Yann Bugeaud

Bernard De Mathan

On a mixed problem in Diophantine approximation

Let d be a positive integer. Let p be a prime number. Let α be a real algebraic number of degree d + 1. We establish that there exist a positive constant c and infinitely many algebraic numbers ξ of degree

Here, H(ξ) and Norm(ξ) denote the naïve height of ξ and its norm, respectively. This extends an earlier result of de Mathan and Teulié that deals with the case d = 1.

Introduction

In analogy with the Littlewood conjecture, de Mathan and Teulié [START_REF] De Mathan | Problèmes diophantiens simultanés[END_REF] proposed recently a 'mixed Littlewood conjecture'. For any prime number p, the usual p-adic absolute value | • | p is normalized in such a way that |p| p = p -1 . We denote by • the distance to the nearest integer.

De Mathan-Teulié conjecture. For every real number α and every prime number p, we have inf

q≥1 q • qα • |q| p = 0. (1.1)
Obviously, the above conjecture holds if α is rational or has unbounded partial quotients in its continued fraction expansion. Thus, it only remains to consider the case when α is an element of the set Bad 1 of badly approximable real numbers, where

Bad 1 = {α ∈ R : inf q≥1 q • qα > 0}.
De Mathan and Teulié [START_REF] De Mathan | Problèmes diophantiens simultanés[END_REF] proved that (1.1) holds for every quadratic real number α (recall that such a number is in Bad 1 ) but, despite several recent results [START_REF] Einsiedler | Measure rigidity and p-adic Littlewood-type problems[END_REF][START_REF] Bugeaud | On a mixed Littlewood conjecture in Diophantine approximation[END_REF], the general conjecture is still unsolved.

2000 Mathematics Subject Classification : 11J04; 11J61, 11J68 .

If we rewrite (1.1) under the form inf a,q≥1, gcd(a,q)=1 q 2 • α -a q

• |q| p = 0 , then we have |q| p = min{|Norm(q/a)| p , 1}. Hence, upon replacing α by 1/α, the de Mathan-Teulié conjecture can be reformulated as follows: For every irrational real number α, for every prime number p and every positive real number ε, there exists a non-zero rational number ξ satisfying

|α -ξ| • min{|Norm(ξ)| p , 1} < εH(ξ) -2 .
Throughout this paper, the height H(P ) of an integer polynomial P (X) is the maximal of the absolute values of its coefficients. The height H(ξ) of an algebraic number ξ is the height of its minimal defining polynomial over the rational integers a 0 + a 1 X + . . . + a d X d , and the norm of ξ, denoted by Norm(ξ), is the rational number (-1) d a 0 /a d . This reformulation suggests us to ask the following question.

Problem 1. Let d be a positive integer. Let α be a real number that is not algebraic of degree less than or equal to d. For every prime number p and every positive real number ε, does there exist a non-zero real algebraic number ξ of degree at most d satisfying

|α -ξ| • min{|Norm(ξ)| p , 1} < εH(ξ) -d-1 ?
The answer to Problem 1 is clearly positive, unless (perhaps) when α is an element of the set Bad d of real numbers that are badly approximable by algebraic numbers of degree at most d, where

Bad d ={α ∈ R : There exists c > 0 such that |α -ξ| > cH(ξ) -d-1 ,
for all algebraic numbers ξ of degree at most d}.

For d ≥ 1, the set Bad d contains the set of algebraic numbers of degree d+1, but it remains an open problem to decide whether this inclusion is strict for d ≥ 2; see the monograph [START_REF] Bugeaud | Approximation by algebraic numbers[END_REF] for more information. The purpose of the present note is to give a positive answer to Problem 1 for every positive integer d and every real algebraic number α of degree d + 1. This extends the result from [START_REF] De Mathan | Problèmes diophantiens simultanés[END_REF] which deals with the case d = 1.

Results

Throughout this paper, for a prime number p, a number field K, and a non-Archimedean place v on K lying above p, we normalize the absolute value

| • | v in such a way that | • | v and | • | p coincide on Q.
Our main result includes a positive answer to Problem 1 when α is a real algebraic number of degree d + 1.

Theorem 1. Let d be a positive integer. Let α be a real algebraic number of degree d + 1 and denote by r the unit rank of Q(α). Let p be a prime number. There exist positive constants c 1 , c 2 , c 3 and infinitely many real algebraic numbers ξ of degree d such that

|α -ξ| < c 1 H(ξ) -d-1 ,
(2.1)

|ξ| v < c 2 (log 3H(ξ)) -1/(rd) , (2.2)
for every absolute value | • | v on Q(ξ) above the prime p, and

|α -ξ| • min{|Norm(ξ)| p , 1} < c 3 H(ξ) -d-1 (log 3H(ξ)) -1/r . (2.3)
Theorem 1 extends Théorème 2.1 of [START_REF] De Mathan | Problèmes diophantiens simultanés[END_REF] that is only concerned with the case d = 1.

Under the assumptions of Theorem 1, Wirsing [START_REF] Wirsing | Approximation mit algebraischen Zahlen beschränkten Grades[END_REF] established that there are infinitely many real algebraic numbers ξ satisfying (2.1).

The proof of Theorem 1 is very much inspired by a paper of Peck [START_REF] Peck | Simultaneous rational approximations to algebraic numbers[END_REF] on simultaneous rational approximation to real algebraic numbers. Roughly speaking, we use a method dual to Peck's to construct integer polynomials P (X) that take small values at α, and we need an extra argument to ensure that our polynomials have a root ξ very close to α.

De Mathan [START_REF] De Mathan | On a mixed Littlewood conjecture for quadratic numbers[END_REF] used the theory of linear forms in non-Archimedean logarithms to prove that Theorem 1 for d = 1 is best possible, in the sense that the absolute value of the exponent of (log 3H(ξ)) in (2.2) cannot be too large. Next theorem extends this result to all values of d.

Theorem 2. Let p be a prime number, d a positive integer and α a real algebraic number of degree d + 1. Let λ be a positive real number. There exists a positive real number κ = κ(λ) such that for every non-zero real algebraic number ξ of degree d satisfying |α -ξ| ≤ λH(ξ) -d-1

(2.4)

we have |ξ| v ≥ (log 3H(ξ)) -κ
for at least one absolute value | • | v on Q(ξ) above the prime p.

As in [START_REF] De Mathan | On a mixed Littlewood conjecture for quadratic numbers[END_REF], the proof of Theorem 2 rests on the theory of linear forms in non-Archimedean logarithms.

Let d be a positive integer. We recall that it follows from the p-adic version of the Schmidt Subspace Theorem that for every algebraic number α of degree d + 1 and for every positive real number ε, there are only finitely many non-zero integer polynomials P (X) = a 0 + a 1 X + . . . + a d X d of degree at most d, with a 0 = 0, that satisfy

|P (α)| • |a 0 | p < H(P ) -d-ε .
Let ξ be a real algebraic number of degree at most d, and denote by P (X) = a 0 + a 1 X + . . . + a d X d its minimal defining polynomial over Z. Then, min{|Norm(ξ)| p , 1} ≥ |a 0 | p and there exists a constant c(α), depending only on α, such that

|P (α)| ≤ c(α) H(ξ) • |ξ -α|.
Let ε be a positive real number. Applying the above statement deduced from the p-adic version of the Schmidt Subspace Theorem to these polynomials P (X), we deduce that

|α -ξ| • min{|Norm(ξ)| p , 1} ≥ H(P ) -d-1-ε
holds if H(P ) is sufficiently large. This implies that if ξ satisfies (2.4) and if H(ξ) is sufficiently large, then we have

|Norm(ξ)| p ≥ H(ξ) -ε , accordingly max v|p |ξ| v ≥ H(ξ) -ε/d .
The result of Theorem 2 is more precise, however we cannot obtain a good lower bound for |Norm(ξ)| p . We conclude this section by pointing out that Einsiedler and Kleinbock [START_REF] Einsiedler | Measure rigidity and p-adic Littlewood-type problems[END_REF] showed that a slight modification of the de Mathan-Teulié conjecture easily follows from a theorem of Furstenberg [START_REF] Furstenberg | Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation[END_REF][START_REF] Boshernitzan | Elementary proof of Furstenberg's Diophantine result[END_REF].

Theorem EK. Let p 1 and p 2 be distinct prime numbers. Then

inf q≥1 q • qα • |q| p 1 • |q| p 2 = 0
holds for every real number α.

In view of Theorem EK, we formulate the following question, presumably easier to solve than Problem 1.

Problem 2. Let d be a positive integer. Let α be a real number that is not algebraic of degree less than or equal to d. For every distinct prime numbers p 1 , p 2 and every positive real number ε, does there exist a non-zero real algebraic number ξ of degree at most d satisfying

|α -ξ| • min{|Norm(ξ)| p 1 , 1} • min{|Norm(ξ)| p 2 , 1} < εH(ξ) -d-1 ?
Theorem EK gives a positive answer to Problem 2 when d = 1. The sequel of the paper is organized as follows. We gather several auxiliary results in Section 3, and Theorems 1 and 2 are established in Sections 4 and 5, respectively.

In the next sections, we fix a real algebraic number field K of degree d + 1. The notation A ≪ B means, unless specific indications, that the implicit constant depends on K. Furthermore, we write A ≍ B if we have simultaneously A ≪ B and B ≪ A.

Auxiliary lemmas

Let K be a real algebraic number field of degree d + 1. Let O denote its ring of integers, and let α 0 = 1, α 1 , . . . , α d be a basis of K. Let D be a positive integer satisfying

D(Z + α 1 Z + . . . + α d Z) ⊂ O ⊂ 1 D (Z + α 1 Z + . . . + α d Z)
and the corresponding inequalities for the dual basis β 0 , . . . , β d defined by

Tr(α i β j ) = δ i,j ,
where Tr is the trace and δ i,j is the Kronecker symbol. We denote by σ 0 = Id, . . . ., σ d , the complex embeddings of K, numbered in such a way that σ 0 , . . . , σ r 1 -1 are real, σ r 1 , . . . , σ d are imaginary and σ r 1 +r 2 +j = σ r 1 +j for 0 ≤ j < r 2 . Set also r = r 1 + r 2 -1, and let ε 1 , . . . , ε r be multiplicatively independent units in K. Lemma 1. Let η be a unit in O such that -1 < η < 1 and define the real number N by |η| = N -1 . The conditions

|σ j (η)| ≍ N 1/d , 0 < j ≤ d , (3.1
)

and |σ i (η)| ≍ |σ j (η)| , 0 < i < j ≤ d , (3.2
a k = Tr(γηβ k ) = γηβ k + d j=1 σ j (η)σ j (γβ k )
implies that if η satisfies (3.1), then

|a k | ≪ N 1/d , 0 ≤ k ≤ d .
Combined with

σ 1 (γ)σ 1 (η) = a 0 + . . . + a d σ 1 (α d ) , this shows that N 1/d ≍ |σ 1 (η)| ≪ max k=0,...,d |a k |.
The proof of the lemma is complete.

Let α be a real algebraic number of degree d + 1. We keep the above notation with the field K = Q(α) and the basis 1, α, . . . , α d of K over Q, and we display an immediate consequence of Lemma 1.

Corollary 1. Let η be a unit in O such that -1 < η < 1 and set N = |η| -1 . Then

D∆γη = P (α),
where P (X) is an integral polynomial of degree at most d satisfying

H(P ) ≍ N 1/d , |P (α)| ≍ N -1 ,
and thus

|P (α)| ≍ H(P ) -d .
Denote by τ j , j = 0, . . . , d the embeddings of K into C p . Recall that the absolute value | • | p on Q has a unique extension to C p , that we also denote by | • | p . In Lemmata 2 to 4 below we work in C p . Let P (X) be an irreducible integer polynomial of degree n ≥ 1. Let ξ be a complex root of P (X) and ξ 1 , . . . , ξ n be the roots of P (X) in C p . We point out that the sets Lemma 2. Assume that γ = α d . Then

|a k | p ≪ max 0≤j≤d |τ j (η) -1| p , 0 ≤ k < d , and
|a d -1| p ≪ max 0≤j≤d |τ j (η) -1| p . Proof. Since Tr(α d β k ) = 0 , for k = 0, . . . , d -1,
we get

a k = Tr(γηβ k ) = Tr(α d (η -1)β k ) = d j=0 (τ j (η) -1)τ j (α d β k ),
and deduce that

|a k | p ≪ max 0≤j≤d |τ j (η) -1| p , 0 ≤ k < d. It follows from Tr(α d β d ) = 1 that a d = 1 + Tr(α d β d (η -1)) = 1 + d j=0 (τ j (η) -1)τ j (α d β d ),
and we derive that

|a d -1| p ≪ max 0≤j≤d |τ j (η) -1| p .
This concludes the proof.

Lemma 3. Let 0 < δ < 1. There exist arbitrarily large positive real numbers H and units

η satisfying η = H -d , σ j (η) σ 1 (η) -1 ≤ δ, 2 ≤ j ≤ d, (3.3
)

and |τ j (η) -1| p ≪ (log H) -1/r , 0 ≤ j ≤ d.
Proof. By taking suitable powers of the units ε 1 , . . . , ε r , we can assume that they are all positive, as well as their real conjugates, and that |τ j (ε i ) -1| p < p -1/(p-1) for i = 1, . . . , r and j = 0, . . . , d. This is possible since |τ j (ε i )| p = 1 for i = 1, . . . , r and j = 0, . . . , d. This allows us to consider the p-adic logarithms of each τ j (ε i ). Our aim is to construct a suitable unit η of the form

η = ε µ 1 p s 1 . . . ε µ r p s r ,
where µ i ∈ Z. The conditions for (3.3) are then

p s µ 1 log |σ j (ε 1 )| |σ 1 (ε 1 )| + . . . + µ r log |σ j (ε r )| |σ r (ε r )| ≤ C 1 , 2 ≤ j ≤ r,
where C 1 = C 1 (δ) > 0 is a constant, and

p s 2π (µ 1 arg σ j (ε 1 ) + . . . + µ r arg σ j (ε r )) ≤ C 2 , r 1 ≤ j ≤ r, with C 2 = C 2 (δ) > 0. Set Y j = p s µ 1 log |σ 1 (ε 1 )| |σ j (ε 1 )| + . . . + µ r log |σ 1 (ε r )| |σ j (ε r )| , 2 ≤ j ≤ r, and
Z k = p s 2π (µ 1 arg σ k (ε 1 ) + . . . + µ r arg σ k (ε r )) ∈ R/Z, r 1 ≤ k ≤ r.
Taking 0 ≤ µ i < M , we have M r points (µ i ) 1≤i≤r . The (Y j , Z k ) 2≤j≤r,r 1 ≤k≤r are in the product of intervals I j , 2 ≤ j ≤ r, of lengths O(M p s ) and of r 2 factors identical to R/Z. This set can be covered by C 3 (M p s ) r-1 sets of diameter at most max{C 1 , C 2 }, where C 3 is a constant depending on δ. By Dirichlet's Schubfachprinzip, choosing M such that

C 3 (M p s ) r-1 < M r ,
which can be done with M ≍ p (r-1)s , we get that there is (µ 1 , . . . , µ r ) ∈ Z r \ {0} such that max

1≤i≤r |µ i | ≪ M , |Y j | ≤ C 1 , 2 ≤ j ≤ r, and
Z k ≤ C 2 , r 1 ≤ k ≤ r.
Set then η = (ε µ 1 1 . . . ε µ r r ) p s in such a way that 0 < η < 1 (if needed, just consider 1/η). This choice implies that

|τ i (η) -1| p = | log p τ i (η)| p ≤ p -s , 0 ≤ i ≤ d,

and

|log η| ≪ p s M ≪ p rs , and the lemma is proved.

Lemma 4. Let P (X) ∈ C p [X] be a polynomial of degree d, and write

P (X) = a 0 + . . . + a d X d . Let ξ i (1 ≤ i ≤ d) be the roots of P (X) in C p . Let c be a real number satisfying 0 ≤ c ≤ 1. If |ξ i | p ≤ c , 1 ≤ i ≤ d, we get |a k | p ≤ c|a d | p , 0 ≤ k < d. (3.4)
Conversely, if (3.4) holds, then we have

|ξ i | p ≤ c 1/d , 1 ≤ i ≤ d. Proof. Since P (X) = a d 1≤i≤d (X -ξ i ) , if |ξ i | p ≤ c ≤ 1 for i = 1, . . . , d, then we have |a k | p ≤ c|a d | p , for k = 0, . . . , d -1. Conversely, if |a k | p ≤ c|a d | p , 0 ≤ k < d,
and if ξ ∈ C p is such that a d ξ d + . . . + a 0 = 0 ,
then, there exists k with 0 ≤ k < d and

|a k ξ k | p ≥ |a d ξ d | p , thus, |ξ| d p ≤ |ξ| d-k p ≤ c .
This completes the proof of the lemma.

We conclude this section with two lemmas used in the proof of Theorem 2. The first of them was proved by Peck [START_REF] Peck | Simultaneous rational approximations to algebraic numbers[END_REF].

Lemma 5. There exists a sequence (η m ) m≥1 of positive units in O such that

η m ≍ e -dm and |σ j (η m )| ≍ e m , 1 ≤ j ≤ d .
Proof. Let us search the unit η m under the form

η m = ε µ 1 1 . . . ε µ r r ,
with µ i ∈ Z. We construct real numbers ν 1 , . . . , ν r such that

ν 1 log ε 1 + . . . + ν r log ε r = -dm (3.5)
and

ν 1 log |σ j (ε 1 )| + . . . + ν r log |σ j (ε r )| = m , 1 ≤ j ≤ d . (3.6)
Taking into account that, by complex conjugation, the equations (3.6) corresponding to an index j with r 1 ≤ j < r 1 + r 2 and to the index j + r 2 are identical, and that the sum of (3.5) and equations (3.6) is zero, we simply have to deal with a Cramer system, since the matrix (σ j (ε i )) 1≤j≤r,1≤i≤r is regular. We solve this system and then replace every ν i by a rational integer µ i such that |µ i -

ν i | ≤ 1/2.
Lemma 6. Let λ ′ be a positive real number. Let (η m ) m≥1 be a sequence of positive units as in Lemma 5. There exists a finite set Γ = Γ(λ ′ ) of non-zero elements of K such that for all integer polynomial P (X) of degree at most d that satisfies

|P (α)| ≤ λ ′ H(P ) -d , (3.7)
there exist a positive integer m and γ in Γ for which

P (α) = γη m .
Proof. Below, all the constants implicit in ≪ depend on K and on λ ′ . Let m be a positive integer such that H(P ) ≍ e m , and set γ = P (α)η -1 m . Since Dα k is an algebraic integer for k = 0, . . . , d, the algebraic number Dγ is an algebraic integer, and, by (3.7), |γ| ≪ 1 .

Furthermore, for j = 1, . . . , d, we have

|σ j (γ)| = |P (σ j (α))| • |σ j (η -1 m )| ≪ H(P )e -m ≪ 1 .
The algebraic integers Dγ ∈ O and all their complex conjugates being bounded, they form a finite set.

Proof of Theorem 1

Let δ be in (0, 1) to be selected later. Apply Lemma 3 with this δ to get a unit η and apply Lemma 1 with this unit and with γ = α d . Since D 2 α d η ∈ Z + . . . + α d Z, we get

D 2 ηα d = a 0 + a 1 α + . . . + a d α d = P (α),
where, by Corollary 1, P (X) is an integer polynomial of degree d and

|P (α)| ≍ H(P ) -d ≍ H -d .
By Lemma 2, each coefficient of P (X) has its p-adic absolute value ≪ (log 3H(P )) -1/r , except the leading coefficient, whose p-adic absolute value equals |D| 2 p . We then infer from Lemma 4 that all the roots of P (X) in C p have their p-adic absolute value ≪ (log 3H(P )) -1/(dr) . This proves (2.2).

It now remains for us to guarantee that P (X) has a root very close to α. To this end, we proceed to check that |P ′ (α)| ≫ H(P ) . Let us write

P ′ (α) = D 2 d i=0 A i σ i (η) with A i = σ i (α d ) d k=1 kα k-1 σ i (β k ) , i = 0, . . . , d.
Observe first that

d i=1 A i = 0 .
Indeed, if this is not the case, then, working with the unit η = 1, that is, with P (X) = D 2 X d and P ′ (α) = dD 2 α d-1 , we get

dα d-1 = A 0 = α d d k=1 kα k-1 β k , hence, d = d k=1 kα k β k .
Taking the trace, and recalling that Tr(α k β k ) = 1, we get

d(d + 1) = d k=1 k , a contradiction. Write P ′ (α) = D 2 d i=1 A i σ i (η) + O(H -d ) = D 2 σ 1 (η) d i=1 A i + B with |B| ≤ D 2 2≤i≤d |A i | • |σ 1 (η)| • σ i (η) σ 1 (η) -1 + O(H -d ) .
Selecting now δ such that

δ 2≤i≤d |A i | ≤ 1 3 d i=1 A i ,
we infer from Lemma 3 that

|P ′ (α)| ≥ 1 2 D 2 σ 1 (η) d i=1 A i ,
when H is sufficiently large. This gives

|P ′ (α)| ≫ |σ 1 (η)| ≫ H .
Consequently, P (X) has a root ξ such that

|α -ξ| ≪ H(P ) -d-1 ≪ H(ξ) -d-1 .
Classical arguments (see at the end of the proof of Theorem 2.11 in [START_REF] Bugeaud | Approximation by algebraic numbers[END_REF]) show that ξ must be real and of degree d if H is sufficiently large. This proves (2.1). Inequality (2.3) follows from (2.1) and (2.2) together with the fact that ξ is of degree d. This completes the proof of the theorem.

Proof of Theorem 2

The constants implicit in ≪ and ≫ below depend on K, p and λ. There exists a positive real number λ ′ , depending on λ and on d, such that the minimal defining polynomial P (X) of any real number ξ of sufficiently large height and for which (2.

)

  are equivalent. Let γ = 0 be in K and let ∆ be a positive integer such that ∆γ ∈ O. If η satisfies (3.1) or (3.2), write γη = a 0 + . . . + a d α d , with a 0 , . . . , a d in Q. We have D∆a k ∈ Z for k = 0, . . . , d and max k=0,...,d |a k | ≍ N 1/d , where the implicit constants depend on γ. Proof. Since η is a unit, we have 0≤j≤d σ j (η) = ±1 . and (3.1) and (3.2) are clearly equivalent. The formula

  {|ξ| v : v is above p on Q(ξ)} and {|ξ i | p : 1 ≤ i ≤ n} coincide, since all the absolute values | • | v and | • | p coincide on Q. Keeping the notation of Lemma 1, we have the following auxiliary result.

Since P ′ (α) = a 1 +

 1 . . . + da d α d-1 , we get P ′ (α) = D 2 Tr(ηα d β 1 ) + 2αTr(ηα d β 2 ) + . . . + dα d-1 Tr(ηα d β d ) , 1 σ i (ηα d β k ) .

1 p 1 p 1 -µ 1 1 .

 11111 4) holds is of degree d and satisfies |P (α)| ≤ λ ′ H(P ) -d .Let (η m ) m≥1 be as in Lemma 5. By Lemma 6, it is sufficient to prove Theorem 2 for the integer polynomials P (X) as above such thatP (α) = γη m = a 0 + a 1 α + . . . + a d α d .Let ξ i be the roots of P (X) in C p and setu := max 1≤i≤d |ξ i | p . Assume that u ≤ 1 . It follows from Lemma 4 that |a k | p ≤ u|a d | p , 0 ≤ k < d .Dividing P (X) by p s = |a d | -1 p if necessary, we can assume that |a d | p = 1, and we obtain that|a k | p ≤ u , 0 ≤ k < d .For j = 1, . . . , d, we then haveγη m α -dτ j (γη m α -d ) = d-1 k=0 a k (α k-dτ j (α k-d )), hence, |γη m α -dτ j (γη m α -d )| p ≪ u .Since |η m | p = 1, we get thatτ j (η m ) η m τ j (γ)α d γτ j (α d ) τ j (ε r ) ε r -µ r,m τ j (γ)α d γτ j (α d ) -If τ j (η m ) η m = γτ j (α d ) τ j (γ)α d holds for j = 1, . . . , d, the number γη m α -dis equal to all its conjuguates, hence is rational, and we haveP (α) = bα dwith b ∈ Q, hence P (X) = bX d , a contradiction. For every m, there thus exists an index j such that 1 ≤ j ≤ d andτ j (ε 1 ) ε ,m . . . τ j (ε r ) ε r -µ r,m τ j (γ)α d γτ j (α d ) =Consequently, by the theory of linear forms in non-Archimedean logarithms (see e.g., Kunrui Yu[START_REF] Yu | p-adic logarithmic forms and group varieties II[END_REF]), there exists a positive constant κ such that u ≫ ( max 1≤i≤r |µ i,m |) -κ . Since η m ≍ H(P ) -d and | log η m | ≍ max 1≤i≤r |µ i,m |,

the matrix (log |σ j (ε i )|) 1≤i≤r,1≤j≤r being regular, we conclude that u ≫ (log 3H(ξ)) -κ . This completes the proof of the theorem.