On the global distance between two algebraic points on a curve

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Journal of Number Theory 104 (2004) 210–254

http://www.elsevier.com/locate/jnt

On the global distance between two algebraic points on a curve Michel Laurenta,
and Dimitrios Poulakisb a

Institut de Mathe´matiques de Luminy, CNRS, 163 Avenue de Luminy, Case 907, 13288 Marseille, Ce´dex 9, France b Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Received 14 May 2002; revised 10 July 2003 Communicated by J.-B. Bost

Abstract We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouville’s inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbert’s irreducibility Theorem and to some classes of diophantine equations in the circle of Runge’s method. r 2003 Elsevier Inc. All rights reserved. MSC: 11J85 Keywords: Diophantine inequalities; Algebraic functions; Hilbert’s irreducibility Theorem; Diophantine equations

1. Introduction and Liouville’s type inequalities on a curve We revisit some classical works on diophantine properties of algebraic functions in the framework of interpolation determinants. See Refs. [1,2,5,16,17, Chapters 5 and 6] as a non-exhaustive list of the topics we have in mind. The common feature of these works consists in applying the machinery of transcendence to the algebraic functions whose expansions in Taylor series are G-functions. Our approach enables


Corresponding author. E-mail address: [email protected] (M. Laurent).

0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.006

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us to establish, what we call ‘‘Liouville’s type inequalities’’ on a curve. These diophantine inequalities are symmetrical in the two points involved, since the origin and the ‘‘extrapolation point’’ play equivalent roles in the context of interpolation determinants. They are applied in Section 2 to Hilbert’s irreducibility Theorem and to some diophantine equations in the circle of Runge’s method, as in [6] or [17]. We specifically have taken care of the dependency upon the partial degrees of the polynomials involved. See [11] for an example of situation related to Hilbert’s irreducibility Theorem, in which this dependency is important. We have also investigated Runge’s method for the set of integral points rational over a totally real number field of arbitrary degree. Theorem 6 below furnishes a new result in this direction. Let us first introduce some notations which are standard. For any embedding of fields % s : Q+C s; where Cs denotes either the field C of complex numbers, or its p-adic analogue Cp ; equiped with their usual norms j:j (normalized by jpj ¼ p1 in the p-adic case), we % defined by jajs ¼ jsaj for any algebraic denote by j  js the induced metric on Q number a: Let L be a number field and let x ¼ ðx1 ; y; xn Þ be any n-tuple in Ln ; we % denote by 1 hðxÞ ¼ ½L : Q %

X

logþ maxðjx1 js ; y; jxn js Þ;

s : L+Cs

where the summation index s ranges over all the distinct embeddings of the number field L into C or Cp ; the (absolute logarithmic) height of x: Recall that the height hðxÞ % the n-tuple x: % is independent of the choice of a field of rationality L for % Now the classical Liouville’s inequality can be stated as follows: for any distinct elements x1 and x2 of L; we have 1 ½L : Q

X s : L+Cs

logþ

1 phðx1 Þ þ hðx2 Þ þ log 2; jx1  x2 js

where s ranges as above along all the distinct embeddings of L into C or Cp : The lefthand side of this inequality can be viewed as a measure of global distance between x1 and x2 involving all the possible metrics. Our goal is to generalize this inequality to an arbitrary algebraic plane curve C; the classical Liouville’s inequality corresponding to the special case where C is a coordinate axis in the plane. It turns out that such extensions are known in arithmetical geometry and will be described in Section 3. We propose here an elementary and completely explicit formulation which enables us to deduce easily effective statements for the indicated applications.

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% and let Let K be a number field, viewed as a subfield of Q; F ðX ; Y Þ ¼ An ðX ÞY n þ ? þ A0 ðX Þ ¼

m X n X i¼0

ai; j X i Y j

j¼0

be an absolutely irreducible polynomial in K½X ; Y : Denote by hðF Þ ¼

1 ½K : Q

X

log

s : K+Cs

max

0pipm; 0pjpn

ðjai; j js Þ;

the (projective) height of the polynomial F : We shall suppose throughout this article that its partial degree n in the variable Y is X2 and that its degree m in X is X1: Let C be the plane curve with equation F ðx; yÞ ¼ 0; and for any field L containing K; denote by CðLÞ the set of L-rational points of C: To formulate our results, we need to introduce some definitions relying on the notion of algebraic functions and of Riemann surfaces. If Cs stands either for the fields C or Cp ; we denote by Bs ðx; rÞ the open disk in Cs with center xACs and radius r40: % be an algebraic point on C such that An ðxÞFY0 ðx; zÞa0: In Let P ¼ ðx; zÞACðQÞ other words, the projection ðx; yÞ/x restricted to C is supposed to be a local formal % isomorphism at the point P: Let s : Q+C s be an embedding as above. The branch yP;s is the meromorphic function yP;s : Bs ðsx; Rs ðPÞÞ-Cs ,fNg defined in a disk centered at sx with maximal radius Rs ðPÞ such that sF ðx; yP;s ðxÞÞ  0

and

yP;s ðsxÞ ¼ sz:

Notice that 0oRs ðPÞoN and that in the complex case Cs ¼ C; we have the lower bound Rs ðPÞXminjsx  aj; where a ranges along the set of branch points in the x-plane of the Riemann surface uniformizing the algebraic function y defined by the equation sF ðx; yÞ ¼ 0: We say % belongs to the branch yP;s if that the point P0 ¼ ðx0 ; z0 ÞACðQÞ jx  x0 js oRs ðPÞ

and

yP;s ðsx0 Þ ¼ sz0 :

Now let L be a number field containing K; let P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ be two points ACðLÞ: Suppose that An ðxi ÞFY0 ðxi ; zi Þa0 for i ¼ 1; 2: We define S ¼ SðL; P1 ; P2 Þ to be the set of embeddings s : L+Cs ¼ C or Cp such that either P2 belongs to the branch yP1 ;s or P1 belongs to the branch yP2 ;s : Roughly speaking, S is the set of embeddings s for which either sP2 is located on a circular patch of Cs ðCs Þ centered at sP1 ; or reversely sP1 lies on a similar disk centered at sP2 : Notice also that when jx1  x2 js ominðRs ðP1 Þ; Rs ðP2 ÞÞ; the assertions yP1 ;s ðsx2 Þ ¼ sz2 and

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yP2 ;s ðsx1 Þ ¼ sz1 are equivalent, since the functions yP1 ;s and yP2 ;s are then equal in the intersection Bs ðsx1 ; Rs ðP1 ÞÞ-Bs ðsx2 ; Rs ðP2 ÞÞ: We can now state our first extension of Liouville’s inequality relative to C:

Theorem 1. Let P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ be two points ACðLÞ: Suppose m ¼ degX F X1; x1 ax2

and

n ¼ degY F X2

An ðx1 ÞAn ðx2 ÞFY0 ðx1 ; z1 ÞFY0 ðx2 ; z2 Þa0:

Then X pffiffiffiffiffiffiffiffiffi 1 1 hðx ; x Þ logþ p 1 2 þ r þ h1 h2 þ log 4; ½L : Q sAS jx2  x1 js n where r¼

X 1 1 logþ ½L : Q sAS maxðRs ðP1 Þ; Rs ðP2 ÞÞ

and for i ¼ 1; 2; hi ¼ mð2n  1Þhðxi Þ þ ð2n  1ÞhðF Þ þ logð8ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 Þ:

Remarks. (1) Since jx2  x1 js omaxðRs ðP1 Þ; Rs ðP2 ÞÞ for any sAS; the same statement remains valid with S replaced by any subset of S: (2) We have the upper bound ( ) X X 1 1 1 1 ; logþ logþ rpmin ½L : Q sAS Rs ðP1 Þ ½L : Q sAS Rs ðP2 Þ so that the term r is bounded for fixed P1 : An inequality of this type was first obtained by Sprindzuck [16]. See also Theorem 1.1 of [6] for a version with explicit constants. (3) We have hðx1 ; x2 Þphðx1 Þ þ hðx2 Þ: The upper bound furnished by Theorem 1 may be better than the classical Liouville’s inequality since nX2: (4) We have measured the distance of P1 and P2 by measuring the distance of their projections x1 and x2 on the x-axis. The process is obviously significant only when the map ðx; yÞ/x is locally smooth in a neighborhood of the two points. The proximity to the singular locus of this map is taken into account on the right-hand side of the inequality through the term r:

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Next we give a projective version of Theorem 1. We shall now measure the distance of the two points using the projective distance ds ðx1 ; x2 Þ ¼

jx2  x1 js ; maxð1; jx1 js Þmaxð1; jx2 js Þ

for any algebraic numbers x1 and x2 ; and any embedding s as above. Notice that this distance is equivalent (see Lemma 8 for a precise statement) with the quantity 1 1 % minð1; jx2  x1 js ; jx1 2  x1 js Þ: If we view the projective space P ðQÞ as the glueing 1 % identifying x with x ; our projective distance turns out to be of two copies of Q essentially the minimum of the distances in the two maps. Remark also that jx2  1 x1 js pjx1 2  x1 js exactly when jx1 x2 js p1; which explains the splitting of our definitions into the three cases below. Besides of the preceding notion of branch, we need its analogue for the map at infinity. For any non-zero xACs and any r40; denote B˜ s ðx; rÞ ¼



    1 1  xACs :   or : x x

Then B˜ s ðx; rÞ is a neighborhood of x which is either a disk or the complementary of a % be a point such that xAn ðxÞFY0 ðx; zÞa0: We define disk in Cs : Let P ¼ ðx; zÞACðQÞ the branch y˜ P;s to be the meromorphic function y˜ P;s : B˜ s ðsx; R˜ s ðPÞÞ-Cs ,fNg defined in a projective disk B˜ s ðsx; R˜ s ðPÞÞ with maximal radius R˜ s ðPÞ such that sF ðx; y˜ P;s ðxÞÞ  0

and

y˜ P;s ðsxÞ ¼ sz:

% belongs to the branch y˜ P;s when Analogously, we say that a point P0 ¼ ðx0 ; z0 ÞACðQÞ    1 1    oR˜ s ðPÞ and  x0 x s

y˜ P;s ðsx0 Þ ¼ sz0 :

Let L be a number field containing K and let P1 and P2 be two points ACðLÞ as before. We shall suppose moreover that x1 x2 a0: For each embedding s : L+Cs denote 8 if jx1 x2 js o1; > < maxðRs ðP1 Þ; Rs ðP2 ÞÞ 0 ˜ ˜ if jx1 x2 js 41; Rs ðP1 ; P2 Þ ¼ maxðRs ðP1 Þ; Rs ðP2 ÞÞ > : ˜ ˜ maxðRs ðP1 Þ; Rs ðP2 Þ; Rs ðP1 Þ; Rs ðP2 ÞÞ if jx1 x2 js ¼ 1:

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We define now S0 to be the set of embeddings s : L+Cs such that 1 0 minðjx2  x1 js ; jx1 2  x1 js ÞoRs ðP1 ; P2 Þ

and either * * *

P1 belongs to the branch yP2 ;s or P2 belongs to the branch yP1 ;s if jx1 x2 js o1; P1 belongs to the branch y˜ P2 ;s or P2 belongs to the branch y˜ P1 ;s if jx1 x2 js 41; P1 belongs to the branch yP2 ;s or to the branch y˜ P2 ;s ; or P2 belongs to the branch y˜ P1 ;s or to the branch y˜ P1 ;s ; if jx1 x2 js ¼ 1:

In other words, S 0 is the set of embeddings s for which sP1 or sP2 lie on some circular patches of the preceding types associated to the coverings ðx; yÞ/x or ðx; yÞ/x1 ; and prescribed by the value of ds ðx1 ; x2 Þ: When jx1 x2 js ¼ 1; both coverings are convenient and we require that at least one of the points P1 or P2 belongs to the union of the two branches associated with the other point. Theorem 2. Under the assumptions of Theorem 1; suppose that x1 x2 a0: Then qffiffiffiffiffiffiffiffiffi X 1 1 hðx Þ þ hðx2 Þ þ r0 þ h01 h02 þ log 8 p 1 logþ ½L : Q sAS0 ds ðx1 ; x2 Þ n where X 1 1 logþ 0 r ¼ ½L : Q sAS0 Rs ðP1 ; P2 Þ 0

and

h0i

  1 ¼ hi þ 2 1  hðxi Þ for i ¼ 1; 2: n

Suppose moreover that the polynomials F ðx1 ; Y Þ ¼ An ðx1 Þ

n Y

ðY  z1; j Þ

and

F ðx2 ; Y Þ ¼ An ðx2 Þ

j¼1

n Y

ðY  z2;k Þ

k¼1

have simple roots. Then we have the lower bound qffiffiffiffiffiffiffiffiffi X 1 1 hðx1 Þ þ hðx2 Þ  nr00  ðn2  1Þ h01 h02 X logþ ½L : Q sAS0 ds ðx1 ; x2 Þ n  ð2n2 þ n  1Þlog 2 with r00 ¼

½L0

1 : Q

X s : L0 +Cs

logþ

1 ; minj;k R0s ðP1; j ; P2;k Þ

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where L0 ¼ Lðz1;1 ; y; z2;n Þ denotes the field generated over L by all the roots z1; j ; z2;k and P1; j ¼ ðx1 ; z1; j Þ;

P2;k ¼ ðx2 ; z2;k Þ

for j; k ¼ 1; y; n:

Instead of the bidegree ðm; nÞ of the polynomial F ; we focus now on its total degree.

Theorem 3. Let n be an integer X2: Suppose that the polynomial F has the form X F ðX ; Y Þ ¼ ai; j X i Y j iþjpn

with a0;n a0: Then Theorems 1 and 2 hold with hi defined now by hi ¼ ð2n  2Þhðxi Þ þ ð2n  1ÞhðF Þ þ logð8ðn þ 1Þ4nþ6 Þ for i ¼ 1; 2: Remark. The above values for hi are smaller than those obtained by putting m ¼ n in Theorem 1. Notice however that the hypotheses of Theorem 3 are more restrictive, since we assume a0;n a0: The proofs of Theorems 1–3 follow the same lines. We shall give a detailed proof for Theorem 1 and will only indicate the modifications of the argumentation for Theorems 2 and 3. The scheme of proof is as follows. We first define some matrix M of evaluation of monomial functions on the curve C at the two points P1 and P2 with high multiplicities. It will be easily verified that the rows of the matrix M are linearly independent. Let D be a non-vanishing minor of maximal order extracted from M: The determinant D is an example of what we call an interpolation determinant. Its various absolute values jDjs can be bounded in different ways, the crucial estimate being provided by the analytical Lemma 7 below. Next, Theorems 1–3 follow easily from the product formula applied to D: In our approach, the multiplicities at P1 and P2 may be arbitrary positive integer parameters and we take advantage of this flexibility in the construction to get a reminder term of the expected size pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O hðx1 Þhðx2 Þ :

2. Arithmetical applications It is known that diophantine inequalities, as those from Section 1, imply Hilbert irreducibility Theorem as well as Runge’s Theorem on Diophantine equations. This approach provides effective results and was developed in the papers [3,6,17]. In this

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section, we apply our estimations in these classical situations. We have not tried to be exhaustive, giving a simple example in the context of Hilbert irreducibility Theorem, and formalizing somehow Runge’s Theorem. In most cases, the method appears to be efficient only when the field of rationality L has degree ½L : Q on: Notice however that Theorem 6 below concerns algebraic points of any degree. It will be convenient to set HðF Þ ¼ ehðF Þ for any polynomial F and HðxÞ ¼ ehðxÞ for any algebraic number x: 2.1. Effective Hilbert irreducibility theorem In this section, we restrict to the case K ¼ L ¼ Q; which leads to a simple statement with a fairly precise upper bound. Compare for instance with the Addition to Corollary 1.3 of [6].

Theorem 4. Let F be an absolutely irreducible polynomial in Q½X ; Y with m ¼ degX F X1 and n ¼ degY F X2: Suppose F ð0; 0Þ ¼ 0 and

An ð0ÞFY0 ð0; 0Þa0:

Let k be a non-zero integer, p a prime number, and t a positive integer. Then the polynomial F ðx; Y Þ is irreducible in Q½Y for (  2 3 1=k jkjXH 3mn ðn1Þ ; x¼ whenever 2 3 pt pt XH 3mn ðn1Þ ; where we have set H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 : Suppose moreover that F satisfies the assumptions of Theorem 3. Then the same 2 3 statement holds for m ¼ n whenever jkj or pt are XH 3n ðn1Þ :

2.2. Diophantine equations Let us first complete the algebraic plane curve C in the two following ways. Homogeneizing the polynomial F ; first into a bihomogeneous polynomial with bidegree ðm; nÞ; and secondly with respect to the total degree in X and Y ; we obtain projective embeddings CDP1  P1

and

CDP2 :

We shall refer to the embedding in P2 only when F satisfies the assumptions of Theorem 3. Denote again by C the Zariski closure of the affine curve C in P1  P1 or

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in P2 : Notice that we can extend the projection ðx; yÞ/x into a morphism p : C-P1 ; since a0;n is supposed to be non-zero in Theorem 3. Set CN ¼ p1 ðNÞ: We shall say that p (or sometimes F ) is regular at N, if CN is made up with n distinct points which are different from ðN; NÞ in the case of an embedding in P1  P1 : This definition means in term of F that the leading polynomial FN ðTÞ ¼

n X

am; j T j ;

j¼0

or

FN ðTÞ ¼

n X

anj; j T j

j¼0

for the embedding in P2 ; has exactly n simple roots. For any number field L; we denote by OL its ring of integers. Theorem 5. Let d be an integer pn  1: Suppose that p is regular at N and that each point QACN is rational over some number field L containing K of degree ½L : Q pd: Let ðx; zÞAOK  K be such that F ðx; zÞ ¼ 0: Then HðxÞpH 3mn

d =ðndÞ2

3 2

with H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 :

Suppose moreover that F satisfies the assumptions of Theorem 3. Then HðxÞpH 3n

d =ðndÞ2

3 2

with H ¼ 8ðn þ 1Þ4nþ6 HðF Þ2n1 :

d Remarks. (1) Bounding nd pn  1; we obtain an effective version of Runge’s theorem, which is consistent with the results of [18] obtained for K ¼ Q: Notice however that we have assumed here that F should be regular at N: The results of [6] (see Corollary 1.3) are more general than our Theorem 5, but the bounds are larger. (2) If we know a priori that the point ðx; zÞ belongs to some specified Archimedean branches at infinity, the upper bound remains valid with d replaced by the maximal degree over Q of the corresponding points QACN :

In order to prove Theorems 4 and 5, we shall apply Theorems 1 and 3 to a subset of S made up with only one embedding, according to Remark 1 after Theorem 1. In some (unfortunately restrictive) situations, the set S turns out to be larger. Here is an example.

Theorem 6. Suppose that K is a totally real number field of degree d over Q and that p is regular at N: Let L be any totally real number field and let ðx; zÞAOL  L satisfy F ðx; zÞ ¼ 0: Suppose that there exists a point Q0 ACN ; such that for each embedding % s : Q+C; sðQ0 Þ is the only real point of sðCN Þ: Then we have the uniform bound HðxÞpH 4dmn

3

=ðn1Þ2

with H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 :

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Suppose moreover that F satisfies the assumptions of Theorem 3. Then 3

HðxÞpH 4dn

=ðn1Þ2

with H ¼ 8ðn þ 1Þ4nþ6 HðF Þ2n1 :

Remarks. (1) Our hypotheses imply that the point at infinity Q0 is necessarily rational over K: (2) Notice that when the sets sðCN Þ have no real point, a similar statement holds with a better upper bound. See [14] for the case of an embedding in P2 : We may apply Theorem 6 to equations of the shape f ðxÞ ¼ gðzÞ: See Chapter 5 of [1] and Theorem 3 of [18] for previous results of this kind in the special case of superelliptic equations zn ¼ f ðxÞ:

Corollary. Let K be a totally real number field of degree d; let n be an odd integer X3; and let f1 ; f2 ; g be three non-zero polynomials AK½X with deg g ¼ n: Suppose that the polynomial F ðX ; Y Þ ¼ f2 ðX ÞgðY Þ  f1 ðX Þ is absolutely irreducible, that f1 ; f2 and g have leading coefficients which are nth powers in K; and that deg f1  deg f2 is divisible by n in Z: Let L be a totally real number field and let ðx; zÞAOL  L be such that F ðx; zÞ ¼ 0: Then HðxÞpð8ðm þ 1Þ2nþ3 ðn þ 1Þ2nþ3 HðF Þ2n1 Þ4dmn

3

=ðn1Þ2

with m ¼ maxðdeg f1 ; deg f2 Þ:

3. Geometrical interpretation Bombieri gave in [3] an alternative proof of Sprindzuck’s inequalities [16], using the theory of heights on the curve C: We extend his arguments and give an interpretation of our Theorems 2 and 3 in terms of height functions associated to divisors on the product C  C: We refer to Chapter 5 of [10] for the results quoted in this section, and more globally to the whole book for the algebraic geometry’s background. For simplicity, let us restrict to polynomials F as in Theorem 3. Embed C into P2 as in Section 2.2 and denote again by C its Zariski closure. Assume that the projective curve C is smooth and that its genus g ¼ ðn  1Þðn  2Þ=2 is X1: Let J be the jacobian of C: We fix a normalized embedding i : C+J; defined by iðPÞ ¼ ðPÞ  a;

% PACðQÞ;

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for some divisor a on C of degree one, chosen in such a way that the theta divisor Y ¼ iðCÞ þ ? þ iðCÞ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ðg1Þ times

on J should be symmetrical. Let P ¼ p
1 Y þ p
2 Y  s
Y be the Poincare´ divisor on J  J; where the maps s; p1 ; p2 : J  J/J denote, respectively, the sum of the two components, the projections on the first and second factors. According to a theorem due to Mumford, we know that its restriction ði  iÞ
ðPÞ on C  C is a divisor linearly equivalent to D  a  C  C  a; where D is the diagonal in C  C: Let hˆY be the Ne´ron–Tate canonical height associated with the divisor Y: Remind that hˆY extends to a positive quadratic form % on JðQÞ#R; and denote by /:; :S the bilinear form associated to the quadratic form hˆY : In terms of height functions, we obtain the following equivalences modulo % bounded functions on ðC  CÞðQÞ: hD ðP1 ; P2 Þ  ha ðP1 Þ  ha ðP2 Þ ¼ hP ðiðP1 Þ; iðP2 ÞÞ þ Oð1Þ ¼  hˆY ðiðP1 Þ þ iðP2 ÞÞ þ hˆY ðiðP1 ÞÞ þ hˆY ðiðP2 ÞÞ þ Oð1Þ ¼  2/iðP1 Þ; iðP2 ÞS þ Oð1Þ: Remark that pffiffiffiffiffiffiffiffiffiffiffi 1 ha ðP1 Þ ¼ hðx1 Þ þ Oð hðx1 ÞÞ n

and

pffiffiffiffiffiffiffiffiffiffiffi 1 ha ðP2 Þ ¼ hðx2 Þ þ Oð hðx2 ÞÞ; n

while the term hD ðP1 ; P2 Þ measures a global distance on C between P1 and P2 ; whenever P1 aP2 : In Theorems 2 and 3, the quantity   X 1 minf1; ds ðx1 ; x2 Þg 1 log ½L : Q sAS0 minf1; R0s ðP1 ; P2 Þg plays the role of hD ðP1 ; P2 Þ: Now Cauchy–Schwarz inequality reads 1 2jhP ðiðP1 Þ; iðP2 ÞÞj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j/iðP1 Þ; iðP2 ÞSjp hˆY ðiðP1 ÞÞhˆY ðiðP2 ÞÞ:

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We also know from Mumford that pffiffiffiffiffiffiffiffiffi g hˆY ðiðPÞÞ ¼ hðxÞ þ O hðxÞ ; n % Thus Theorems 2 and 3 can be viewed as explicit for any point P ¼ ðx; zÞACðQÞ: formulations in terms of naı¨ ve heights of the Cauchy–Schwarz inequality for the quadratic form hˆY : Notice however that in the situation of Theorem 3, we get values h1 and h2 of the shape hi ¼ ð2n  2Þhðxi Þ þ Oð1Þ which are slightly larger than the pffiffiffiffiffiffiffiffiffiffi expected quantities ð2g=nÞhðxi Þ þ O hðxi Þ : It may be enlightening to deduce alternatively the preceding facts from basic properties of arithmetic intersection. Let us fix some arithmetic surface X with generic fiber C K L and for each point PACðLÞ; denote by EP the horizontal divisor on X having P ¼ ðEP ÞL for generic point. Suppose for simplicity that the divisor a is rational over L: Then we can choose the Arakelov’s intersection product

ha ðPÞ ¼

ðEP  Ea Þ ½L : Q

as a height function associated to the divisor a: The main observation is that the bilinear form 2/DL ; EL S connected with the Ne´ron–Tate height coincides with the intersection pairing ðD  EÞ=½L : Q for any Arakelov’s divisors D and E on X which are orthogonal to the subgroup V generated by the irreducible components of the vertical fibers. See [8,9] for a proof. Select now bounded divisors F1 and F2 in V such that EP1  Ea þ F1 and EP2  Ea þ F2 are orthogonal to V : Then, by bilinearity of the intersection pairing, we can write ðEP1  Ea þ F1  EP2  Ea þ F2 Þ ½L : Q ðEP1  EP2 Þ ¼  ha ðP1 Þ  ha ðP2 Þ þ Oð1Þ: ½L : Q

2/iðP1 Þ; iðP2 ÞS ¼

Thus we can choose

hD ðP1 ; P2 Þ ¼

ðEP1  EP2 Þ : ½L : Q

Now the adjunction formula ðEP  EP Þ ¼ ðEP  oX Þ; see [8] for more informations, pffiffiffiffiffiffiffiffiffiffiffiffi provides us with the estimate hˆY ðiðPÞÞ ¼ gha ðPÞ þ O ha ðPÞ :

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4. Auxiliary results We collect in this section the necessary lemmas. Most of the material is concerned with the properties of iterated derivatives on C: We shall also need to estimate the various radii involved in Theorems 1 and 2, which is the content of Section 4.2. 4.1. Derivations on a curve Let KðCÞ be the field of rational functions on the irreducible curve C: Thus KðCÞ ¼ Kðx; yÞ where x is transcendental over K and F ðx; yÞ ¼ 0: We shall be concerned with the two derivations @

and

@* : KðCÞ-KðCÞ

of the field KðCÞ which are the unique extensions to KðCÞ of the derivations @=@x and @=@x1 operating on the subfields KðxÞ and Kðx1 Þ; respectively. We also % % extend the derivations @ and @* to the field QðCÞ :¼ KðCÞ#K Q: % We are concerned with points P ¼ ðx; zÞACðQÞ such that An ðxÞFY0 ðx; zÞa0: The projections p : ðX ; Y Þ/X and p* : ðX ; Y Þ/X 1 ; whenever xa0; are then local % formal isomorphisms at P: To each rational function jAQðCÞ regular at P; we associate the two local functions fP ¼ j3p1

and

gP ¼ j3p* 1 :

In other words, fP and gP are the formal series in X  x and X 1  x1 ; respectively, which represent the function j viewed as an algebraic function of the variable X or X 1 around x or x1 : Notice that formally fP ðzÞ ¼ gP ðz1 Þ: Then for each embedding % s : Q+C s we have the equalities sðjðP0 ÞÞ ¼ sfP ðsx0 Þ ¼ sgP ðsx01 Þ % belonging to the intersection of the branches yP ; s for any point P0 ¼ ðx0 ; z0 ÞACðQÞ 0 01 and y˜ P ; s; such that sx and sx lie in the disks of convergence of the series sfP and sgP : * For tX1 we Let us begin by some formal properties of the derivations @ and @: 1 shall denote by @ ½t ¼ ðt!Þ @ ðtÞ the operator @ iterated t times and divided by t!: % Lemma 1. Let jAQðCÞ be regular at the smooth point P for p and p: * Then for every tX0; we have ½t

@ ½t j3p1 ¼ fP

and

½t @*½t j3p* 1 ¼ gP ;

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where ½t fP

  1 @ t ¼ fP t! @X

and

½t gP

 t 1 @ ¼ gP t! @X 1

denote the divided derivatives of the one variable functions fP ðX Þ and gP ðX 1 Þ:

Proof. For t ¼ 0; these are the definitions of fP and gP : Notice that the first equality % is obvious for jAQðxÞ: For j ¼ y and t ¼ 1; this equality is also valid since @y ¼ 

FX0 ðx; yÞ and FY0 ðx; yÞ

ðy3p1 Þ0 ðX Þ ¼ 

FX0 ðX ; y3p1 ðX ÞÞ ; FY0 ðX ; y3p1 ðX ÞÞ

both relations coming by differentiating the identities F ðx; yÞ ¼ 0

and

F ðX ; y3p1 ðX ÞÞ  0:

The first equality is then satisfied for any j when t ¼ 1: Next we argue by induction for tX1: The proof for @* is similar. & Lemma 2. For any integer tX1; we have the formula  t  X t1 @ ¼ ð1Þ xtt @*½t : t  1 t¼1 ½t

t

* 1 Þ; observe that Proof. Since @ðx1 Þ ¼ x2 ¼ x2 @ðx * @ ¼ x2 @; which proves the lemma for t ¼ 1: From this relation and Lemma A3 from [13], we can write ½t

@ ¼

t X t¼1

at;t @*½t

with at;t

X  t  Y @ ½i1 ðx2 Þsi ¼ : P i s iX1 si ¼t % P isi ¼t

Since @ ½i1 ðx2 Þ ¼ @ ½i ðx1 Þ ¼ ð1Þi xi1 ; i

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we get

1

0

B X  C   B t C t1 C ¼ ð1Þt xtt C BP t1 @ si ¼t s A % P

t tt B

at;t ¼ ð1Þ x

isi ¼t

according to formula 3h from [4, p. 146]. & Let us now give some rational expression for @ ½t yv which is based on an explicit version of the formal implicit functions theorem, contained in the appendix of [13]. Write m X n X F ðX ; Y Þ ¼ ai; j X i Y j i¼0

j¼0

and denote by a the ðm þ 1Þðn þ 1Þ-tuple of its coefficients which we view for the % moment as indeterminates. Denote as usual by LðÞ the length of any polynomial with integral coefficients, that is the sum of the absolute value of its coefficients.

Lemma 3. For each integer tX0; vX1; there exists a polynomial Gv;t AZ½a; X ; Y such % that G ð a; x; yÞ v;t @ ½t yv ¼ % FY0 ðx; yÞ2t and

dega Gv;t p2t;

degX Gv;t pð2m  1Þt; degY Gv;t pð2n  2Þt þ v;   tþv1 ðm þ 1Þ5t ðn þ 1Þ6t : LðGv;t Þp v1

%

Assume moreover that m ¼ n and that the total degree in X and Y of the polynomial F is pn: Then the total degree in X and Y of the polynomial Gv;t is pð2n  3Þt þ v:

Proof. The result is obvious for t ¼ 0: For brevity write Fi; j ¼ ði!j!Þ1 and for any tX1; set G1;t ¼

X t¼ðti; j Þi; j %

@ iþj F ðX ; Y Þ; @X i @Y j

ð2t

qðtÞF0;1 %

P

ti; j Þ

Y

ðFi; j Þti; j ;

ði;jÞ

where the summation index t ranges over the set of ðmn þ m þ n  1Þ-tuples of % integers t ¼ ðti; j Þ with 0pipm; 0pjpn and ði; jÞað0; 0Þ; ð0; 1Þ; %

ARTICLE IN PRESS M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254

satisfying

X

iti; j ¼ t;

i; j

X

ð j  1Þti; j ¼ 1;

1p

X

i; j

225

ti; j p2t  1;

i; j

and where 1

qðtÞ ¼ P % i; j ti; j

P

i; j

ti; j

 :

t % It is shown in Lemma A2 from [13] that @ ½t y ¼ G1;t =FY0 2t: Moreover the coefficients qðtÞ are integers, so that G1;t has integral coefficients. Now we easily check the % following estimations of degrees: ( ) X X dega G1;t p max 2t  ti; j þ ti; j ¼ 2t; %

( degX G1;t p max m 2t  t %

t %

i; j

X

ðn  1Þ 2t 

t %

ti; j

þm

X

i; j

( degY G1;t p max

!

i; j

X i; j

! ti; j

þn

ti; j 

X

i; j

i; j

X

X

i; j

ti; j 

) iti; j

¼ ð2m  1Þt; )

jti; j

¼ ð2n  2Þt þ 1:

i; j

On the other hand, the polynomial    m X n X m n am;n Fi; j ¼ X mi Y nj ; i j m¼0 n¼0 when viewed as a polynomial in Z½a; X ; Y ; has length %   m X n    X m n mþ1 nþ1 LðFi; j Þ ¼ ¼ pðm þ 1Þiþ1 ðn þ 1Þ jþ1 : i j i þ 1 j þ 1 m¼0 n¼0 We deduce that P Y X qðtÞLðF0;1 Þð2t ti; j Þ LðFi; j Þti; j LðG1;t Þp % t ði;jÞ % P P P P X 2t t þ ðiþ1Þti; j Þ i; j ðn þ 1Þð2ð2t ti; j Þþ ð jþ1Þti; j Þ p qðtÞðm þ 1Þð % t % ! X qðtÞ ðm þ 1Þ3t ðn þ 1Þ4t1 : p % t %

Since X t %

qðtÞ ¼ %

0

2X t1 k¼1

1   t1 X k C 2X 1B 1 B Cp ðmn þ m þ n  1Þk pððm þ 1Þðn þ 1ÞÞ2t1 ; @ A k Pt k t k¼1 % % ti; j ¼k

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we obtain the upper bound LðG1;t Þpðm þ 1Þ5t1 ðn þ 1Þ6t2 pðm þ 1Þ5t ðn þ 1Þ6t ; thus proving the lemma when v ¼ 1: For vX1; we use Leibnitz formula @ ½t yv ¼

X

@ ½t1 y  ?  @ ½tv y ¼

t1 þ?þtv ¼t

Gv;t FY0 2t

with X

Gv;t ¼

G1;t1  ?  G1;tv :

t1 þ?þtv ¼t

The upper bounds for the partial degrees of Gv;t are then obvious and   X tþv1 LðGv;t Þp LðG1;t1 Þ  ?  LðG1;tv Þp ðm þ 1Þ5t ðn þ 1Þ6t : v  1 t1 þ?þtv ¼t Suppose now that m ¼ n and degðX ;Y Þ F pn: Since degðX ;Y Þ Fi; j pn  i  j; we deduce from the explicit formula for G1;t that ( ! ) X X degðX ;Y Þ G1;t p max ðn  1Þ 2t  ti; j þ ðn  i  jÞti; j ¼ ð2n  3Þt þ 1: t %

i; j

i; j

For any vX1; Leibnitz formula then implies the upper bound degðX ;Y Þ Gv;t pð2n  3Þt þ v:

&

4.2. Around Eisenstein’s Theorem % We are concerned here with the Taylor coefficients of a function jAQðCÞ at a % Eisenstein’s Theorem provides us with an upper bound for these point PACðQÞ: % Our results from coefficients relatively to any p-adic or Archimedean metric on Q: Section 4.1 imply a fairly precise version of this theorem for any smooth point P for p: See [7,15] for previous results of this kind. % Let s : Q+C s be an embedding as before and define  0 if Cs ¼ Cp ; e¼ 1 if Cs ¼ C: Denote by jF js ¼

max

0pipm;0pjpn

ðjai; j js Þ;

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where we recall that a ¼ ðai; j Þ stands for the tuple of the coefficients of the % polynomial F : % with An ðxÞFY0 ðx; zÞa0: Denote Lemma 4. Let P ¼ ðx; zÞ be a point ACðQÞ Hs ðPÞ ¼ 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 maxð1; jxjs Þ2mnm1 : s For any integers tX0; vX1; we have the upper bound j@ ½t yv ðPÞjs pcvs;P



tþv1

e

!t

Hs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s

v1

with cs;P ¼

ðm þ 1Þe ðn þ 1Þe jF js maxð1; jxjs Þm : jAn ðxÞjs

Moreover if we suppose xa0; we have *½t v

j@ y

ðPÞjs pcvs;P



tþv1 v1

e

jxjs Hs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s

!t :

Proof. Let us first bound jzjs in term of jxjs : Since F ðx; zÞ ¼ 0; we can write the formula z¼

An1 ðxÞ þ ? þ A0 ðxÞznþ1 An ðxÞ

from which follows the upper bound maxð1; jzjs Þp

ððm þ 1Þðn þ 1ÞÞe jF js maxð1; jxjs Þm : jAn ðxÞjs

The lemma is then obvious for t ¼ 0: Suppose now v ¼ 1 and tX1 so that @ ½t yðPÞ ¼

G1;t ða; x; zÞ : % FY0 ðx; zÞ2t

We combine the estimations of Lemma 3 with (1). Remind the formula P X ð2t ti; j Þ Y qðtÞF0;1 ðFi; j Þti; j : G1;t ¼ % t¼ðt Þ ði;jÞ %

i; j

ð1Þ

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228

We start with the obvious upper bound jFi; j ðx; zÞjs p LðFi; j Þe jF js maxð1; jxjs Þmi maxð1; jzjs Þnj p ðm þ 1Þeðiþ1Þ ðn þ 1Þeð jþ1Þ jF js maxð1; jxjs Þmi maxð1; jzjs Þnj :

ð2Þ

For i ¼ 0 and jX1; we refine this inequality into jF0;j ðx; zÞjs p2e ðm þ 1Þe ðn þ 1Þeð jþ1Þ jF js maxð1; jxjs Þm maxð1; jzjs Þnj1 :

ð3Þ

Let us check this estimation in the Archimedean case,  the p-adic case being similar n nj and simpler. We separate the leading term An ðxÞ j z from the others. Using (1) we find jF0;j ðx; zÞjs p ðm þ 1Þðn þ 1ÞjF js maxð1; jxjs Þm þ ðm þ 1ÞjF js maxð1; jxjs Þm

  n

maxð1; jzjs Þnj1 j ! n1   X n maxð1; jzjs Þnj1 j n¼j

p 2ðm þ 1Þðn þ 1Þ jþ1 jF js maxð1; jxjs Þm maxð1; jzjs Þnj1 : Now from (2) and (3), we deduce an upper bound for the term indexed by t in the % sum defining G1;t : P ð2t ti; j Þ Y jFi; j ðx; zÞjtsi; j jF0;1 ðx; zÞjs 

p2

e 2t

P

t iX1;j i; j





 ðn þ 1Þ

ði;jÞ



ðm þ 1Þ

e 2 2t

e 2t



i; j

ti; j þ

m 2t

maxð1; jxjs Þ 

 maxð1; jzjs Þ

i; j

 P

P



jF j2t s

P

ðn2Þ 2t

i; j

ti; j þ

P i; j

ð jþ1Þti; j

i; j

ti; j þ

i; j

 P

P i; j

ti; j þ

jX2





 P

P

ðiþ1Þti; j

ðmiÞti; j

ðnj1Þt0;j þ

P iX1

ðnjÞti; j

ð2m1Þt p4et ðm þ 1Þe3t ðn þ 1Þeð4t1Þ jF j2t maxð1; jzjs Þ s maxð1; jxjs Þ

Notice that X iX1;j

ti; j p

X i; j

iti; j ¼ t

ð2n4Þtþ1þ

P iX1;j

ti; j

:

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229

so that the last exponent is pð2n  3Þt þ 1: Using again (1), we find

ð2t jF0;1 ðx; zÞjs

P

ti; j Þ

Y

jFi; j ðx; zÞjtsi; j

ði;jÞ

p

et

4 ðm þ 1Þ

eð2ntþ1Þ

ðn þ 1Þeð2nþ1Þt jF jð2n1Þtþ1 maxð1; jxjs Þð2mnm1Þtþm s jAn ðxÞjð2n3Þtþ1 s

:

P Since t qðtÞpððm þ 1Þðn þ 1ÞÞ2t1 ; we obtain the expected upper bound in the case % % v ¼ 1: For vX1 , we use Leibnitz formula as in Lemma 3. The argumentation is similar for the second inequality, changing X into X 1 : Let ˜ ; Y Þ ¼ X m F ðX 1 ; Y Þ ¼ A˜ n ðX ÞY n þ ? þ A˜ 0 ðX Þ FðX ˜ 1 ; yÞ ¼ 0 and Fðx ˜ 1 ; zÞ ¼ 0; so that with A˜ n ðX Þ ¼ X m An ðX 1 Þ: Then Fðx Gv;t ð*a; x1 ; zÞ : @*½t yv ðPÞ ¼ % F˜0Y ðx1 ; zÞ2t Using the previous upper bound, we obtain !t    ½t v  tþv1 e H˜ s ðPÞ @* y ðPÞ p˜cv s;P s v1 jF˜0Y ðx1 ; zÞj2s jA˜ n ðx1 Þj2n3 s with ˜ 2n1 H˜ s ðPÞ ¼ 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jFj maxð1; jx1 js Þ2mnm1 s ¼ jxjð2mnm1Þ Hs ðPÞ s and

c˜s;P ¼

m ˜ s maxf1; jxj1 ðm þ 1Þe ðn þ 1Þe jFj s g ¼ cs;P ; jA˜ n ðx1 Þjs

˜ s ¼ jF js and A˜ n ðx1 Þ ¼ xm An ðxÞ: Noting now that F˜0 ðx1 ; zÞ ¼ since jFj Y m 0 x FY ðx; zÞ; we obtain the desired upper bound. &

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Corollary. Let rs ðPÞ and r˜s ðPÞ be the convergence’s radii of the Taylor’s series of the analytical functions yP;s ðzÞ and y˜ P;s ðz1 Þ at the points x and x1 ; respectively. Then 1 1 Hs ðPÞ p p ; Rs ðPÞ rs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s 1 1 jxjs Hs ðPÞ p : p 0 ˜ r ˜ ðPÞ Rs ðPÞ s jFY ðx; zÞj2s jAn ðxÞj2n3 s

Proof. On a neighborhood of x in Cs ; we find the Taylor’s expansions yP;s ðxÞ ¼ sz þ

þN X

s@ ½t yðPÞðx  sxÞt ;

t¼1

y˜ P;s ðxÞ ¼ sz þ

þN X

s@*½t yðPÞðx1  sx1 Þt

t¼1

using Lemma 1 applied to j ¼ y: The convergence’s radii rs ðPÞ and r˜s ðPÞ of the series yP;s ðxÞ and y˜ P;s ðx1 Þ satisfy   1 Hs ðPÞ ¼ lim sup j@ ½t yðPÞj1=t ; p s rs ðPÞ t/þN jFY0 ðx; zÞj2s jAn ðxÞj2n3 s   1 jxjs Hs ðPÞ ¼ lim sup j@*½t yðPÞj1=t : p s 0 r˜s ðPÞ t/þN jFY ðx; zÞj2s jAn ðxÞj2n3 s The corollary follows immediately since Rs ðPÞXrs ðPÞ and R˜ s ðPÞX˜rs ðPÞ:

&

5. Proofs of Liouville’s type inequalities The proof of the inequalities differs only by various estimates concerning the same determinant D: We shall keep common notations in the proof of the three theorems. Let U; T1 ; T2 ; T be four positive integers satisfying nU ¼ T1 þ T2 ¼ T: Recall that P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ are the two points involved in our theorems. Denote for brevity c1 ¼ ðm  1Þðn  1Þ and consider the matrix M ¼ ð@ ½t1 xu yv ðP1 Þ j @ ½t2 xu yv ðP2 ÞÞ whose rows are labelled by the couples of integers ðu; vÞ with 0puoU; 0pvon; and whose first T1 columns are indexed by the integer t1 with 0pt1 oT1 ; while the

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231

following T2 þ c1 columns are indexed by t2 with 0pt2 oT2 þ c1 : Thus the matrix M contains T rows and T þ c1 columns. As usual when using interpolation determinants, we start with a zero lemma.

Lemma 5. The matrix M has maximal rank equal to T:

Proof. Suppose on the contrary that the rows of M are linearly dependent. Then there exist algebraic numbers gu;v ; 0puoU; 0pvon; not all zero, such that the polynomial GðX ; Y Þ ¼

U 1 X n1 X u¼0

gu;v X u Y v

v¼0

satisfies @ ½t1 GðP1 Þ ¼ 0 for 0pt1 oT1

and

@ ½t2 GðP2 Þ ¼ 0 for 0pt2 oT2 þ c1 :

Let RðX Þ be the resultant with respect to Y of the polynomials F and G: Since there exist polynomials A and B such that R ¼ AF þ BG; observe that the polynomial R vanishes with multiplicity XT1 at the point P1 ; and multiplicity XT2 þ c1 at the point P2 : Since deg RpðU  1Þn þ ðn  1Þm ¼ T þ c1  1; we deduce that R is the zero polynomial. Reminding now that F is absolutely irreducible, it follows that F divides G; which contradicts the upper bound degY Gpn  1on ¼ degY F :

&

Fix now a non-zero minor D extracted from M with maximal format T  T: Thus D ¼ detð@ ½t1 xu yv ðP1 Þ j @ ½t2 xu yv ðP2 ÞÞ0puoU;0pvon t1 AE1 ;t2 AE2

for some subsets E1 and E2 made up, respectively, with integers oT1 and oT2 þ c1 corresponding to the selected columns from M: Then their cardinalities T10 :¼ Card E1

and

T20 :¼ Card E2

satisfy

T1  c1 pT10 pT1 ;

T2 pT20 pT2 þ c1 ;

T10 þ T20 ¼ T:

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5.1. Proof of Theorem 1 We shall bound jDjs for each embedding s : L+Cs ¼ C; Cp in two different ways according as sAS or not. For i ¼ 1; 2; denote for simplicity hi; s ¼ logð2e maxð1; jxi js ÞHs ðPi ÞÞ ¼ logð8e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 maxð1; jxi js Þ2mnm Þ; s fi; s ¼ logðjFY0 ðxi ; zi Þj2s jAn ðxi Þj2n3 Þ: s Our first upper bound is valid for any such embedding s:

Lemma 6. For large T; we have logjDjs p

T1 T2 ðlog maxð1; jx1 js ; jx2 js Þ þ e log 2Þ þ n   T2 þ ðh2;s  f2;s Þ þ OðT log TÞ: 2



T1



2

ðh1;s  f1;s Þ

Moreover, the remainder term OðT log TÞ vanishes for almost all s:

Proof. Observe first that if P ¼ ðx; zÞ stands either for P1 or P2 jFY0 ðx; zÞj2s jAn ðxÞj2n3 pmaxf1; jxjs gHs ðPÞ s

ð5Þ

since, combining (1) and (3), we get jFY0 ðx; zÞjs p 2e ðm þ 1Þe ðn þ 1Þ2e jF js maxf1; jxjs gm maxf1; jzjs gn2 p

2e ðm þ 1Þeðn1Þ ðn þ 1Þen jF jn1 maxf1; jxjs gmnm s jAn ðxÞjn2 s

Write now

½t u v

@ x y ¼

minðt;uÞ X  t¼0

u t

 xut @ ½tt yv

:

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233

from Leibnitz formula. Using Lemma 4 and (5), we obtain !e minðt;uÞ X u ½t u v maxð1; jxjs Þut max fmaxð1; jxjs Þt j@ ½t yv ðPÞjs g j@ x y ðPÞjs p 0ptpt t t¼0 p cvs;P 2eu maxð1; jxjs Þut ( !t )  t þ v  1 e maxð1; jxjs ÞHs ðPÞ  max 0ptpt v1 jFY0 ðx; zÞj2s jAn ðxÞj2n3 s !t  e Hs ðPÞ u tþv1 v eu p cs;P 2 maxð1; jxjs Þ v1 jFY0 ðx; zÞj2s jAn ðxÞj2n3 s for tX0 and vX1: For v ¼ 0; we obviously bound j@ ½t xu ðPÞjs p2eu maxð1; jxjs Þu : Let us enumerate consecutively the two sets of columns E1 ¼ ft1;1 ; y; t1;T10 g

and

E2 ¼ ft2;T10 þ1 ; y; t2;T g

involved in the minor D and expand 0

T1 T X Y Y D¼ 7 @ ½t1; j xuj yvj ðP1 Þ @ ½t2; j xuj yvj ðP2 Þ; j¼T10 þ1

j¼1

where the sum is taken over all the bijections j/ðuj ; vj Þ of the set f1; y; Tg onto f0; y; U  1g  f0; y; n  1g: From the above upper bounds, we obtain   !e U  Y  tþn1 Y tþn1 e ðn1ÞT en jDjs p ðT!Þ maxðcs;P1 ; cs;P2 Þ 2 2 n1 n1 t1 AE1 t2 AE2 !P !P t t t1 AE1 1 t2 AE2 2 Hs ðP1 Þ Hs ðP2 Þ  jFY0 ðx1 ; z1 Þj2s jAn ðx1 Þj2n3 jFY0 ðx2 ; z2 Þj2s jAn ðx2 Þj2n3 s s ( ) PT PT 0 uj 1 u 0  sup maxð1; jx1 js Þ j¼1 j maxð1; jx2 js Þ j¼T1 þ1 ; where the supremum is taken over all the bijections j/ðuj ; vj Þ: Suppose for instance PT10 that jx1 js Xjx2 js : The maximal value in the supremum is then reached when j¼1 uj is P maximal and Tj¼T 0 þ1 uj is minimal. This occurs exactly when the sequence u1 ; y; uT10 1 contains the value U  1 repeated n times,y, U  ½T10 =n repeated n times and U  ½T10 =n  1 repeated T10  n½T10 =n times. Then 

0

T1 X j¼1

uj ¼ n

U 2



 

U  ½T10 =n 2

 þ OðTÞp

T1 T2 þ n



T1 2

 þ OðTÞ

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T X

 uj ¼ n

½T20 =n 2

j¼T10 þ1

Notice also that   X T1 t1 p ; 2 t1 AE1

X

t2 p



t2 AE2



T2 2

  T2 þ OðTÞp þ OðTÞ: 2



 þ OðTÞ;

n

     T1 U T2 T1 T2 þ p þ : n 2 2 2

Putting these estimates in the upper bound for jDjs ; we obtain the lemma. The proof is similar when jx1 js pjx2 js : & Now we give an other upper bound for jDjs which involves the distance of the points P1 and P2 : It will be used for sAS: Here is the point where interpolation determinants are used. Lemma 7. Let Z40 and let s : L+Cs be an embedding such that jx2  x1 js o2e minð1; Rs ðP1 Þ  ZÞ and

yP1 ;s ðsx2 Þ ¼ sz2 :

Then log jDjs pT1 T2 log jx2  x1 js þ



T



2

 1 þ e log 2 þ OZ ðT log TÞ: log Rs ðP1 Þ  Z þ

Proof. Denote for brevity R ¼ minð1; Rs ðP1 Þ  ZÞ: By multilinearity on the rows of the determinant D; we can write D ¼ det ð@ ½t1 ðx  x1 Þu yv ðP1 Þ j @ ½t2 ðx  x1 Þu yv ðP2 ÞÞ0puoU;0pvon : t1 AE1 ;t2 AE2

In a first step we eliminate the poles, thanks to some process reminiscent of the Borel–Dwork Criterion. Set QðxÞ ¼ An ðxÞn1 and observe that the functions QðxÞðx  x1 Þu yv ¼ ðx  x1 Þu ðAn ðxÞyÞv An ðxÞn1v

with 0puoU; 0pvon

are regular on the algebraic curve C: Using Leibnitz formula we find the formula 0

D¼

T1 XY j¼1

@ ½t1; j t1; j Q1 ðP1 Þ

T Y

@ ½t2; j t2; j Q1 ðP2 Þ

j¼T10 þ1

 det ð@ ½t1 QðxÞðx  x1 Þu yv ðP1 Þ j @ ½t2 QðxÞðx  x1 Þu yv ðP2 ÞÞ

t1 ¼t1;1 ;y;t1;T 0 ; 1 t2 ¼t2;T 0 þ1 ;y;t2;T 1

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where the sum is taken over all the sequences of integers t1;1 ; y; t1;T10 ; t2;T10 þ1 ; y; t2;T with 0pt1;i pt1;i and 0pt2; j pt2; j : Remark that we can restrict the summation index to the sequences for which the t1;i and the t2; j are pairwise distinct, since otherwise the determinant on the right hand side vanishes. Then we have 0

T1 X j¼1

t1; j  t1; j pc1 T10

T X

and

t2; j  t2; j pc1 T20 :

j¼T10 þ1

Note that there exists c2 40; depending only upon F ; x1 ; x2 ; s such that j@ ½t Q1 ðPi Þjs pctþ1 2 for any tX0 and i ¼ 1; 2: We deduce an upper bound of the shape   0  Y T Y   T1 ½t t 1 ½t2; j t2; j 1 1; j 1; j  @ Q ðP1 Þ @ Q ðP2 Þ pcT3    j¼1 j¼T10 þ1 s

for all the factors occurring in the above expression of D: We are thus reduced to bound determinants of the shape D0 ¼ det ð@ ½t1 QðxÞðx  x1 Þu yv ðP1 Þ j @ ½t2 QðxÞðx  x1 Þu yv ðP2 ÞÞ: Since the functions yP1 ;s and yP2 ;s are equal in a neighborhood of sx2 ; on account of Lemma 1 we can write ½t1 ½t2 ðsx1 Þ j fu;v ðsx2 ÞÞ sD0 ¼ det ð fu;v

as an interpolation determinant made up with the power series X fu;v;k ðz  sx1 Þk fu;v ðzÞ :¼ sQðzÞðz  sx1 Þu yP1 ;s ðzÞv ¼ kX0

converging in the disk Bs ðsx1 ; Rs ðP1 ÞÞ: Then Theorem 1 of [12] provides us with the expansion      X ki ki t1  ki ki t2 0 sD ¼ det ð fu;v;kj Þdet 0 :  t ðsx2  sx1 Þ t1 2 0pk1 o?okT Set for simplicity R ¼ minð1; Rs ðP1 Þ  ZÞ: Cauchy’s formula applied to the functions fu;v on the disk Bs ðx1 ; RÞ implies the upper bounds j fu;v;k jpc4 Rk ;

kX0;

where c4 is an upper bound on the disk Bs ðsx1 ; RÞ of all the functions sQðzÞyP1 ;s ðzÞv for 0pvon: Observe that non-zero contributions in the above sum come from sequences 0pk1 o?okT which contain necessarily all the values t1;1 ; y; t1;T10 :

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     Bounding now  tk2  p2ek ; we find s

jD0 js p ðT!T20 !Þe cT4 R



PT 0

X



0pk1 o?okT

1 t  j¼1 1; j

PT

t j¼T 0 þ1 2; j 1

PT

2

e

t j¼T 0 þ1 2; j 1

 e PT kj PT10 t1; j PT 0 t2; j j¼1 j¼1 j¼T þ1 2 jx2  x1 js 1 : R

By assumption r :¼ 2e jx2  x1 js =R is o1; so that PT

X

k j¼1 j

r

T  ¼r 2

X

r

PT

k j¼1 j

0pk1 p?pkT

0pk1 o?okT

T T   1 pr 2 : 1r

Estimating 

0

T1 X

t1; j ¼

j¼1

T1 2

 þ OðT1 Þ and



T X

t2; j ¼

j¼T10 þ1

we find the required upper bound since T1 T2 ¼

T  2



T  1

2

T2

 þ OðT2 Þ;

2 

T  2

2

: &

We are now able to prove Theorem 1. For any embedding s : L+Cs denote Rs ¼ maxfRs ðP1 Þ; Rs ðP2 Þg; and for any ZX0; set SZ ¼ fsAS: jx2  x1 js o2e minð1; Rs  ZÞg: Notice that when Z40 and sASZ ; we can apply Lemma 7 either to the point P1 or to P2 ; so that the estimation for jDjs provided by this lemma is valid for any sASZ : For the moment fix Z40; and write the product formula: X

logjDjs ¼ 0:

s : L+Cs

We bound each term logjDjs ; using Lemma 6 when seSZ ; or Lemma 7 when sASZ :       Note that T2 ¼ T1 T2 þ T21 þ T22 and gather the respective coefficients of     T1 T2 ; T21 ; T22 :

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We obtain the basic inequality 8 
Z

Divide now this inequality by T1 T2 and choose T1 and T2 tending to þN with pffiffiffiffiffiffiffiffiffiffiffiffi limðT2 =T1 Þ ¼ h1 =h2 : This process enables us to eliminate the term OZ ðT log TÞ: Next select values of Z tending to 0, in order to replace Z by 0 in the inequality. Finally, we find  X  X 1 log maxð1; jx1 js ; jx2 js Þ e log 2 þ logjx2  x1 js þ logþ þ e log 2 þ Rs n n sAS0 seS0 sffiffiffiffiffi( )   X 1 h2 X þ 1 þ log þ e log 2 þ ðh1;s  f1;s Þ 2 h1 sAS Rs seS0 0 sffiffiffiffiffi( )   X 1 h1 X 1 þ logþ þ e log 2 þ ðh2;s  f2;s Þ X0: 2 h2 sAS Rs seS 0

0

Using the corollary of Lemma 4, we bound

logþ

1 1 Hs ðPi Þ plogþ plogþ phi;s  fi;s  e log 2 0 Rs Rs ðPi Þ jFY ðxi ; zi Þj2s jAn ðxi Þj2n2 s

for i ¼ 1; 2 (observe that hi;s Xfi;s þ e log 2 is equivalent to (5)). Replace now in the previous inequality logþ ð1=Rs Þ by its upper bound h1;s  f1;s  e log 2

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on the second line and by h2;s  f2;s  e log 2 on the third line. We obtain X

logjx2  x1 j1 s p

sAS0

1 X ðlog maxð1; jx1 js ; jx2 js Þ þ e log 2Þ n seS 0  X  þ 1 þ log þ e log 2 Rs sAS0 sffiffiffiffiffi sffiffiffiffiffi ! ! 1 h2 X 1 h1 X þ ðh1;s  f1;s Þ þ ðh2;s  f2;s Þ : 2 h1 s 2 h2 s

P Now for i ¼ 1; 2; we have s fi;s ¼ 0 by the product formula and hi ¼ ½L : P Q 1 s hi;s by definition of the height. It follows that X X 1 1 hðx ; x Þ 1 1 log p 1 2 þ logþ ½L : Q sAS jx2  x1 js n ½L : Q sAS Rs 0

! þ

pffiffiffiffiffiffiffiffiffi h1 h2 þ log 2:

0

On the other hand, the definition of S0 shows that  X X  1 1 1 þ þ 1 log p log þ e log 2 ½L : Q sAS\S jx2  x1 js ½L : Q sAS\S Rs 0 0 ! X 1 þ 1 p log þ log 2: ½L : Q sAS\S Rs 0

Adding the two last inequalities gives Theorem 1. 5.2. Proof of Theorem 2 Let us first compare the projective distance with the usual distance in the two maps covering P1 : % we have the % Lemma 8. For any embedding s : Q+C s and any non-zero x1 ; x2 AQ; inequalities         1 1 1 1 2e min 1; jx1  x2 js ;    pds ðx1 ; x2 Þpmin 2e ; jx1  x2 js ;    : x1 x2 s x1 x2 s

Proof. Let us begin with the right-hand side. The triangle inequality implies ds ðx1 ; x2 Þp2e :

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Since     1 jx1  x2 js 1   ¼ min jx1  x2 js ;    ds ðx1 ; x2 Þp x1 x2 s maxf1; jx1 x2 js g we obtain the upper bound. The proof of the lower bound splits into several cases.  When jx1 js p1 and jx2 js p1; we have     1 1  e  ds ðx1 ; x2 Þ ¼ jx1  x2 js X2 min 1; jx1  x2 js ;    : x1 x2 s  When jx1 js X1 and jx2 js X1; we have       1 1 1  1  e   ds ðx1 ; x2 Þ ¼    X2 min 1; jx1  x2 js ;    : x1 x2 s x1 x2 s  When jx1 js p1pjx2 js ; we have ds ðx1 ; x2 Þ ¼

jx1  x2 js : jx2 js

Suppose jx2 js p2e : Then     1 jx1  x2 js jx1  x2 js 1  e  X X2 min 1; jx1  x2 js ;    : x1 x2 s jx2 js 2e Suppose jx2 js 42e : Since jx1 js p1; we can write     1 jx1  x2 js jx2 js  jx1 js 1 X X21 X21 min 1; jx1  x2 js ;    ; x1 x2 s jx2 js jx2 js in the Archimedean case and     1 jx1  x2 js 1 ¼ 1Xmin 1; jx1  x2 js ;    ; x1 x2 s jx2 js in the non-Archimedean case.

&

In addition to Lemma 7, we shall also use an analytical argument in the map at N on P1 : Lemma 9. Suppose x1 x2 a0: Let Z40 and let s : L+Cs be an embedding such that   1    1  o2e minð1; R˜ s ðP1 Þ  ZÞ and y˜ P ;s ðsx2 Þ ¼ sz2 : 1 x x1  s 2

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Then    1 1  1 1  logjDjs p T1 T2 log   þlogþ þ logjx1 x2 js þ e log 2 x2 x1 s R˜ s ðP1 Þ  Z n      T1 1 1 þ  2 1  logjx1 js þ e log 2 logþ n 2 R˜ s ðP1 Þ  Z      T2 1 1 þ þ  2 1  logjx2 js þ e log 2 log n 2 R˜ s ðP1 Þ  Z 

þ OZ ðT log TÞ:

Proof. We first transform D into a linear combination of analogous determinants involving @* instead of @: Using multilinearity on the columns, we deduce from Lemma 2 the formula D ¼7

 T10  X Y t1; j s %

j¼1

s1; j

 T  Y t2; j

t s x1 1; j 1; j

s2; j

j¼T10 þ1

t2; j s2; j

x1

 det ð@*½s1 xu yv ðP1 Þ j @*½s2 xu yv ðP2 ÞÞ

s1 ¼s1;1 ;y;s1;T 0 1 s2 ¼s2;T 0 þ1 ;y;s2;T 1

where the sum is taken over the sequences of integers s1;1 ; y; s1;T10 ; s2;T10 þ1 ; y; s2;T with 0psi; j pti; j and si; j X1 whenever ti; j X1: We may also suppose that the integers s1;1 ; y; s1;T10 and s2;T10 þ1 ; y; s2;T are pairwise distinct. Next we use the same process as in Lemma 7 in order to eliminate the poles. Set Q ¼ xUþ1 ðxm An ðxÞÞn1 AK½x1 so that the functions Qxu yv ¼ ðx1 ÞU1u ðxm An ðxÞÞn1v ðxm An ðxÞyÞv are integral over the ring K½x1 : Using Leibnitz formula and multilinearity on the columns, we obtain the expression X Cs;t det ð@*½t1 Qxu yv ðP1 Þ j @*½t2 Qxu yv ðP2 ÞÞ t1 ¼t1;1 ;y;t1;T 0 D¼7 s;t %%

%%

1

t2 ¼t2;T 0 þ1 ;y;t2;T 1

with coefficients Cs;t %%

  T10  T  Y Y t1; j t2; j t1; j s1; j *½s1; j t1; j 1 t s ¼ Q ðP1 Þ @ x1 x2 2; j 2; j @*½s2; j t2; j Q1 ðP2 Þ; s1; j s2; j j¼1 j¼T 0 þ1 1

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where the summation’s indices t ¼ ðti; j Þi; j satisfy ti; j psi; j pti; j : Again we can % restrict to indices t for which the t1; j and the t2;k are pairwise distinct. Then % 0

T1 X

s1; j  t1; j pc1 T10

T X

and

s2; j  t2; j pc1 T20 :

j¼T10 þ1

j¼1

For such s and t; let us prove an upper bound of the shape % %       T1 T2 logjx1 x2 js 2 2 logjCs;t js p T1 T2  2  logjx1 js  2  logjx2 js n n n %% 2 2 þ OðT log TÞ: Using again Leibnitz formula @*½st Q1 ðX Þ ¼

st X

ð1Þk



U 2þk k

k¼0

and writing



X U1þk



@ @X 1

½stk 

 1 A˜ n ðX 1 Þ

t st  t  ts x ; we obtain easily ¼ x2t ts s x 0

0

logjCs;t js p logjx1 js @2@ %%

0

X

1

t1 A þ UT10 A

t1 AT10

0

þ logjx2 js @2@

1

X

1

1

t2 A þ UT20 A þ OðT log TÞ

t2 AT20

which implies the expected upper bound. As in Lemma 7, we are thus reduced to bound determinants of the shape D0 ¼ det ð@*½t1 Qxu yv ðP1 Þ j @*½t2 Qxu yv ðP2 ÞÞ: Using now multilinearity on the rows, we replace in D0 the functions Qxu yv ¼ xðU1uÞ ðxm An ðxÞÞn1 yv by u m ðx1  x1 An ðxÞÞn1v ðxm An ðxÞyÞv 1 Þ ðx

which are clearly integral over K½x1 : Denote R ¼ minð1; R˜ s ðP1 Þ  ZÞ and set u m sAn ðzÞÞn1v ðzm sAn ðzÞy˜ P1 ;s ðzÞÞv : f˜u;v ðzÞ ¼ ðz1  sx1 1 Þ ðz

ARTICLE IN PRESS 242

M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254

The functions f˜u;v are analytical and uniformly bounded in the projective disk B˜ s ðsx1 ; RÞ; so that the power series X k gu;v ðzÞ :¼ f˜u;v ðz1 Þ ¼ gu;v;k ðz  sx1 1 Þ kX0

converge and are uniformly bounded in the disk Bs ðsx1 1 ; RÞ: Using now the second formula from Lemma 1, we can again write 1 ½t2 sD0 ¼ 7det ðg½tu;v1 ðsx1 1 Þ j gu;v ðsx2 ÞÞ

as an interpolation determinant. Then, the same argumentation as in Lemma 7 leads to the upper bound      1 T 1 1 logjD0 js p T1 T2 log   þ þ e log 2 logþ x1 x2 s 2 R˜ s ðP1 Þ  Z þ OZ ðT log TÞ: Collecting our upper bounds for jD0 js and jCs;t js ; we obtain Lemma 9. %%

&

The proof of the first part of Theorem 2 follows the same steps as for Theorem 1. Let us detail only the points which differ. Remind the notation 8 if jx1 x2 js o1; > < maxfRs ðP1 Þ; Rs ðP2 Þg 0 ˜ ˜ if jx1 x2 js 41; Rs ðP1 ; P2 Þ ¼ maxfRs ðP1 Þ; Rs ðP2 Þg > : maxfRs ðP1 Þ; Rs ðP2 Þ; R˜ s ðP1 Þ; R˜ s ðP2 Þg if jx1 x2 js ¼ 1 and define for any ZX0 the subset       1 1  0 0 e 0  SZ ¼ sAS : min jx2  x1 js ;    o2 minð1; Rs ðP1 ; P2 Þ  ZÞ : x2 x1 s Observe that 8    < jx2  x1 js 1    1 min jx2  x1 js ;    ¼  1 1 :    x2 x1 s x2 x1 s 

if jx1 x2 js p1; if jx1 x2 js X1:

Let Z40 and let sASZ0 : In order to estimate jDjs ; we use either Lemma 7 when jx1 x2 js o1; or Lemma 9 when jx1 x2 js 41: For the third case jx1 x2 js ¼ 1; it may happen that both lemmas are valid; in this situation among the 4 possibilities, we use any point (P1 or P2 ) and any lemma (7 or 9) for which the corresponding radius reach the maximal value R0s ðP1 ; P2 Þ: pffiffiffiffiffiffiffiffiffiffiffiffi Select now T1 and T2 tending to N with limðT2 =T1 Þ ¼ h01 =h02 and next choose Z tending to 0: Taking again the same estimations as in Theorem 1, we find

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the basic inequality X 1 þ log minðjx2  x1 js ; jx1 2  x1 js Þ þ log sAS00

1 1 þ logþ jx1 x2 js þ e log 2 R0s ðP1 ; P2 Þ n



X log maxð1; jx1 j ; jx2 j Þ e log 2 s s þ n n 0 seS0 8 9 sffiffiffiffiffi   X = 1 h02 < X 1 þ þ  l log logjx j þ e log 2 þ ðh  f Þ s 1;s 1;s 1 s ; 2 h01 :sAS0 R0s ðP1 ; P2 Þ seS00 0 9 sffiffiffiffiffi8   X = 1 h01 < X 1 þ  l þ log logjx j þ e log 2 þ ðh  f Þ X0; s 2;s 2;s 2 s ; 2 h02 :sAS0 R0s ðP1 ; P2 Þ seS 0 þ

0

0

where ls ¼ 0 if we apply Lemma 7 and ls ¼ 2ð1  1=nÞ if we use Lemma 9 relatively to the embedding s: Thanks to the usual Liouville’s inequality, we bound



X sAS00



 1 ls logjxi js p2 1  ½L : Q hðxi Þ n

for i ¼ 1; 2: Next, we bound obviously X sAS00

logþ jx1 x2 js þ

X

log maxð1; jx1 js ; jx2 js Þp½L : Q ðhðx1 Þ þ hðx2 ÞÞ:

seS00

Using again the corollary of Lemma 4, notice that logþ

1 phi;s  fi;s  e log 2 R0s ðP1 ; P2 Þ

for i ¼ 1; 2; and replace in the above basic inequality logþ ð1=R0s ðP1 ; P2 ÞÞ by its upper bound h1;s  f1;s  e log 2 on the third line and by h2;s  f2;s  e log 2 on the fourth line. Noting that X 1 1 logþ 1 ½L : Q sAS0 \S0 minðjx2  x1 js ; jx1 2  x1 js Þ 0  X  1 1 þ þ e log 2 ; p log ½L : Q sAS0 \S0 R0s ðP1 ; P2 Þ 0

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we obtain the upper bound X 1 1 logþ 1 ½L : Q sAS0 minðjx2  x1 js ; jx1 2  x1 js Þ ffiffiffiffiffiffiffiffiffi q hðx Þ þ hðx2 Þ þ r0 þ h01 h02 þ log 4: p 1 n The first part of Theorem 2 follows immediately using Lemma 8. Let us now prove the lower bound. It turns out that the lower bound is a formal consequence of the upper bound, as was remarked by Debes in Theorem 2 of [5]. P First, observe that the global distance ½L : Q 1 sAS0 logþ ds ðx1 ; x2 Þ1 remains unchanged by extension of the field L: Thus may assume without loss of generality that L0 ¼ L: Let S 00 be the set of embeddings s : L+Cs such that 1 00 0 minðjx2  x1 js ; jx1 2  x1 js ÞoRs :¼ min Rs ðP1; j ; P2;k Þ: j;k

0 For any j; k ¼ 1; y; n; denote Sj;k the set S0 associated to the pair of points 00 00 0 ðP1; j ; P2;k Þ: Set now Sj;k ¼ S -Sj;k and

dj;k

0 1 1 @X 1 A  hðx1 Þ  hðx2 Þ: ¼ logþ 1 1 ½L : Q sAS00 n n minðjx2  x1 js ; jx2  x1 js Þ j;k

00 Let sAS 00 : Observe that the number of pairs ð j; kÞ for which sASj;k is exactly n: To 00 see this, fix sAS and suppose for simplicity that jx1 x2 js o1: Write j-k if P2;k belongs to the branch yP1; j ;s and j’k if P1; j belongs to the branch yP2;k ;s : Then s 00 belongs to Sj;k whenever j-k or j’k: Since jx2  x1 js omaxðRs ðP1; j Þ; Rs ðP2;k ÞÞ for each pair ð j; kÞ; there exists either j 0 or k0 such that j-k0 or j 0 ’k: Moreover, the right and left arrows are injective since all the roots z1; j and z2;k are supposed to be simple. It follows easily from these properties that there are exactly n pairs ð j; kÞ for which j and k are linked either by a right arrow or a left arrow. In the remaining cases jx1 x2 js o1 or jx1 x2 js ¼ 1; we argue in a similar way, using branches of the type y˜ P;s ; or unions of the type yP;s ,y˜ P;s : Thus,

n n X X j¼1

k¼1

! ! X 1 1 dj;k ¼ n log  hðx1 Þ  hðx2 Þ 1 ½L : Q sAS00 minð1; jx2  x1 js ; jx1 2  x1 j s Þ ! ! X 1 1  e log 2  hðx1 Þ  hðx2 Þ Xn log ½L : Q sAS00 ds ðx1 ; x2 Þ

ARTICLE IN PRESS M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254

X X n 1   ¼ log e log 2 ½L : Q ds ðx1 ; x2 Þ sAS00 seS 00

245

!

! X n 1 X log  n log 2 1 ½L : Q seS00 minð1; jx2  x1 js ; jx1 2  x1 j s Þ ! X n 1 logþ 00  n log 2 X ½L : Q seS00 Rs by Lemma 8 and the product formula. On the other hand, for any pair ð j; kÞ with j; k ¼ 1; y; n; we have the upper 0 bound 1 qffiffiffiffiffiffiffiffiffi 1 @X 1 dj;k p logþ 00 A þ h01 h02 þ log 4 ½L : Q sAS00 Rs j;k

by the proof of the first part of Theorem 2 (use also Remark 1 following Theorem 1). Putting together the upper and lower bounds, we deduce for each fixed pair a; b the lower bound qffiffiffiffiffiffiffiffiffi  h01 h02 þ log 4  n log 2; da;b X  nr00  ðn2  1Þ 00 for n values of ð j; kÞ: Now ðP1 ; P2 Þ is one noting again that each sAS 00 appears in Sj;k of the pairs ðP1;a ; P2;b Þ: Using again Lemma 8, we obtain the required lower bound 00 DS 0 ðP1 ; P2 Þ: since Sa;b

5.3. Proof of Theorem 3 Assume now that m ¼ n and that the polynomial F has total degree pn: Let X F ðX ; Y Þ ¼ ai; j Xi Y j iþjpn

with a0;n a0: The underlying idea is to embed the curve C into P2 ; rather than P1  P1 ; which produces better estimates in our situation. The scheme of proof remains formally identical with the one of Theorems 1 and 2. We shall only display the points which differ. In relation with Sections 4, 5.1 and 5.2, define henceforth Hs ðPÞ ¼ 4e ðn þ 1Þeð4nþ6Þ jF j2n1 maxð1; jxjs Þ2n3 ; s hi;s ¼ logð2e maxð1; jxi js ÞHs ðPi ÞÞ ¼ logð8e ðn þ 1Þeð4nþ6Þ jF j2n1 maxð1; jxi js Þ2n2 Þ; s fi;s ¼ logðjFY0 ðxi ; zi Þj2s ja0;n j2n3 Þ; s for i ¼ 1; 2:

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Let us first bound jzjs in term of jxjs : Now the refined inequality ðn þ 1Þ2e jF js maxf1; jxjs g maxf1; jzjs gp ja0;n js

ð6Þ

holds. Notice that the factor maxð1; jxjs Þ was raised to a power m ¼ n in (1). Let us write Pn1 Pnj j j¼0 i¼0 ai; j xi z n z ¼ : a0;n We obtain maxð1; jzjs Þn p

ðn þ 1Þ2e jF js max0pjpn1 fmaxð1; jxjs Þnj maxð1; jzjs Þ j g : ja0;n js

Remark now that ja0;n js pjF js pðn þ 1Þ2e jF js so that we may suppose without restriction that maxð1; jzjs ÞXmaxð1; jxjs Þ: Then the above maximum is reached for j ¼ n  1; which implies (6). In relation with Lemma 4, we prove now upper bounds of the shape !t   tþv1 e Hs ðPÞ ½t v v ; ð7Þ j@ y ðPÞjs pcs;P v1 jFY0 ðx; zÞj2s ja0;n j2n3 s *½t v

j@ y

ðPÞjs pcvs;P



tþv

e 

tþv1 v1

v

(

e

max jxjs ;

jxjs Hs ðPÞ

)t

jFY0 ðx; zÞj2s ja0;n j2n3 s

ð8Þ

for tX0 and vX1: According to Lemma 3, the polynomial Gv;t has a total degree in X and Y bounded by ð2n  3Þt þ v: Thus we deduce from Lemma 3 and (6) that j@ ½t yv ðPÞjs p

ð2n3Þtþv LðGv;t Þe jF j2t s maxð1; jxjs ; jzjs Þ

jFY0 ðx; zÞj2t s

ðn þ 1Þ2e jF js maxð1; jxjs Þ p ja0;n js

!v 

tþv1 v1

e

Hs ðPÞ

!t

jFY0 ðx; zÞj2s ja0;n j2n3 s

which proves (7). For (8), we change again X into X 1 : Set ˇ ; YÞ ¼ FðX

X iþjpn

anij; j Xi Y j ¼

n X

X nj Aj ðX 1 ÞY

j

j¼0

ˇ 1 ; YX 1 Þ ¼ X n F ðX ; Y Þ: Then Fðx ˇ 1 ; yx1 Þ ¼ 0 and Fðx ˇ 1 ; zx1 Þ ¼ 0: so that FðX

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Thus we can write the formula ˇ x1 ; zx1 Þ Gv;t ða; : @*½t ðy=xÞv ðPÞ ¼ Fˇ0Y ðx1 ; zx1 Þ2t Applying (7) to the polynomial Fˇ we obtain an upper bound of the shape v j@*½t ðy=xÞv ðPÞjs pðjxj1 s cs;P Þ



tþv1

e

jxjs Hs ðPÞ jFY0 ðx; zÞj2s ja0;n j2n3 s

v1

!t ;

observing that Fˇ0Y ðx1 ; zx1 Þ ¼ xnþ1 FY0 ðx; zÞ;

aˇ 0;n ¼ a0;n

and

Hs ððx1 ; zx1 ÞÞ ¼ jxjsð2n3Þ Hs ðPÞ:

Write now Leibnitz formula @*½t yv ¼

t X

ð1Þ

t¼0

t



 t þ v  1 vþt *½tt ðy=xÞv : x @ t

We deduce the upper bound j@*½t yv ðPÞjs pcvs;P

!e   t  X vþt1 tþv1 e t¼0

v1

t

( max jxjs ;

jxjs Hs ðPÞ

)t

jFY0 ðx; zÞj2s ja0;n j2n3 s

 vþt P  ¼ v : which gives (8) since tt¼0 vþt1 t Let us now verify the inequality hi;s Xfi;s  e log 2:

ð9Þ

It is equivalent to the previous upper bound jFY0 ðx; zÞj2s ja0;n j2n3 pmaxð1; jxjs ÞHs ðPÞ s

ð5Þ

with our new value of Hs ðPÞ: We restrict to the Archimedean case, the p-adic case leading to similar computations. As for (3), we isolate the leading term a0;n Y n in F : We deduce now from (6) that jFY0 ðx; zÞjs p nðja0;n js jzjs Þjzjn2 þ ðn þ 1ÞjF js s

n1 X j¼1

p

2ðn þ 1Þ

2n1

jF jn1 maxð1; jxjÞn1 s : ja0;n jn2

Noting that ja0;n js pjF js ; we obtain (5).

maxð1; jxjs Þnj jjzjsj1

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Finally, the corollary of Lemma 4 becomes 

 1 1 maxð1; jxjs ÞHs ðPÞ ; max ; p rs ðPÞ r˜s ðPÞ jFY0 ðx; zÞj2s ja0;n j2n3 s

ð10Þ

which follows immediately from (7) and (8). Now we can read again the proofs of Theorems 1 and 2. Lemmas 5 and 7, and their proofs, remain valid mutatis mutandis with our new values of the parameters hi;s and fi;s ; on account of estimations (7)–(10). As an example, the inequality logþ

1 R0s ðP1 ; P2 Þ

pminfh1;s  f1;s  e log 2; h2;s  f2;s  e log 2g;

which was used in the proof of Theorem 2, follows immediately from (5), (9) and (10). In conclusion, Theorems 1 and 2 hold with hi ¼

X

hi;s ¼ ð2n  2Þhðxi Þ þ ð2n  1ÞhðF Þ þ logð8ðn þ 1Þ4nþ6 Þ

s

for i ¼ 1; 2:

6. Deduction of the arithmetical applications Their proofs follow the same principle. We fix some point P1 and look at the points P2 which are close to P1 : Under suitable conditions, Theorems 1–3 provide a lower bound for the degree ½L : Q of the field L ¼ KðP1 ; P2 Þ: For Hilbert’s theorem, we choose P1 ¼ ð0; 0ÞACðQÞ and P2 ¼ ðx; zÞ; where z is some root of F ðx; Y Þ ¼ 0: We expect that L ¼ QðzÞ would have the maximal degree n; meaning that the polynomial Pðx; Y Þ is irreducible in Q½Y : In the case of Runge’s Theorem, the origin P1 is an algebraic point ACN ; with degree don; and we expect that P2 ¼ ðx; zÞ cannot belong to CðKÞ: Thus the two situations may be viewed as reverse from each other, and lead to parallel computations. 6.1. Proof of Theorem 4 Recall that K ¼ Q in this section and denote by s the natural embedding  s : QDCs ¼

C

if x ¼ 1=k;

Cp

if x ¼ pt

of Q into Cs : Let us consider the point P1 ¼ ð0; 0ÞACðQÞ and the corresponding meromorphic function yP1 ;s : Notice first that rs ðP1 Þ4H 1 ; since by corollary

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249

of Lemma 4 we know that 1 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 s p oH; 2 2n3 rs ðP1 Þ ja0;1 js ja0;n js the last upper bound coming from the product formula Y Y Y Y ja0;1 js ¼ ja0;1 j1 jF j1 and ja0;n js ¼ ja0;n j1 jF j1 t X t t X t : tas

tas

tas

tas

It follows from the assumptions that 3

jxjs pH 3mn

ðn1Þ2

oH 1 ors ðP1 Þ:

Define now ( z ¼ yP1 ;s ðxÞA

% Q-R % Q-Q p

if x ¼ 1=k; if x ¼ pt

and set L ¼ QðzÞ;

P2 ¼ ðx; zÞACðLÞ:

Let us verify that An ðxÞFY0 ðx; zÞa0: Denote by R the resultant of F ðX ; Y Þ and FY0 ðX ; Y Þ with respect to Y : Since R is equal to the product of An ðX Þ with the discriminant of F ; we have to prove that RðxÞa0: Assuming the contrary, we obtain  pffiffiffi2n1 HðF Þ2n1 oH 2 HðxÞp2HðRÞp2 ðm þ 1Þðn þ 1Þ n using Lemma 4 of [15]. Noting that HðxÞ is equal either to jkj or to pt ; we find a contradiction with our hypotheses. The number field L is contained in Cs by its definition. Denote again by s the natural inclusion s : LDCs ; which obviously extends the preceding embedding s : QDCs : By construction, the point P2 belongs to the branch yP1 ;s : Then we may apply Theorem 1 with S ¼ fsg; according to Remark 1 after Theorem 1. We obtain the inequality pffiffiffiffiffiffiffiffiffi 1 1 hðx; 0Þ 1 1 logþ þ logþ þ h1 h2 þ log 4: p ½L : Q jxjs n ½L : Q Rs ðP1 Þ Noting that logþ jxj1 s ¼ hðxÞ in both cases, we find pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ mð2n  1ÞhðxÞÞ þ log 4: ½L : Q n Suppose on the contrary that the polynomial Pðx; Y Þ is reducible in Q½Y ; so that ½L : Q pn  1: Writing now hðxÞ ¼ lmn3 ðn  1Þ2 log H with lX3; we deduce the

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inequality log 4 pffiffiffiffiffi 1 þ log 1 H o ; l  2lp 2 mn ðn  1Þ 2

which is incompatible with the lower bound lX3: Suppose now that the polynomial F is as in Theorem 3. Applying this last result leads by the same argument to the sharper inequality pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ ð2n  2ÞhðxÞÞ þ log 4: ½L : Q n Then we obtain in this case the same exponent divided by m:

&

6.2. Proof of Theorem 5 First we perform a change of coordinates ðx; yÞ/ðx1 ; yÞ;

or

ðx; yÞ/ðx1 ; yx1 Þ

whether we stay on P1  P1 or on P2 : Let QACN : In this new frame, the point Q has coordinates ð0; ZÞ for some root Z of FN : In order to unify the proof in both cases, denote by G the polynomial ˜ ; Y Þ ¼ X m F ðX 1 ; Y Þ; FðX

or

ˇ ; Y Þ ¼ X n F ðX 1 ; YX 1 Þ FðX

in the case of an embedding in P2 : The curve C has local equation Gðx; yÞ ¼ 0 in the new coordinates. With some abuse of notations set Q ¼ ð0; ZÞ; regarded now as a % point on the curve Gðx; yÞ ¼ 0; and for any embedding s : Q+C; denote by rs ðQÞ be the convergence’s radius of the Taylor’s series of the function yQ;s ðzÞ at the origin. Let us verify that rs ðQÞ4H d : Denote by Bn ðX Þ the coefficient of Y n in the polynomial GðX ; Y Þ; which is equal either to A˜ n ðX Þ; or to the constant a0;n : Notice that Gð0; Y Þ ¼ FN ðY Þ: Since F is regular at infinity, it follows that both GY0 ð0; ZÞ ¼ 0 ðZÞ and Bn ð0Þ are non-zero. For each embedding t : L+Ct of the field L ¼ KðZÞ; FN recall from (5) the upper bound jGY0 ð0; ZÞj2t jBn ð0Þj2n3 pð4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 ÞeðtÞ jGj2n1 : t t It follows from the corollary of Lemma 4, joined with the product formula, that 1 4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 jGj2n1 s p rs ðQÞ jGY0 ð0; ZÞj2s jBn ð0Þj2n3 s Y p ð4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 ÞeðtÞ jGj2n1 oH d : t t : L+Ct

Suppose now that jxjs 4H maxfd;2dg where d :¼ ½K : Q : Let us first verify that Bn ðx1 ÞGY0 ðx1 ; oÞa0 for any algebraic number o with Gðx1 ; oÞ ¼ 0: Assuming the

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251

contrary and arguing as in Theorem 4, we obtain   d pffiffiffi2n1 jxjs pHðxÞd p 2 ðm þ 1Þðn þ 1Þ n HðF Þ2n1 oH 2d which contradicts our assumption. Now jx1 js ors ðQÞ for any QACN by the preceding remarks. Define oQ;s ¼ yQ;s ðsx1 Þ

PQ;s ¼ ðsx1 ; oQ;s Þ:

and

Thus when Q ranges along CN ; the complex numbers oQ;s are the n distinct roots of the polynomial sGðsx1 ; Y Þ: On the other hand we have Gðx1 ; zÞ ¼ 0;

or

Gðx1 ; zx1 Þ ¼ 0

in the case of an embedding in P2 ; since F ðx; zÞ ¼ 0: Denote P ¼ ðx1 ; zÞ; or P ¼ ðx1 ; zx1 Þ in the second case. It follows that there exists a unique Qs ¼ ð0; Zs ÞACN such that sP ¼ PQs ;s whenever jxjs 4H maxfd;2dg : Then the point P belongs to the branch yQs ;s and L ¼ KðZs Þ is a field of rationality for both P and Qs : Applying Theorems 1 and 3 to the points P and Qs ; we obtain the respective inequalities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logþ jxjs hðxÞ þ log H þ log Hðlog H þ mð2n  1ÞhðxÞÞ þ log 4 p n d and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logþ jxjs hðxÞ þ log H þ log Hðlog H þ ð2n  2ÞhðxÞÞ þ log 4: p n d Notice that both upper bounds remain obviously true when jxjs pH maxfd;2dg : Choose % now d embeddings s : Q+C which restrict on K to distinct embeddings and sum the corresponding inequalities. Dividing by d; we find pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ mð2n  1ÞhðxÞÞ þ log 4 d n and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ ð2n  2ÞhðxÞÞ þ log 4; d n from which follow the respective upper bounds hðxÞp3mn3 d2 =ðn  dÞ2 log H as in the proof of Theorem 4.

and

hðxÞp3n3 d2 =ðn  dÞ2 log H

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6.3. Proof of Theorem 6 We use the same setting as in Section 6.2, and we recall the changes of coordinates ðx; yÞ/ðx1 ; yÞ or ðx; yÞ/ðx1 ; x1 yÞ: Observe that each point QACN is rational % and over some number field containing K of degree over Q bounded by nd: Let xAQ nd 1 % let s : Q+C be such that jxjs 4H : It follows from Section 6.2 that sðp ðxÞÞ is made up with the n distinct points whose (new) coordinates are PQ;s ¼ ðsx1 ; oQ;s Þ with oQ;s ¼ yQ;s ðsx1 Þ for QACN : Suppose now that jxjs 42H nd : Since minQACN rs ðQÞ4H nd ; we deduce from the triangle inequality that min

PAp1 ðxÞ

rs ðPÞX min rs ðQÞ  12H nd 412H nd : QACN

Then, we can interchange the roles of x and N; so that sðCN Þ is the set of the n points QP;s ¼ ð0; yP;s ð0ÞÞ

for PAp1 ðxÞ:

The two applications P/Q and Q/P thus defined are bijections which preserve real points when sxAR: By construction, corresponding points P and Q belong to the same branch relative to the embedding s: Now let L be a totally real number field containing K (this is not restrictive) and let P ¼ ðx; zÞAOL  L with F ðx; zÞ ¼ 0: We assume that for any embedding % s : Q+C; the set sðCN Þ contains a unique real point sðQ0 Þ: It follows from the preceding remarks that P belongs to the branch yQ0 ;s for all the embeddings % s : Q+C such that jxjs 42H nd ; since then sðPÞ turns out to be the unique real point located on sðp1 ðxÞÞ: The points P and Q are both rational over L: Now Theorem 1 gives the inequality 0 1 ½L : Q

X s : L+C jxjs X2H nd

logjxjs p

1

X C hðxÞ 1 B B þ logðH nd ÞC @ A n ½L : Q s : L+C jxjs X2H nd

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ log Hðlog H þ mð2n  1ÞhðxÞÞ þ log 4:

We deduce that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞp þ nd log H þ log Hðlog H þ mð2n  1ÞhðxÞÞ þ log 8: n

ARTICLE IN PRESS M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254

253

3

mn Write hðxÞ ¼ lðn1Þ 2 log H and suppose lX1: It follows that

l

log 8 pffiffiffiffiffi nd þ log H 2lp mn2 pd; n1

which implies lp4d: When F satisfies the assumptions of Theorem 3, we can replace in the above inequapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lities the term log Hðlog H þ mð2n  1ÞhðxÞÞ by log Hðlog H þ ð2n  2ÞhðxÞÞ; which leads to exponents divided by m: This concludes the proof of Theorem 6. For the proof of the corollary, suppose for instance that deg f1 Xdeg f2 ; the case deg f1 odeg f2 being similar. Write f1 ðX Þ ¼ anm1 X m1 þ ? þ a0 ;

f2 ðX Þ ¼ bnm2 X m2 þ ? þ b0

and gðX Þ ¼ cnn X n þ ? þ c0 with am1 bm2 cn a0: The equation F ðx; zÞ ¼ 0 is then equivalent to Gðx; zxðm1 m2 Þ=n Þ ¼ 0

with GðX ; Y Þ ¼ f2 ðX ÞgðX ðm1 m2 Þ=n Y Þ  f1 ðX Þ:

Notice that G is the product of a power of X by an absolutely irreducible polynomial, since F is absolutely irreducible. Dividing eventually by some power of X ; we may assume without loss of generality that G is absolutely irreducible. For any % complex embedding s : Q+C; the polynomial sGN ðY Þ ¼ sðbm2 cn Þn Y n  sanm1 ¼

n  Y

sðbm2 cn ÞY  e2p

pffiffiffiffiffi 1n=n

sam1



n¼0

has an unique real root sðam1 =bm2 cn Þ: Thus Theorem 6 can be applied, which gives the required upper bound since HðGÞ ¼ HðF Þ:

References [1] Y. Andre´, G-Functions and Geometry, Vieweg, Braunschwieg, 1989. [2] E. Bombieri, On G-functions, Recent Progress in Analytic Number Theory, Vol. 2, Academic Press, New York, 1981, pp. 1–67. [3] E. Bombieri, On Weil’s The´ore`me de de´composition, Amer. J. Math. 105 (1983) 295–308. [4] L. Comtet, Analyse combinatoire, tome premier, Collection Sup. Le mathe´maticien, Presses universitaires de France, 1970. [5] P. Debes, G-fonctions et the´ore`me d’irre´ductibilite´ de Hilbert, Acta Arith. 47 (1986) 371–401. [6] P. Debes, Hilbert subsets and s-integral points, Manuscripta Math. 89 (1996) 107–137. [7] B. Dwork, A. Van der Poorten, The Eisenstein constant, Duke Math. J. 65 (1992) 23–43 (corrigendum 76 (1994) 669–672). [8] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984) 387–424.

ARTICLE IN PRESS 254

M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254

[9] P. Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1) (1985) 23–38. [10] S. Lang, Fundamentals of Diophantine Geometry, Springer, Berlin, 1983. [11] M. Laurent, Some remarks on the approximation of complex numbers by algebraic numbers, Bull. Greek. Math. Soc. 42 (1999) 49–557. [12] M. Laurent, Interpolation determinants of exponential polynomials, Publ. Math. Debrecen 56 (3–4) (2000) 457–473. [13] M. Laurent, D. Roy, Sur l’approximation alge´brique en degre´ de transcendance un, Ann. Inst. Fourier 49 (1999) 27–55. [14] D. Poulakis, Polynomial bounds for the solution of a class of Diophantine equations, J. Number Theory 66 (1997) 271–281. [15] W. Schmidt, Eisenstein’s theorem on power series expansion of algebraic functions, Acta Arith. 56 (1990) 161–179. [16] V.G. Sprindzuck, Arithmetic specializations of polynomials, J. Reine. Angew. Math. 340 (1983) 26–52. [17] V.G. Sprindzuck, Classical Diophantine Equations, Springer, Berlin, 1993. [18] P.G. Walsh, A quantitative version of Runge’s theorem on diophantine equations, Acta Arith. 62 (1992) 157–172.