On the Algebraic Points of a Definable

20080627 On the algebraic points of a deﬁnable set

Jonathan Pila

Abstract This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. It observes a refinement of the main theorem of the author’s recent paper with A J Wilkie, and uses it to deduce a strong result on the density of algebraic points of such sets.

2000 Mathematics Subject Classiﬁcation: 11G99, 03C64 Keywords: o-minimal structure, deﬁnable set, rational point, algebraic point

1. Introduction This paper builds on [15] in considering diophantine properties of certain non-algebraic subsets n X ⊂ R . We use geometric means to control the rational or algebraic points of X, so it is natural to restrict to sets X with suitably nice geometric properties. The sets definable in an o-minimal structure (over R) provide a natural setting for the methods employed in [15] and pursued here. A definition is given below in Section 2. Examples of o-minimal structures (over R) include : the structure comprising all semialgebraic sets (the paradigm example, whose o-minimality follows from the Tarski-Seidenberg Theorem); the structure comprising globally subanalytic sets Ran (for which o-minimality is afforded by Gabrielov’s Theorem [6]); the structure Rexp of sets definable using the exponential function (Wilkie [20]); the structure Rpfaff generated by pfaffian sets (Wilkie [21]). Thus o-minimal structures provide rich and flexible categories to work in while maintaining strong geometric finiteness properties. For a description of these and further examples see [4, 5, 17, 18, 19] in addition to the references above, for a development of the theory of o-minimal structures see [3, 5], and for the context of the diophantine results pursued here see [15]. n 1.1. Definition. Let S be an o-minimal structure over R, which will remain fixed. We call X ⊂ R definable if it definable in S. In studying the diophantine properties of such sets, our guiding idea is that a transcendental set should contain only few rational points, in a suitable sense. However, a non-algebraic set X may contain semi-algebraic subsets of positive dimension. Such semi-algebraic subsets may contain many rational points (where “many” means >> T δ points of height ≤ T ). To isolate the truly transcendental part of a set X, we make the following definition. n 1.2. Definition. Let X ⊂ R . The algebraic part of X, denoted Xalg, is the union of all connected semi-algebraic subsets of X of positive dimension. Our primary interest is in diophantine properties of the transcendental part X − Xalg of X. The transcendental part of a definable set may contain infinitely many rational points (e.g. y = 2x definable in Rexp). So we define a height function on the points of interest and a function counting points up to given height. We then seek upper estimates for this counting function. Excluding Xalg is thus crudely analogous to excluding the special set when counting rational points up to height T on an algebraic variety. By H(α) we denote the (multiplicative) height of an algebraic number α, i.e. H(α) = exp h(α), where h(α) is the absolute logarithmic height, as defined in [1, p16] (see the characterization of h(α) in Section 5 below). As h(α) ≥ 0, H(α) ≥ 1. For a rational number a/b in lowest terms, H(a/b) = max(|a|, |b|). For an n-tuple (α1, . . . , αn) of algebraic numbers we define H(α1, . . . , αn) = maxi H(αi). Note that this is not the usual projective height Hproj of an n-tuple, but simply coordinate-wise (affine) height, which seems more natural when considering non-algebraic sets. As H ≤ Hproj ([1, p16 cf p15], e.g. Hproj of a tuple of rational numbers involves a least common denominator for all the coordinates), our results are marginally stronger than if stated using Hproj, though this would be seen only in the implicit constants.

1 n n 1.3. Deﬁnition. For a set X ⊂ R let X(Q) = X ∩ Q . For a real number T ≥ 1, set

X(Q,T ) = {x ∈ X(Q): H(x) ≤ T }, and deﬁne the counting function N(X,T ) = #X(K,T ).

With these deﬁnitions, the basic result of [15] is as follows. n 1.4. Theorem. Let X ⊂ R be deﬁnable and let > 0. Then, for T ≥ 1, alg N(X − X ,T ) = OX, T .

This result can be adapted to estimate points of X with coordinates in a number field up to height T in a straightforward way (for the graph of a transcendental real analytic function on a compact interval n this is already in [10]). For a set X ⊂ R and a number field K ⊂ R let X(K) = X ∩ Kn and define, for a real number T ≥ 1,

X(K,T ) = {x ∈ X(K): H(x) ≤ T },NK (X,T ) = #X(K,T ).

n 1.5. Theorem. Let X ⊂ R be deﬁnable, let K ⊂ R be a number ﬁeld of degree k, and let > 0. Then alg NK (X − X ,T ) = OX,k, T .

Theorem 1.4 is established in a refined form in [15]. The main point of this paper is that the proof in [15] actually yields a further strengthening. We defer stating this result to Section 3, but it enables us to obtain a result about algebraic points that is much stronger than 1.5 in which we consider, given a positive integer k, points x of X that are defined over any algebraic number field of degree k — indeed n different coordinates of x may be defined over different fields. We define, for a set X ⊂ R , a positive integer k, and a real number T ≥ 1,

X(k, T ) = {x = (x1, . . . , xn) ∈ X : max[ (xi): ] ≤ k, max H(xi) ≤ T }, i Q Q i

Nk(X,T ) = #X(k, T ).

n 1.6. Theorem. Let X ⊂ R be definable, let k be a positive integer, and let > 0. Then alg Nk(X − X ,T ) = OX,k, T . The results of [15] have been used to prove diophantine results. A new proof of the Manin-Mumford conjecture by the present author and Zannier [16], and affirmation of a special case of the Zilber- Pink conjecture by Masser and Zannier [9] both make essential use of [15] or its predecessors. It is anticipated that Theorem 1.6 will also have applications in this circle of diophantine problems. Our strengthening of 1.4 also yields a result for points of X whose coordinates lie in some finite- k P dimensional Q vector subspace of R. For λ = (λ1, . . . , λk) ∈ R let Qhλi = { i qiλi : qi ∈ Q}. For x ∈ P Qhλi set Hλ(x) = min{maxi H(qi): q1, . . . , qk ∈ Q, i qiλi = x}, with Hλ(x1, . . . , xn) = maxi Hλ(xi) n n for x = (x1, . . . , xn) ∈ Qhλi . We let X(Qhλi) = X ∩ Qhλi , and define

X(λ, T ) = {x ∈ X(Qhλi): Hλ(x) ≤ T },Nλ(X,T ) = #X(λ, T ).

n d 1.7. Theorem. Let X ⊂ R be deﬁnable, k a positive integer, λ ∈ R and > 0. Then alg Nλ(X − X ,T ) = OX,k, T .

While in all the statements above we exclude the algebraic part Xalg in its entirety, it is the exploration of precisely which parts of Xalg need to be removed from X to achieve the O(T ) estimate that is the focus of our strengtheneing of 1.4, and the key to proving 1.6 and 1.7. Theorem 1.7 is a somewhat easier consequence of our main result than 1.6 and we prove it ﬁrst, in Section 4, before establishing 1.6 in Section 5.

2 As remarked in [15], the O(T ) estimate in 1.4 cannot be much improved for a general o-minimal structure, in view of one-dimensional examples in Ran constructed in [12, 7.5]. However, Wilkie has conjectured ([15, 1.11]) that a much better estimate holds for sets X deﬁnable in Rexp, namely, for T ≥ e, alg OX (1) N(X − X ,T ) = OX (log T )

One would expect such estimates to hold likewise for the strengthenings of 1.1 obtained in [15] and here, and also for algebraic points. In [14] I established such an estimate for the rational points on a pfaff curve.A pfaff curve X is the graph of a pfaffian function f of one variable on some connected subset of its domain. A definition of pfaffian function is given in Section 6. Pfaffian functions and the sets defined by them arose in the work of Khovanskii [8] on “fewnomials”. The reader is referred there and to [7] for further information, and to [21] for o-minimal aspects. This paper concludes with the observation that an adaptation of the method of [14] obtains a result for points on a pfaff curve defined over a number field in which the exponent of log T is independent of the field. 2 1.8. Theorem. Let X ⊂ R be a non-algebraic pfaff curve and K ⊂ R a number field of degree k. Then, for T ≥ e, OX (1) NK (X,T ) = OX,k (log T ) .

The proof of 1.8 is carried out in Section 6. It is straightforward but unfortunately seems to require revisiting several results of [14], and some earlier papers, in order to make the requisite minor modifications. I have tried to make the ideas of the proof and the modifications required clear without too much repetition. In this paper, #A denotes the cardinality of a set A, the inclusion A ⊂ B need not be strict, and N denotes the set of non-negative integers. By f(T ) = O(g(T )), for real functions f, g defined on some specified domain, we mean that there is an absolute constant C such that |f(T )| ≤ Cg(T ) for all T in the domain. If f(T ) = fa,b,c,...,α,β,γ,...(T ) depends on some parameters then by f(T ) = Oa,b,c,...(g(T )) we indicate that the implied constant C = Ca,b,c,... is permitted to depend on those exhibited parameters a, b, c, . . . but not on α, β, γ, . . .. By Oa,b,c,...(1) we denote a constant that depends on a, b, c, . . ..

2. Deﬁnable sets and families

We give a deﬁnition of o-minimal structure over R, set up some notation and recall some well-known properties of sets deﬁnable in o-minimal structures, and explicate a consequence of the Trivialization Theorem.

2.1. Definition. A pre-structure is a sequence S = (Sn : n ≥ 1) where each Sn is a collection of subsets n of R . A pre-structure S is called a structure (over the real field) if, for all n, m ≥ 1, the following conditions are satisfied:

(1) Sn is a boolean algebra (under the usual set-theoretic operations) n (2) Sn contains every semi-algebraic subset of R (3) if A ∈ Sn and B ∈ Sm then A × B ∈ Sn+m m n (4) if m ≥ n and A ∈ Sm then π[A] ∈ Sn, where π : R → R is projection onto the ﬁrst n coordinates. n If S is a structure and X ⊂ R , we say X is deﬁnable in S if X ∈ Sn. If S is a structure and, in addition,

(5) the boundary of every set in S1 is finite then S is called an o-minimal structure (over the real field). n Let X ⊂ R be definable. Then X has finitely many connected components and each of them is definable ([5, 4.3]). p For each pair κ, p ∈ N we define the p-regular points of dimension κ of X, denoted regκ(X), to be the set of x ∈ X such that there is an open neighbourhood U of x with U ∩ X a Cp (embedded) n p submanifold of R of dimension κ. Each regκ(X) is definable ([5, B.9]). A regular point of dimension κ will mean a 1-regular point of dimension κ.

3 The dimension of X is the maximum κ ∈ N such that X has a regular point of dimension κ. 1 Therefore, if X has dimension κ, then X − regκ(X) has dimension ≤ κ − 1. n m We need to consider families of definable sets. In considering subsets of R × R , let π1 denote n m projection on the first factor, and π2 projection on the second factor. For a set Z ⊂ R × R put n m Y = YZ = π2(Z) and, for y ∈ Y , put Zy = {z ∈ R × R : π2(z) = y} and XZ,y = {π1(z): z ∈ Zy}. Thus, for any y ∈ YZ , the restriction of π1 to Zy identifies Zy with XZ,y. n m 2.2. Definition. A definable family of sets is a definable set Z ⊂ R × R considered as the family of n fibres XZ,y ⊂ R for y ∈ YZ . 2.3. Proposition. ([5, B.10]) Let Z be a definable family, and κ, p ∈ N. Then

p {z = (x, y) ∈ Z : x ∈ regκ(XZ,y}

is deﬁnable.

n m 2.4. Proposition. Let Z ⊂ R × R be a deﬁnable family. There is a positive integer J and deﬁnable n m families Zj ⊂ R × R , j = 1,...,J, with the following property.

For each j, every fibre of Zj is connected. The set Yj = YZj is the subset of YZ such that the fibre XZ,y has at least j connected components, and for each y ∈ YZ , the fibres (Zj)y are disjoint and

J [ Zy = (Zj)y. j=1

Proof. We apply the Trivialization Theorem [3, Ch. 9, 1.2] to the restriction of π2 to Z to obtain: a finite partition A1 ∪ ...A` of YZ (i.e. the Ai are disjoint and their union is YZ ) into definable sets Ai, n m and definable sets Fi, and, for each i, a definable map θi : Ai × Fi → R × R that, for each y ∈ Ai, the map θi(y, η) gives a definable homeomorphism of Fi with the fibre Zy, and hence with XZ,y. Now each Fi is a union of finitely many connected components Fij, each of which is definable. By restricting to these components, and taking the family of images of θi restricted to Fij on Ai we obtain 0 a finite collection Zi of families whose fibres are all connected, and, at each y ∈ Y , the union of the fibres is Zy. As θi(y.η) restricted to y is a homeomorphism, the images of the Fij are the connected components of Zy, and the projections of these under π1 are the connected components of XZ,y. We now need to re-organize these families to achieve the properties required. Let J be the maximum number of connected components in a fibre Zy (the finiteness of which follows from the trivialization). For j = 1,...,J let Yi be the union of the sets Ai such that Fi has at least i connected components. Over Y1, choosing arbitrarily one of the components of Fi for each Ai, we form the family Z1. Over Y2, choosing from each of the constituent Ai a component different to the one chosen for Y1, we construct Z2. Continue to define Y3,Y4,... until all components of all Fi are exhausted.

3. A further strengthening of Theorem 1.4 Theorem 1.4 is strengthened in [15] in two ways. First, it is made uniform for sets in definable families. Indeed this is an artifact of the proof. Such uniformity will also be obtained for 1.6 and 1.7. alg Second, there is some refinement concerning the subset X of X that needs to be excluded in order to get a OX,(T ) estimate for N(X − X,T ). The following is the final version of 1.4 established in [15]. 3.1. Theorem. ([15, 1.10]) Let Z be a definable family, and > 0. There is a definable family W = alg W (Z, ) with the following property. Let y ∈ YZ , put X = XZ,y and X = XW,y. Then X ⊂ X and

N(X − X,T ) = OZ,(T ).

As remarked in [15, p597], Theorem 3.1 makes a non-empty assertion in certain situations where Theorem 1.4 does not. This and the further reﬁnement that we will obtain are best illustrated by some simple examples. Each of the following sets are deﬁnable in Ran (or alternatively in Rexp).

4 3 y alg Let X1 = {(x, y, z) ∈ R : 2 < x, y < 3, z = x . Then X1 consists of a union of segments q alg of algebraic curves: for each q ∈ Q, 2 < q < 3, the curve y = q, z = x . Thus (X1 − X1 )(Q) is alg empty, and the conclusion of 1.4 is trivial. However, X1 is not definable in Ran, or indeed in any o-minimal structure, as it has infinitely many connected components – a definable set has only finitely many connected components. So 3.1 makes the non-trivial (though not difficult) assertion that a O(T ) alg estimate for N(X1 − X1,,T ) can be obtained by removing a definable subset X1, of X1 , depending on , which will consist of finitely many curve segments. 3 x+y Let X2 = {(x, y, z) ∈ R : 2 < x, y < 3, z = 2 . Here every point lies on some line segment c alg z = 2 , x + y = c contained in X2. So X2 = X2. Now both 1.4 and 3.1 make trivial assertions, because alg X2 is definable. However, to achieve a O(T ) estimate for N(X2 − X2,,T ) one need remove only those log T = O(T ) segments with c ∈ Z, 0 ≤ c ≤ log T/ log 2. Our strengthening of 3.1 will make a non-trivial (though not difficult) assertion in this example. 2 x Let X3 = {(x, y) ∈ R : 0 < x < 1, 0 < y < e }. Here every (x, y) ∈ X3 is contained in some small 4 open disk also contained in X3. The disk is semialgebraic, also it contains >> T rational points up alg to height T . Therefore, X3 = X3 and definable. The assertions of 1.4 and 3.1 are trivial, but now we cannot hope to recover a O(T ) estimate for N(X3 − X,T ) without removing essentially all of X3. The set X3 is not semi-algebraic (if it were then its bounding curves would also be semialgebraic), but it is a “definable piece of a semi-algebraic set” — which is made precise below. Examination of the proof of 3.1 in [15] shows that all the sets excluded in the course of the proof have this form, and come from a finite number of definable families.

3.2. Definition. n 1. A basic definable semialgebraic block or basic block of dimension κ in R is a connected definable n set U ⊂ R of dimension κ contained in some semialgebraic set A of dimension κ such that every point x of U is regular of dimension κ in U and in A and in U ∩ A. Dimension zero is allowed: a point is a basic block. 2. A definable semi-algebraic block, or block, is the image of a basic block U under a semialgebraic n m map φ : R → R defined and continuous on a semialgebraic set containing U. 3. A basic definable semialgebraic block family, or basic block family, is a definable family Z whose fibres are all basic blocks. ` m 4. A definable semi-algebraic block family, or block family, is the family W ⊂ R × R of images of n m n ` a basic block family Z ⊂ R × R under a semialgebraic map φ : R → R defined and continuous on a semialgebraic set containing all the fibres.

3.3. Remark. We have framed the above definitions to meet our requirements. Presumably a block is always a union of finitely many basic blocks (even if the semialgebraic map is not continuous), and then we could dispense with this distinction and restriction to continuous semialgebraic maps. This is presumably entirely a question about semialgebraic sets and maps. In defining a block family, one could allow the semialgebraic map to vary with the parameters.

We observe the following facts.

n 3.4. Proposition. Let X ⊂ R be definable. 1. Let U ⊂ X be a basic block of dimension κ > 0. Then U ⊂ Xalg. 2. Let U ⊂ X be a block of dimension κ > 0. Then U ⊂ Xalg. 3. Let W be a basic block family such that each fibre of W is a subset of X, and all the fibres have dimension κ > 0. Then the union of all the fibres is definable and a subset of Xalg. 4. Let W be a block family such that every fibre of W is a subset of X. Then the union of fibres of positive dimension is definable and is a subset of Xalg.

Proof. 1. Let A be a semialgebraic set with the properties set out in 3.2.1. Each point x ∈ U has a neighbourhood Bx such that Bx ∩ U is regular of dimension κ in U, Bx ∩ A is regular of dimension κ in X, Bx ∩ U ∩ A is regular of dimension κ in U ∩ A. These three intersection sets therefore coincide. Now alg Bx ∩ A is a connected semialgebraic set of positive dimension contained in X. So x ∈ Bx ∩ A ⊂ X .

5 2. The block U is the image of some basic block V under a continuous semialgebraic map φ. Let x, y ∈ U be distinct points, and let P,Q be preimages of them in U. As U is connected, it is path- connected [5, 4.21], and their is a definable continuous map of g : [0, 1] → U with G(0) = P, g(1) = Q. Then g([0, 1]) is compact and covered by finitely many balls Bx as in part 1. On each such Bx we may replace the corresponding portion of g([0, 1]) by a semialgebraic path. Thus there is a semialgebraic path joining P,Q. Then the image of this path is a connected semialgebraic set of positive dimension containing x. n m n 3. Suppose W ⊂ R × R . The union of the fibres is the projection on R of W , and so definable. Call this set A. If x ∈ X is a point in A, then for some y, the fibre Wy contains x, and so contains a alg semialgebraic set Bx ∩ Wy that is connected of positive dimension contained in X. So x ∈ X , and so A ⊂ Xalg. 4. The union of fibres of positive dimension is definable (it is the projection of W minus the set of fibres that are single points). Each fibre of positive dimension is contained in Xalg by part 2. We can now state our further refinement of 3.1. n m 3.5. Theorem. Let Z ⊂ R × R be a definable family, and > 0. There exist J = J(Z, ) ∈ N and a n m µj collection of basic block families Wj ⊂ R × (R × R ), j = 1,...,J, such that each point in each fibre of Wj is regular of dimension wj, and with the following properties. m µj 1. For each j and (y, η) ∈ R × R , XWj ,(y,η) ⊂ XZ,y. 2. If X = XZ,y is a fibre Z and T ≥ 1 then X(Q,T ) is contained in OZ,(T ) basic blocks, each of m µj which is a fibre of one of the Wj at some (y, η) ∈ R × R . 3. Let W = W (Z, ) be the family whose fibre at y is the union of all fibres of the Wj over y over all alg j with wj > 0. Then W is definable. If X is the fibre of W at y then X ⊂ X and N(X − X,T ) = OZ,(T ). Proof. Examining the proof of [15, Theorem 1.10] shows that, given Z and , one considers various families obtained by taking regular points of highest dimension in the fibres previously defined families and intersecting with suitable families of semialgebraic sets of the same dimension. Only finitely many such families arise. Among them are a finite number of basic block families, Wj, whose fibres are all contained in the corresponding fibres of Z, and the proof shows that, given T and y ∈ YZ , XZ,y(Q,T ) is contained in OZ,(T ) fibres of the Wj. This gives the first two statements of 3.5. If wj > 0 then the alg union of fibres over y is a definable subset of Zy , by 3.4.3. Let W be the family whose fibre is the union of these definable sets over j with wj > 0. Since XZ,y(Q,T ) is contained in OZ,(T ) fibres of the Wj, the same estimate applies to the number of fibres coming from Wj with wj = 0. This gives the final statement.

4. Proof of Theorem 1.3 We state a reﬁned version of 1.7 for a deﬁnable family Z and then prove it by applying 3.5 to a suitable family obtained from Z.

Note that the family Zk constructed in the proof has the property that every setXZk,y in Zk is fibred by (k − 1)-planes, so that Xalg = X for all y and Theorems 1.4 and 3.1 make a trivial assertion. Zk,y Zk,y n m 4.1. Theorem. Let Z ⊂ R × R be a definable family, k a positive integer, and > 0. There exists n m k µj J = J(Z, k, ) ∈ N and a collection Wj ⊂ R × (R × R × R ), j = 1,...,J of block families with the following properties. m k µj 1. For each j and (y, λ, η) ∈ R × R × R , XWj ,(y,λ,η) ⊂ XZ,y. k 2. If X = XZ,y is a fibre of Z and λ ∈ R then X(λ, T ) is contained in OZ,k,(T ) blocks contained µj in X, each a fibre of one of the families Wj at some (y, λ, η), η ∈ R . Hence alg Nλ(X − X ),T ) = OZ,k,(T ). n m 3. Further, let W ⊂ R × R be the family whose fibre at y is the union over all j of all fibres m of Wj at (y, λ, η) of positive dimension. Then W is definable, and, for any y ∈ R , if X = XZ,y and alg k X = XW,y then X ⊂ X and, for any λ ∈ R , Nλ(X − X,T ) = OZ,k,(T ).

6 k k n Proof. Deﬁne π :(R )n × R → R by

k k X π(x1, . . . , xn, λ) = ξ, where xi = (xi1, . . . , xik) ∈ R , and ξi = xijλj, i = 1, . . . , n. j=1 Put k n k m Zk = {(x1, . . . , xn, λ, y) ∈ (R ) × R × R :(π(x1, . . . , xn, λ), y) ∈ Z}. k n k We view Zk, which is evidently deﬁnable, as a family of sets in (R ) parameterized by R × YZ , and apply 3.5. Thus, given > 0, we have J = J(Zk, ) ∈ N and a collection of basic block families

k n k m µj Vj ⊂ (R ) × R × R × R with the properties asserted in 3.5. k If now y ∈ Y and λ ∈ R , the set of points XZ,y(λ, T ) is precisely the image of the set of points

XZk,y,λ(Q,T ) under the projection π. The map π is evidently semialgebraic and continuous on all its domain. The image of a basic block family is therefore a block family. We put

Wj = {(π(x1, . . . , xn, λ), y, η):(x1, . . . , xn, λ, y, η) ∈ Vj}.

The asserted properties of Wj follow immediately from the properties of Vj.

5. Algebraic points and proof of Theorem 1.2 The construction required to prove 1.6 is an elaboration of the one used to prove 1.7. We will parameterize algebraic coordinates α ∈ R with [Q(α): Q] ≤ k by means of their minimal polynomials. poly 5.1. Deﬁnition. Let k be a positive integer. For a real number α we deﬁne Hk (α) as follows. If poly [Q(α): Q] > k we set Hk (α) = ∞. Otherwise we set

k poly k X j−1 Hk (α) = min{H(ξ): ξ = (ξ1, . . . , ξk) ∈ Q − {(0,..., 0), ξjα = 0}. j=1

poly (Recall that H(ξ) is the maximum height of the coordinates.) Thus Hk (α) < ∞ when [Q(α): Q] ≤ k. n poly poly For x = (x1, . . . , xn) ∈ R we set Hk (x) = maxi Hk (xi). Now suppose α ∈ Q of degree k with minimal polynomial (over Z)

f(t) = ak(t − α1) ... (t − αk) ∈ Z[t]. Then ([1, 1.6.5 and1.6.6]) k X log |ak| + max(0, log |αi|) = kh(α) i=1 (= log M(f), the Mahler measure of f(t)). Hence

Y k |ak| max(1, |αi|) = H(α) .

Thus we havef(α) = 0 where f has integer coeﬃicients of maximum absolute value at most k Y k k max( | αj| |ak|) ≤ 2 H(α) i i the product being the largest absolute value of the product of i distinct roots of f. Therefore

poly k k Hk (α) ≤ 2 H(α) ,

poly and it suﬃces to prove 1.6, which we establish in a strengthened form for families below, using Hk rather than H.

7 5.2. Lemma. Let n and k be positive integers. Deﬁne

k k n n X j−1 An,k = {(ξ, x) = (ξ1, . . . , ξn, x) ∈ (R − {0}) × R : ξi = (ξi1, . . . , ξik), ξij(xi) = 0, i = 1, . . . , n}. j=1

π k n An,k = π1(An,k) ⊂ (R − {0}) ,

k n n k n where π1 :(R ) × R → (R ) is projection on the first factor. k n k n n There are a finite number of semialgebraic maps φi :(R ) → (R ) × R , sections of π1, defined k n and continuous on semialgebraic sets Ui ⊂ (R ) , with the following property. n poly π If (x1, . . . , xn) ∈ R with Hk (x) ≤ T then there is an index i such that Ui ∩ An,k contains a k n preimage ξ = (ξ1, . . . , ξn) of x under the composition π2φi such that ξ ∈ (Q − {0}) with H(ξ) ≤ T .

n poly k n Proof. Given x = (x1, . . . , xn) ∈ R with Hk (x) ≤ T we can, by definition, find ξ ∈ (Q − {0}) P j with H(ξ) ≤ T and ξijxi = 0 for each i. Therefore ξ is in π1 of the preimage of x under π2 in d n (R − {0})n × R . Now π1 has finite fibres, as a non-zero polynomial of degree ≤ k has at most k roots. We can therefore construct semialgebraic sections. Say ψ1 is the largest root, where a root exists, ψ2 is the second largest root, where there are 2 or more roots, etc up to ψk. These maps are evidently definable in the structure of semialgebraic sets, and therefore they are defined on some semialgebraic sets Vi. Then each Vi may be decomposed into a finite number of semialgebraic sets on which the restriction of ψi is continuous. We take the collection of maps so contructed as our φj. Theorem 1.6 follows from the following version. (For the record, 1.5 also follows a fortiori.) Note again that 1.4 and 3.1 would yield a trivial conclusion when applied to the family Zn,k constructed in the proof.

n m 5.3. Theorem. Let Z ⊂ R × R be a definable family, k a positive integer, and > 0. There exists n m µj J = J(Z, k, ) ∈ N and block families Wj ⊂ R × (R × R ), j = 1,...,J, with the following properties. m µj 1. For each j and (y, η) ∈ R × R , XWj ,(y,η) ⊂ XZ,y. 2. If X = XZ,y is a fibre of Z then X(k, T ) is contained in OZ,k,(T ) blocks contained in X, each µj a fibre of one of the families Wj at some (y, η), η ∈ R . Hence

alg Nk(X − X ,T ) = OZ,k,(T ).

n m 3. Further, let W ⊂ R × R be the family whose fibre at y is the union over all j of all the fibres of Wj at (y, η) of positive dimension. Then W is definable, and, for any y ∈ YZ , if X = XZ,y and alg X = XW,y then X ⊂ X and N(X − X,T ) = OZ,k,(T ).

Proof. Deﬁne

k n n m Zn,k = {(ξ, x, y) ∈ (R − {0}) × R × R :(ξ, x) ∈ An,k, (x, y) ∈ Z},

k n m k n k a deﬁnable family in ((R )n × R ) × R . Let π be the projection of (R )n × R onto (R )n, and

π Zn,k = {(ξ, y): ξ ∈ π(XZn,k,y)},

k m a deﬁnable family in (R )n × R . π We apply Theorem 3.5 to Zn,k with , to obtain a ﬁnite collection of basic block families having the requisite properties.

Applying the semialgebraic maps φi of Lemma 5.2 we get a finite number of block families whose fibres are contained in the fibres of Z, and which satisfy the required properties by 5.2.

8 5.4. Remarks. poly 1. In defining Hk in 5.1 we did not stipulate that the polynomials involved be irreducible, though this will be true when the degree of α is equal to k. It may be possible that an algebraic α of degree poly poly ` < k has Hk (α) < H` (α). module 2. Let us define Hk (α), for α ∈ R of degree ≤ k as the minimum height of a k × k matrix (qij) 2 of rational numbers (considered as a k -tuple) such that there exist complex numbers z1, . . . , zk, not P all zero, such that αzi = qijzj. This height, constructed directly from the definition of an algebraic number as one which preserves a finite (positive) dimensional Q-subspace of C, seems quite natural. If α 6= 0 then multiplication by α is an invertible Q-linear map on the Q-subspace of C generated by zi, module module and so 1/α also preserves this space, so that Hk (1/α) = Hk (α). Of course Theorem 5.3 restricts to algebraic numbers of degree ≤ k, and one can ask whether it is possible to get some comparable result about all algebraic points of height ≤ T on a definable set X. 5.5. Question. Let X : y = f(x), x ∈ [0, 1] be definable and non-algebraic – or even more specifically real analytic and non-algebraic . Let X(Q,T ) denote the set of algebraic points of X of (coordinate-wise) height ≤ T , and N (X,T ) = #X(Q,T ). Is it always true that N (X,T ) = OX,(T )? Q Q 6. Algebraic points on a pfaff curve n 6.1. Definition. ([7, 2.1]) Let U ⊂ R be an open domain. A pfaffian chain of order r ≥ 0 and degree α ≥ 1 in U is a sequence of real analytic functions f1, . . . , fr in U satisfying differential equations n X dfj = gij(x, f1(x), f2(x), . . . , fj(x))dxi i=1

for j = 1, . . . , r, where x = (x1, . . . , xn) and gij ∈ R[x1, . . . , xn, y1, . . . , yj] of degree ≤ α. A function f on U is called a pfaffian function of order r and degree (α, β) if f(x) = P (x, f1(x), . . . , fr(x)), where P is a polynomial of degree at most β ≥ 1. The order and degree of a pfaff curve X will be taken to be the order and degree of the pfaffian function f whose graph is X. In this section we prove Theorem 1.8 by establishing the following somewhat more precise version. 2 6.2. Theorem. Let X ⊂ R be a non-algebraic pfaff curve of order r and degree (α, β). Let K ⊂ R be a number field of degree k. Then 5(r+2) NK (X,T ) = Or,α,β,k((log T ) ).

It is convenient to work with yet another height on algebraic points, one that is often used in transcendence theory. 6.3. Deﬁnition. For an algebraic number α, we let den(α) denote the denominator of α, namely the smallest positive integer δ such that δα is an algebraic integer. If α1, . . . , αk are the conjugates of α we size set H (α) = max(den(α), |α1|,..., |αk|).

Now suppose α ∈ Q of degree k with minimal polynomial (over Z) f(t) = ad(t − α1) ... (tαk) ∈ Z[t] then size Y k H (α) ≤ |ak| max(1, |αi|) = H(α) . So it suﬃces to prove 6.2 using Hsize rather than H, and in this section we will work throughout with Hsize. Accordingly we deﬁne size size size size X (K,T ) = {x ∈ X(K): H (x) ≤ T },NK (X,T ) = #X (K,T ).

6.4. Deﬁnition. Let I be an interval (which may be closed, open or half-open; bounded or unbounded), k ∈ N = {0, 1, 2,...}, L > 0 and f : I → R a function with k continuous derivatives on I. Set AL,0(f) = 1 and, for positive k, |f (i)(x)|Li−1 1/i AL,k(f) = max 1, sup . 1≤i≤k x∈I i!

(so possibly AL,k(f) = ∞ if a derivative of order i, 1 ≤ i ≤ k, is unbounded).

9 6.5. Lemma. Let K ⊂ R be a number ﬁeld with [K : Q] = k. Let d ≥ 1,T ≥ 1,L ≥ 1/T 4k. Let I be an interval of length ≤ L. Suppose that the function f possess D − 1 continuous derivatives on I, with |f 0| ≤ 1, and let X be the graph of f on I. Put D = (d + 1)(d + 2)/2. Then Xsize(K,T ) is contained in the union of at most k 4k4/(3(d+3)) 6 LT AL,D−1 plane algebraic curves of degree ≤ d (possibly reducible).

Proof. This is a variant of [14, 2.2]. Suppose that x1 < x2 < . . . < xD and that (x1, y1),..., (xD, yD) are points of Xsize(K,T ) that do not lie on any plane algebraic curve of degree ≤ d. Then 0 6= ∆ = det φj(xi)

a b where φj, j = 1,...,D ranges over all the functions x f(x) with 0 ≤ a, b ≤ a + b ≤ d. Let si, ti ≤ T be the denominators of xi, yi, and σ1, . . . , σk the embeddings of K into C. Since Q d Q d si ti ∆ is an algebraic integer, and non-zero, the product of its conjugates is a non-zero rational integer. Hence  σi k Y Y d Y d 1 ≤  sj tj ∆ .

i=1 j j

We pursue an upper bound for this quantity. For any σ, using H(xi),H(yi) ≤ T , a straightforward estimation gives |∆σ| ≤ D!T 2dD/3.

Now ∆ is an alternant formed by evaluating the D functions φj, which have D − 1 continuous derivatives, at the D points xi, so by the “Mean Value Theorem” for alternants (see e.g. [2, Prop. 2]) (i−1) ! φ (ξij) ∆ = V (x , . . . , x ) det j 1 D (i − 1)!

where V (x1, . . . , xD) is the Vandermonde determinant and ξij are suitable intermediate points. Since a b the functions φj are the monomial functions x f(x) , a + b ≤ d, we can use column operations to replace the points xi by translations xi − b, so that xi ∈ [0,L], and f by a translation f(x) − c for suitable c so that, since |f 0| ≤ 1, we have (i) 1−i i |f (x)| ≤ i!L (AL,k(f)) for i = 0 as well as i = 1, . . . , k, and all x ∈ I. Then ([11, Lemma 1]) for 0 ≤ a, b and 2, i + 1 ≤ D |(xaf(x)b)(i)| ≤ La+b−i2a(i + 1)b(A (f))i ≤ La+b−iDa+b(A (f))i. i! L,k L,k We use this to estimate the entries in ∆. Expand ∆ into D! products of entries. In each term the multiplicands run over 0 ≤ a, b ≤ a + b ≤ d and over 0 ≤ i ≤ D − 1. Since P(a + b) = 2dD/3 and P i = D(D − 1)/2 over these ranges we ﬁnd, as in [13, 2.4] and the surrounding discussion, that

2dD/3−D(D−1)/2 2dD/3 D(D−1)/2 |∆| ≤ |V (x1, . . . , xD)|D!L D (AL,D−1(f)) . We therefore have

D(D−1)/2 k 2dD/3 2dD/3−D(D−1)/2 8kdD/3 D(D−1)/2 1 ≤ |xD − x1| (D!) D L T (AL,D−1(f)) . Now computation shows (D!D2dD/3)2/(D(D−1)) ≤ 5 for all d, so that the above implies

−2/(D(D−1)) −k 2dD/3 8kdD/3 −1 xD − x1 ≥ 5 L L T (AL,D−1) .

Since the interval I has length ≤ L, it is thus covered by at most

k 4k4/(3(d+3)) k 4k4/(3(d+3)) 5 LT AL,D−1 + 1 ≤ 6 LT AL,D−1 (here we use LT 4k ≥ 1) subintervals on each of which the points of Xsize(K,T ) must lie on a single algebraic curve of degree ≤ d (possibly reducible).

10 6.6. Proposition. Let k ∈ N, L > 0,A ≥ 1 and let I be an interval of length ≤ L. Suppose g : I → R has k continuous derivatives on I. Suppose that |g0| ≤ 1 throughout I and that (a) |g(i)(x)| ≤ i!AiL1−i, all 1 ≤ i ≤ k − 1, x ∈ I, and (b) |g(k)(x)| ≥ k!AkL1−k all x ∈ I. Then I has length ≤ 2L/A. Proof. Let a, b ∈ I. By Taylor’s formula, for a suitable intermediate point ξ,

k−1 X g(i)(a) g(k)(ξ) g(b) − g(a) = (b − a)i + (b − a)k. i! k! i=1 Therefore

k k−1 k−1 i (b − a)A X X (b − a)A L ≤ (b − a)kAkL1−k ≤ (b − a)iAiL1−i + L ≤ L . L L i=1 i=0

k Pk−1 i Thus, if q = (b − a)A/L, then q ≤ i=0 q , whence q ≤ 2, completing the proof. 6.7. Proposition. Let d ≥ 1, k ≥ 1,T ≥ e, L ≥ 1/T 4k and I ⊂ R an interval of length ≤ L. Let K ⊂ R be a numberﬁeld of degree k. Let f : I → R have D continuous derivatives, with |f 0| ≤ 1 and f (j) either non-vanishing in the interior of I or identically vanishing, for j = 1,...,D. Let X be the graph of f. Then Xsize(K,T ) is contained in the union of at most

9 · 6kD(LT 4k)4/(3(d+3)) log(eLT 4k) real algebraic curves of degree ≤ d (possibly reducible). Proof. This follows closely the scheme of proof of [14], and we just sketch the key points of the argument. The interval I may or may not include either of its endpoints. Let us write I = ha, bi allowing h to be either ( or [ and similarly for i, and for the other intervals that arise. For a subinterval J of I write f|J for the restriction of f to J, and denote its graph by X|J . Let size G(f, J) be the minimal number of algebraic curves of degree ≤ d required to contain X|J (K,T ). We pursue an upper bound on G(f, I) by a recursion argument, using the fact that f and its derivatives up to order D are monotonic to subdivide the given interval into subintervals on which AL,K (f) is controlled, the residual intervals being short, as aﬀorded by 6.6, and few, due to the hypothesis on the derivatives of f up to order D. Let A ≥ 2D. Subdivide I at suitable points (i.e. where f and its derivatives assume suitable values) to obtain a “good” subinterval I0 = [s, t] of I (possibly empty!) on which AL,D−1(f) ≤ A, and such that the residual L R 0 0 “bad” intervals J1 = ha, s) and J1 = (t, bi each comprise at most D intervals of the form ha, s ) or (t , bi L R (by the monotonicity) of length ≤ 2L/A (by 6.6), so that J1 ,J1 each have length ≤ 2DL/A. Thus, setting λ = 2D/A, σ = 4/(3(d + 3)), τ = 16k/(3(d + 3)), we have

L R k τ σ L R G(f, I) ≤ G(f, I0) + G(f, J1 ) + G(f, J1 ) ≤ 6 AT L + G(f, J1 ) + G(f, J1 )

L R where J1 ,J1 have length ≤ 2DL/A. We now repeat the subdivision using the same A but with λL instead of L. The monotonicity of L L each derivative up to order D − 1 ensures that the “bad” part of J1 is a single subinterval of J1 of the R form ha, u), and likewise the “bad” subinterval of J1 is of the form (v, bi. In particular, we have only 2 “bad” intervals, and 2 further “good” intervals of length ≤ λL on n−1 4k which AλL,D−1(f) ≤ A. After n iterations of this process we have (provided λ LT ≥ 1 to retain the assumptions of 6.5)

k τ σ n−1 L R G(f, I) ≤ 6 AT L 1 + 2λ + ... + 2λ + G(f, Jn ) + G(f, Jn )

k τ σ L R ≤ 6 AT L (2n − 1) + G(f, Jn ) + G(f, Jn ) L R n (since λ ≤ 1) where Jn ,Jn have (each) length ≤ λ L.

11 Now consider α, β ∈ K with α 6= β and Hsize(α),Hsize(β) ≤ T . Then

k Y σi den(α)den(β)(α − β) i=1

σi 2 is a non-zero rational integer. Since |(α − β) | ≤ 2T for each σi and den(α)den(β) ≤ t we see that

1 1 |α − β| ≥ ≥ . (2T )k−1T 2k T 4k

(since T ≥ e). Take n such that λ/(LT 4k) ≤ λn < 1/(LT 4k). Then an interval of length ≤ λnL contains at most one point α ∈ K with Hsize(α) ≤ T , while the assumption λn−1LT 4k ≥ 1 allows the subdivision process to be repeated n times. We take A = 2eD, i.e. λ = 1/e. We then have n ≤ log(eLT 4k) and

G(f, I) ≤ 6k2eD(LT 4k)4/(3(d+3))(2 log(eLT 4k) − 1) + 2 ≤ 9 · 6kD(LT 4k)4/(3(d+3)) log(eLT 4k)

as required. 6.8 Corollary. With the hypotheses of 6.7, if also L ≤ 2T and T ≥ e then Xsize(K,T ) is contained in at most 12 · (4k + 3) · 6kDT 4(4k+1)/(3(d+3)) log T real algebraic curves of degree ≤ d (possibly reducible). The key point behind the uniform exponent in 6.2 is here in the Corollary. Eventually we will take d = [log T ]. Then T 1/(d+3) ≤ e, and the effect of the number field K is reduced to a constant factor c(k) in the number of curves required. Proof of 6.2. As already observed it suffices to prove the result with the height Hsize. The proof follows exactly the proof of [14], save that we appeal to 6.8 rather than the corresponding result (Corollary 3.3) in [14]. Since f is a pfaffian function, its derivatives are also pfaffian. Indeed, f (i) is pfaffian of order r and degree (α, β + i(α − 1)) ([7, 3.3 or 14, 4.1]). Using the bounds from [7], as quoted in [14, 4.1], we can 2r+3 subdivide I into at most c1(α, β, r)D subintervals on which either f or its inverse (is defined and) satisfies the hypotheses of 6.6. We intersect these intervals with the interval [−T,T ] of the relevant axis, in which all points α ∈ K size size with H (α) ≤ T lie. In each such interval J, X|J (K,T ) is, by Corollary 6.8, contained in at most 12 · (4k + 3) · 6kDT 4(4k+1)/(3(d+3)) log T algebraic curves of degree ≤ d, while, using again the bounds r+1 from [7, or 14, 4.1], the intersection of X with such a curve comprises at most c2(r, α, β)d points. Combining, we have

size 5r+7 4(4k+1)/(3(d+3)) NK (X,T ) ≤ c3(r, α, β, k)d DT log T and taking d = [log T ] completes the proof.

12 References 1. E. Bombieri and W. Gubler, Heights in diophantine geometry, CUP 2006. 2. E. Bombieri and J. Pila, The number of integral points on arcs and ovals, DMJ 59 (1989), 337–357. 3. L. van den Dries, Tame topology and o-minimal structures, CUP 1998. 4. L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. Math. 140 (1994), 183–205. 5. L. van den Dries and C. Miller, Geometric categories and o-minimal structures, DMJ 84 (1996), 497–540. 6. A. Gabrielov, Projections of semi-analytic sets, Funct. Anal. Appl. 2 (1968), 282–291. 7. A. Gabrielov and N. Vorobjov, Complexity of computations with pfaffian and noetherian functions, in Normal Forms, Bifurcations and Finiteness problems in Differential Equations, Kluwer, 2004. 8. A. G. Khovanskii, Fewnomials, Translations of Mathematical Monographs 88, AMS, Providence, 1991. 9. D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 491–494. 10. J. Pila, Geometric postulation of a smooth function and the number of rational points, DMJ 63 (1991), 449–463. 11. J. Pila, Density of integer points on plane algebraic curves, IMRN (1996), 903–912. 12. J. Pila, Integer points on the dilation of a subanalytic surface, QJM 55 (2004), 207–223. 13. J. Pila, Rational points on a subanalytic surface, AIF 55 (2005), 1501–1516. 14. J. Pila, The density of rational points on a pfaff curve, Ann. Fac. Sci. Toulouse, 16 (2007), 635–645. 15. J. Pila and A. J. Wilkie, The rational points of a definable set, DMJ 133 (2006), 591–616. 16. J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Rend. Lincei Mat. Appli. 19 (2008), 149–162. 17. J.-P. Rolin, P.Speissegger and A.J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), 751–777. 18. P. Speissegger, The pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. 19. P. Speissegger, O-minimal expansions of the real field, in Model theory and applications, Quad. Math. 11 (2002), 379–408. 20. A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051– 1094. 21. A. J. Wilkie, A theorem of the complement and some new o-minimal structures, Selecta Math. N.S. 5 (1999), 397–421.

School of Mathematics University of Bristol Bristol, BS8 1TW UK