQ be a minimal polynomial in Z[t] of some complex zero ofP and letã denote the leading coefficient ofQ, thenQ =ã σ :K →C σ (P). SinceQ is primitive and divides Q γ in Sinceã divides a, it follows from Lemma 3.4 thatq i ∈ Z. SinceQ divides Q γ , we have |q i | ≤ c 6 a Q γ (ζ )=0 max{1, |ζ |}. Thanks to (7.4), (7.5), and (7.6), this implies that SinceQ divides Q γ , we can also use (7.4), (7.5), and (7.6) to estimate It then follows from (7.7), (7.8), (7.9), and Lemma 3.5 that the number of possibilities for p i ∈ O K is bounded from above by c 8 H d+ l (i = 1, . . . ,l). The leading coefficient of aP is of course always equal to a. Furthermore, suppose that P γ = aP 1 · · ·P m withP 1 =P and allP i monic and irreducible in K [t]. The above argument forP shows that aP i ∈ O K [t] for all i. Since a m−1 P γ = (aP 1 ) · · · (aP m ), we deduce thatp˜l divides a m β in O K . As m ≤ l, it follows that N K /Q (p˜l) divides a [K :Q](l−1) N K /Q (aβ) in Z. Lemma 3.5 then shows together with (7.4), (7.9), and elementary bounds for the divisor function that there are at most c 9 H l possibilities forp˜l.
This implies that the number of possibilities forP is bounded from above by c 10 H d(l−1)+ l l . If γ satisfies conditions (1) to (4), but not (5), then a −1 P γ is equal to a product of at least two such factorsP. Furthermore, γ is uniquely determined by P γ , so it follows that the number of such γ is less than or equal to c 11 H d(l−2)+ . This completes the proof of Lemma 7.4 and thereby completes the proof of Theorem 7.1.
It is now easy to show that a(k, d) does not exist if 0 < k < d and gcd(k, d) > 1. We can even determine the corresponding limit superior and limit inferior. Proof Set l = gcd(k, d). The limit superior is less than or equal to d(l − 1) by Theorem 5.2(ii). If l = 1, this already proves the theorem, so let us assume that l ≥ 2. We want to show that the limit superior is also greater than or equal to d(l − 1). We fix a totally real number field L of degree d l that is a Galois extension of Q. Such an L can be constructed as a subfield of Q cos 2π p , where p is prime and p ≡ 1 mod 2d l . We will prove the following analogue of Theorem 1.1 for the Mahler measure instead of the height:  We see that e ≤ max i {k i − 1} ≤ k − 1 < max{k, d − k} and f ≤ max{k, d − k}. It follows that the number of possibilities for A is bounded from above by CM max{k,d−k}+(s+2) . This proves that the limit superior in (8.4) is less than or equal to max{k, d − k}.
For the inequality in the other direction, we consider M ∈ N such that M We can deduce from Theorem 4.1(ii) that the limit in (8.5) has to be greater than or equal to d + 1 (if it exists). For the inequality in the other direction (which will also imply the existence of the limit), we can use that for M ∈

Dynamics of the height function
In this section, we study the dynamics of the restriction of the height function toQ ∩ R.
We start by classifying the periodic points. We define inductively H 0 = id and H n = H • H n−1 (n ∈ N). Theorem 9.1 If n ∈ N and α ∈Q are such that H n (α) = α, then α = a b for some a ∈ N and b ∈ Q, b > 0, and H(α) = α. Conversely, H(a b ) = a b for all a ∈ N and b ∈ Q, b > 0.
The proof of this theorem will be essentially achieved by the following lemma: We can now prove Theorem 9.1.