Alina Firicel

RATIONAL APPROXIMATIONS TO ALGEBRAIC LAURENT SERIES WITH COEFFICIENTS IN A FINITE FIELD

In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.

Introduction

One of the basic question in Diophantine approximation is how well real numbers can be approximated by rationals. The theory of rational approximation of real numbers has then been transposed to function fields after the pioneering works of Maillet [START_REF] Maillet | Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions[END_REF] in 1906 and Gill [START_REF] Gill | An analogue for Algebraic Functions of the Thue-Siegel Theorem[END_REF] in 1930. In the present work, we are interested in the way algebraic Laurent series with coefficients in a finite field can be approximated by rational functions.

Given a field K, we let K(T ), K[[T -1 ]] and K((T -1 )) denote, respectively, the field of rational functions, the ring of formal series and the field of Laurent series over the field K. We also consider the absolute value defined on K(T ) by 2 , where |T | is a fixed real number larger than 1. The field of Laurent series in 1/T , usually denoted by K((T -1 )), should be seen as a completion of the field K(T ) for this absolute value. Thus, if f is a nonzero element of K((T -1 )) defined by

|P/Q| = |T | deg P -deg Q , for (P, Q) ∈ K[T ]
f (T ) = a i 0 T -i 0 + a i 0 +1 T -i 0 -1 + • • • ,
where i 0 ∈ Z, a i ∈ K, a i 0 = 0, we have |f | = |T | -i 0 . We also say that a Laurent series is algebraic if it is algebraic over the field of rational functions K(T ). The degree of an algebraic Laurent series f in K((1/T )) is defined as [K(T )(f ) : K(T )], the degree of the field extension generated by f . We recall that the irrationality exponent (or measure) of a given Laurent series f , denoted by µ(f ), is the supremum of the real numbers τ for which the inequality

f - P Q < 1 |Q| τ
has infinitely many solutions (P, Q) ∈ K[T ] 2 , Q = 0. Thus, µ(f ) measures the quality of the best rational approximations to f . Throughout this paper, p denotes a prime number and q is a power of p. Our aim is to study the irrationality exponent of algebraic Laurent series in the field F q ((1/T )).

In 1949, Mahler [START_REF] Mahler | On a theorem of Liouville in fields of positive characteristic[END_REF] observed that the analogue of the fundamental Liouville's inequality holds true for algebraic Laurent series over a field of positive characteristic.

Theorem 1.1. [START_REF] Mahler | On a theorem of Liouville in fields of positive characteristic[END_REF] Let K be a field of positive characteristic and f ∈ K((T -1 )) be an algebraic Laurent series over K(T ) of degree d > 1. Then, there exists a positive real number C such that

f - P Q ≥ C |Q| d , for all (P, Q) ∈ K[T ] 2 , with Q = 0.
In other words, Mahler's theorem tells us that the irrationality exponent of an algebraic irrational Laurent series is at most equal to its degree.

In the case of real numbers, Liouville's theorem was superseded by the works of Thue [START_REF] Thue | Uber Annaherungswerte algebraischer Zahlen[END_REF], Siegel [START_REF] Siegel | Approximation algebraischer Zahlen[END_REF], Dyson [START_REF] Dyson | The approximation to algebraic numbers by rationals[END_REF] and others, leading to the famous Roth's theorem [START_REF] Roth | Rational approximations to algebraic numbers[END_REF], which tells that the irrationality exponent of an algebraic real number is equal to 2. In 1960, Uchiyama obtained an analogue of Roth's theorem [START_REF] Uchiyama | Rational approximation to algebraic function[END_REF], for the case of Laurent series with coefficients in a field of characteristic 0.

When the base field has positive characteristic, it is well-known that there is no direct analogue of Roth's theorem. In fact, the Liouville-Mahler theorem turns out to be optimal. In order to see this, it is sufficient to consider the element f q ∈ F q ((T -1 )) defined by f q (T ) = i≥0 T -q i . It is not difficult to see that f q is an algebraic Laurent series of degree q (since it verifies the equation f qf + T -1 = 0) while the irrationality exponent of f is equal to q. Note that these examples are sometimes referred to as Mahler's algebraic Laurent series. In the same direction, Osgood [START_REF] Osgood | An effective lower bound on the "Diophantine approximation" of algebraic functions by rational functions[END_REF] and Baum and Sweet [START_REF] Baum | Continued fractions of algebraic power series in characteristic 2[END_REF] gave examples of algebraic Laurent series of various degrees for which Liouville's bound is the best possible. For a special class of algebraic Laurent series, the bound given by Liouville for the irrational exponent was improved by Osgood [START_REF] Osgood | An effective lower bound on the "Diophantine approximation" of algebraic functions by rational functions[END_REF][START_REF] Osgood | Effectives bounds on the "Diophantine approximation" of algebraic functions over fields of arbitrary characteristic and applications to differential equations[END_REF]. In 1976, this author proved an analog of Thue's theorem for algebraic Laurent series which are not solutions of a rational Ricatti differential equation. In 1996, de Mathan and Lasjaunias [START_REF] Lasjaunias | Thue's theorem in positive characteristic[END_REF] proved that Thue's theorem actually holds for every algebraic Laurent series in K((T -1 )), K being an arbitrary field of characteristic p, which satisfies no equation of the form f = (Af p s + B)/(Cf p s + D), where A, B, C, D ∈ K[T ], not all zero, and s ∈ N * . Laurent series satisfying such an equation are called hyperquadratic and they were studied by many authors [START_REF] Lasjaunias | A survey of Diophantine Approximation in Fields of Power Series[END_REF][START_REF] Schmidt | On continued fractions and Diophantine approximation in power series fields[END_REF][START_REF] Thakur | Diophantine approximation exponents and continued fractions for algebraic power series[END_REF][START_REF] Voloch | Diophantine approximation in positive characteristic[END_REF]. Note that every hyperquadratic Laurent series does also satisfy a Ricatti differential equation.

Except the results obtained by Mahler, Lasjaunias-DeMathan or Osgood, we do not know other general method in order to bound up the irrationality exponent of algebraic Laurent series over F q (T ). It is worth mentionning that the situation for function fields totally differs from the one of real numbers. For instance Schmidt [START_REF] Schmidt | On continued fractions and Diophantine approximation in power series fields[END_REF] and Thakur [START_REF] Thakur | Diophantine approximation exponents and continued fractions for algebraic power series[END_REF] independently proved that the set of irrationality exponents of algebraic Laurent series contains all rational real numbers greater than or equal to 2.

The aim of this paper is to introduce a new approach in order to bound up the irrationality exponent of an algebraic Laurent series, which is based on the use of the Laurent series expansion. As a starting point, we use a theorem of Christol [START_REF] Christol | Ensembles presque periodiques k-reconnaissables[END_REF] which characterize in terms of automata the algebraic Laurent series with coefficient in a finite field. More precisely, we recall that

f (T ) = i≥0 a i T -i ∈ F q [[T -1 ]
] is algebraic if and only if the sequence (a i ) i≥0 is generated by a p-automaton. Furthermore, we recall that by a classical result of Eilenberg [START_REF] Eilenberg | Automata, languages and machines[END_REF] the so-called p-kernel of a p-automatic sequence is always finite (see Section 3.1).

Our main result is the following explicit general upper bound for the irrationality exponent of algebraic Laurent series in F p ((1/T )).

Theorem 1.2. Let f (T ) = i≥-k a i T -i be an algebraic Laurent series with coefficients in a finite field of characteristic p. Let s be the cardinality of the p-kernel of a = (a i ) i≥0 and e be the number of states of the minimal automaton generating a (in direct reading). Then the irrationality exponent µ(f ) satisfies

(1) µ(f ) ≤ p s+1 e.
The approach we use to prove Theorem 1.2 already appears in a different framework in [START_REF] Adamczewski | Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance[END_REF][START_REF] Adamczewski | Diophantine properties of real numbers generated by finite automata[END_REF][START_REF] Adamczewski | Irrationality measures for some automatic reals numbers[END_REF]. It is essentially based on repetitive patterns occurring in automatic sequences. More precisely, each algebraic formal series f (T ) = i≥0 a i T -i is identified with a p-automatic sequence a := (a i ) i≥0 over F q . Then we use a theorem of Cobham which characterizes p-automatic sequences in terms of p-uniform morphisms of free monoids (see Section 2.2). As a consequence of this result and of the pigeonhole principle, we are able to find infinitely many pairs of finite words (U n , V n ) and a number ω > 1 such that U n V ω n is prefix of a for every positive integer n. Hence, there exists an infinite sequence of pairs of polynomials (P n , Q n ) such that the Laurent series expansion of the rational function

P n /Q n is the (ultimately periodic) sequence c n := U n V ∞
n . The sequence of rational functions P n /Q n provides good rational approximations to f since the words a and c n have the common prefix U n V ω n . Furthermore, the length of U n and V n are respectively of the form kp n and ℓp n . Using such approximations we are able to prove the following result (see Theorem 3.1):

(2)

k + ωℓ k + ℓ ≤ µ(f ) ≤ p s+1 (k + ℓ) (ω -1)ℓ .
In practice, it may happen that we can choose U n and V n such that a and c n have the same first (k + ωℓ)p n digits, while the (k + ωℓ)p n + 1th are different. In this case, the upper bound we obtain for the irrationality exponent may be a much better one and in particular does not depend anymore on the cardinality of the p-kernel. Furthermore, in the case where we can prove that (P n , Q n ) = 1 for all n large enough, we obtain a significative improvement on the upper bound, as it will be explained in Remark 3.1, that sometimes leads to the exact value µ(f ). Note that, when working with similar constructions involving real numbers, it is well-known that this coprimality assumption is usually difficult to check (see [START_REF] Adamczewski | Irrationality measures for some automatic reals numbers[END_REF]).

In the second part of this paper, we introduce a new approach in order to overcome this difficulty. We provide an algorithm that allows us to check, in a finite amount of time, wether the polynomials P n and Q n , associated with an algebraic Laurent series f , are relatively prime for all n large enough. In order to do this, we observed that the rational approximations we obtained have a very specific form: the roots of Q n can only be 0 or the ℓth roots of unity (see Section 3.4). Then we have to develop a calculus allowing to compute the polynomials P n (T ). In order to do this, we introduce some matrices associated with p-morphisms. These matrices generalize the so-called incidence matrix of the underlying morphism (see Section 4) and their study could also be of independent interest.

In the last part of this paper, we illustrate the relevance of our approach with few examples. We give in particular several algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.

Terminology and basic notions

A word is a finite, as well as infinite, sequence of symbols (or letters) belonging to a nonempty set A, called alphabet. We usually denote words by juxtaposition of their symbols.

Given an alphabet A, we let A * := ∪ ∞ m=0 A m denote the set of finite words over A. Let V := a 0 a 1 • • • a m-1 ∈ A * ; then, the integer m is the length of V and is denoted by |V |. The word of length 0 is the empty word, usually denoted by ε. We also let A m denote the set of all finite words of length m and by A N the set of all infinite words over A. We typically use the uppercase italic letters U, V, W to represent elements of A * . We also use bold lowercase letters a, b, c to represent infinite words.

For any nonnegative integer n, we denote by U n := U U • • • U (n times concatenation of the word U ). More generally, for any positive real number ω we let U ω denote the word U ⌊ω⌋ U ′ , where U ′ is the prefix of U of length ⌈(ω -⌊ω⌋)|U |⌉. Here, ⌊ζ⌋ and ⌈ζ⌉ denote, respectively, the integer part and the upper integer part of the real number ζ. We also denote

U ∞ := U U • • • , that is U concatenated (with itself) infinitely many times.
An infinite word a is periodic if there exists a finite word V such that a = V ∞ . An infinite word is ultimately periodic if there exist two finite words U and V such that a = U V ∞ .

Throughout this paper, we set A m := {0, 1, . . . , m -1}, for m ≥ 1, which will serve as a generic alphabet.

2.1. Automatic sequences and Christol's theorem. Let k be a positive integer, k ≥ 2. An infinite sequence a = (a i ) i≥0 is k-automatic if, roughly speaking, there exists a finite automaton which produces the term a i as output, when the input is the k-ary expansion of i.

For a formal definition of an automatic sequence, let us define a k-deterministic finite automaton with output or, shortly, k-DFAO. This is a 5-tuple

M = (Q, δ, q 0 , ∆, ϕ)
where Q is a finite set of states, δ : Q × A k → Q is the transition function, q 0 is the initial state, ∆ is the output alphabet and ϕ : Q → ∆ is the output function. For a finite word W = w r w r-1 • • • w 0 ∈ A r k , we let [W ] k denote the number r i=0 w i k i . We now say that a sequence a = (a i ) i≥0 is k-automatic if there exists a k-DFAO such that a i = ϕ(δ(q 0 , w)) for all i ≥ 0 and all words W with [W ] k = i.

A classical example of automatic sequence is the so-called Thue-Morse sequence: t = (t i ) i≥0 = 01101001100 • • • , which counts the number of 1's (mod 2) in the base-2 representation of i. It is generated by the following automaton. More references on automatic sequences can also be found in the monograph [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF].

q 0 /0 q 1 /1 0 0 1 1 Figure 1.
Automaton generating the Thue-Morse sequence If we now consider the Laurent series f t (T ) = i≥0 t i T -i as an element of F 2 ((T -1 )), one can check that f t satisfies the algebraic equation:

(T + 1) 3 f 2 t (T ) + T (T + 1)f t (T ) + 1 = 0. Hence, f t is an algebraic Laurent series over F 2 (T ) whose sequence of coefficients is a 2-automatic sequence. Actually, this is not an isolated case. Indeed, there is a famous theorem of Christol [START_REF] Christol | Ensembles presque periodiques k-reconnaissables[END_REF] which precisely describes the algebraic Laurent series over F q (T ) as follows.

Theorem 2.1 [START_REF] Christol | Ensembles presque periodiques k-reconnaissables[END_REF]. Let f a (T ) = i≥-i 0 a i T -i be a Laurent series with coefficients in a finite field of characteristic p. Then f a is algebraic over F q (T ) if and only if the sequence a = (a i ) i≥0 is p-automatic.

We also mention that a well-known result of Eilenberg proves that a sequence is p-automatic if and only if is q-automatic, for any q, power of p.

Morphisms and Cobham's theorem.

Let A (respectively B) be a finite alphabet and let A * (respectively B * ) be the corresponding free monoid. A morphism σ is a map from A * to B * such that σ(U V ) = σ(U )σ(V ) for all words U, V ∈ A * . Since the concatenation is preserved, it is then possible to define a morphism defined on A.

Let k be a positive integer. A morphism σ is said to be k

-uniform if |σ(a)| = k for any a ∈ A. A k-uniform morphism will also be called k-morphism. If k = 1, then σ is simply called a coding. If A = B we can iterate the application of σ. Hence, if a ∈ A, σ 0 (a) = a, σ i (a) = σ(σ i-1 (a)), for every i ≥ 1. Let σ : A → A * be a morphism. The set A * ∪ A N is endowed with its natural topology.
Roughly, two words are close if they have a long common prefix. We can thus extend the action of a morphism by continuity to

A * ∪ A N . Then, a word a ∈ A N is a fixed point of a morphism σ if σ(a) = a. A morphism σ is prolongable on a ∈ A if σ(a) = aX, for some X ∈ A + := A * \{ε} such that σ k (X) = ε, for any k ∈ N. If σ is prolongable then the sequence (σ i (a)) i≥0 converges to the infinite word σ ∞ (a) = lim i→∞ σ i (a) = aXσ(X)σ 2 (X)σ 3 (X) • • • .
With this notation, we now can cite an important theorem of Cobham, which gives a characterization of k-automatic sequences in terms of k-uniform morphisms.

Theorem 2.2 [START_REF] Cobham | Uniform tag sequences[END_REF]. Let k ≥ 2. Then a sequence a = (a i ) i≥0 is kautomatic is and only if it is the image, under a coding, of a fixed point of a kuniform morphism.

Proof of Theorem 1.2

Theorem 1.2 is an easy consequence of the more precise result established in Theorem 3.1. In order to prove Theorem 3.1, we first establish our approximation lemma which is the analog of a classical result in Diophantine approximation (Lemma 3.2). Then we show how to construct, starting with an arbitrary algebraic Laurent series f with coefficients in a finite field, an infinite sequence of rational approximations of f satisfying the assumptions of our approximation lemma. We thus deduce the expected upper bound for the irrationality exponent of f . All along this section, we provide comments and remarks allowing one to improve in most cases this general upper bound (see in particular Remark 3.1 and Section 3.4).

3.1.

Maximal repetitions in automatic sequences. Before stating our approximation lemma, we first recall a useful result, which will allow us later to control repetitive patterns occurring as prefixes of automatic sequences. The proof of the following lemma can be found in [START_REF] Adamczewski | Diophantine properties of real numbers generated by finite automata[END_REF] (Lemma 5.1, page 1356). Before stating it, we recall that the kernel K k of a k-automatic sequence a = (a i ) i≥0 is defined as the set of all subsequences of the form (a k n i+l ) i≥0 , where n ≥ 0 and 0 ≤ l < k n . Furthermore, we recall that by a result of Eilenberg a sequence a is k-automatic if and only if K k (a) is finite. Lemma 3.1. Let a be a non-ultimately periodic k-automatic sequence defined on a alphabet A. Let U ∈ A * , V ∈ A * \ {ε} and ω ∈ Q be such that U V ω is a prefix of the sequence a. Let s be the cardinality of the k-kernel of a. Then we have the following inequality:

|U V ω | |U V | < k s .
3.2. An approximation lemma. We start with the following result which is, in fact, an analog version of Lemma 4.1 in [START_REF] Adamczewski | Irrationality measures for some automatic reals numbers[END_REF] for Laurent series with coefficients in a finite field. We also recall the proof, since it is not very long and it may also be of independent interest. Lemma 3.2. Let f (T ) be a Laurent series with coefficients in F q . Let δ, ρ and θ be real numbers such that 0 < δ ≤ ρ and θ ≥ 1. Let us assume that there exists a sequence (P n /Q n ) n≥1 of rational fractions with coefficients in F q and some positive constants c 0 , c 1 and c 2 such that (i)

|Q n | < |Q n+1 | ≤ c 0 |Q n | θ (ii) c 1 |Q n | 1+ρ ≤ |f - P n Q n | ≤ c 2 |Q n | 1+δ .
Then, the irrationality measure µ(f ) satisfies

(3) 1 + δ ≤ µ(f ) ≤ θ(1 + ρ) δ .
Furthermore, if we assume that there is N ∈ N * such that for any n ≥ N , (P n , Q n ) = 1, then, we have

1 + δ ≤ µ(f ) ≤ max(1 + ρ, 1 + θ δ ).
In this case, if ρ = δ and θ ≤ δ 2 , then µ(f ) = 1 + δ.

Proof. The left-hand side inequality is clear. We thus turn our attention to the second inequality. Let P/Q ∈ F q (T ) such that |Q| is large enough. Then there exists a unique integer

n = n(Q) ≥ 2 such that (4) |Q n-1 | < (2c 2 |Q|) 1 δ ≤ |Q n |. If P/Q = P n /Q n then P Q - P n Q n ≥ 1 |QQ n | ,
and using Eq. ( 4) and (ii) we get that

f - P n Q n ≤ c 2 |Q n | 1+δ = c 2 |Q n ||Q n | δ ≤ 1 2|Q||Q n | .
By the triangle inequality, we have that

f - P Q ≥ P Q - P n Q n -f - P n Q n .
Eq. (i) together with Eq. ( 4) imply that

|Q n | ≤ c 0 |Q n-1 | θ < c 0 (2c 2 |Q|) θ/δ . Thus, f - P Q ≥ 1 2|Q||Q n | ≥ 1 2|Q|c 0 (2c 2 |Q|) θ/δ ≥ c 3 |Q| θ(1+ρ) δ since 1 + θ/ρ ≤ θ + θ/ρ (because θ ≥ 1) and c 3 := 1/2c 0 (2c 2 ) θ/δ .
On the other hand, if

P/Q = P n /Q n , then f - P Q = f - P n Q n ≥ c 1 |Q n | 1+ρ ≥ c 1 (c 0 (2c 2 |Q|) θ/ρ ) 1+ρ = c 4 |Q| θ(1+ρ) δ
,

where c 4 = c 1 /c 1+ρ 0 (2c 2 ) θ(1+ρ) δ
. This ends the proof.

The case where (P n , Q n ) = 1 is treated in a similar way and we refer the reader to Lemma 4.1 in [START_REF] Adamczewski | Irrationality measures for some automatic reals numbers[END_REF] (page 10). The proof consists, as previously, of two cases, but, when P n /Q n = P/Q and P/Q is reduced, then Q n = Q; this permits to obtain an improved upper bound.

Note that the second part of Lemma 3.2 is also known as a Voloch's Lemma and for more details, we refer the reader to the original paper of [START_REF] Voloch | Diophantine approximation in positive characteristic[END_REF], but also to Thakur's monograph [START_REF] Thakur | Function Field Arithmetic[END_REF], page 314.

Construction of rational approximations via Christol

's theorem. Let f (T ) := i≥0 a i T -i ∈ F q [[T -1 ]]
be an irrational algebraic Laurent series over F q (T ).

We recall that, by Christol's theorem, the sequence a := (a i ) i≥0 is p-automatic. According to Cobham's theorem, there exist m ≥ 1,

σ : A m → A * m a p-morphism and ϕ : A m → F q a coding such that a = ϕ(σ ∞ (a)) where a ∈ A m .
In all that follows, we let f a (T ) = i≥0 a i T -i denote the Laurent series associated with the infinite word a = (a i ) i≥0 .

We also give the following definition for a polynomial associated with a finite word.

Definition 3.1. Let U = a 0 a 1 • • • a k-1 be
a finite word over a finite alphabet. We associate with U the polynomial

P U (T ) := k-1 j=0 a k-1-j T j . If U = ε, we set P U (T ) = 0.
For example, if we consider the word U = 1020031 ∈ A 5 then

P U (T ) = T 6 + 2T 4 + 3T + 1 is a polynomial with coefficients in F 5 .
Using this notation, we have the following two lemmas.

Lemma 3.3. Let U, V be two finite words such that |U | = k ∈ N and |V | = ℓ ∈ N and let a := U V ∞ . Then, f a (T ) = P U (T )(T ℓ -1) + P V (T ) T k-1 (T ℓ -1)
.

If k = 0, we have

f a (T ) = T P V (T ) T ℓ -1 . Proof. Let U := a 0 a 1 • • • a k-1 and V := b 0 b 1 • • • b ℓ-1 .
Writing the associated Laurent series with a := U V ∞ , we have k+2ℓ) , . . . and using the definition of P U (T ) and P V (T ) (see def. 3.1), we obtain

f a (T ) = (a 0 + a 1 T -1 + • • • + a k-1 T -(k-1) ) + (b 0 T -k + • • • + b ℓ-1 T -(k+l-1) ) + • • • and then, factorizing T -(k-1) , T -k , T -(k+ℓ) , T -(
f a (T ) = T -(k-1) P U (T ) + T -k (b 0 + b 1 T -1 + • • • + b ℓ-1 T -(ℓ-1) )+ + T -(k+ℓ) (b 0 + • • • + b ℓ-1 T -(ℓ-1) ) + • • • = T -(k-1) P U (T ) + T -(k+ℓ-1) P V (T )(1 + T -ℓ + T -2ℓ + T -3ℓ + • • • ) = P U (T )(T ℓ -1) + P V (T ) T k-1 (T ℓ -1)
.

When k = 0, that is, when U = ε, we just have to replace k = 0 in the identity given above. Proof. This lemma immediately follows from the definition of an ultrametric norm.

We now construct a sequence of rational fractions (P n /Q n ) n≥0 satisfying the assumptions of Lemma 3.2. The approach we use appears in [START_REF] Adamczewski | Diophantine properties of real numbers generated by finite automata[END_REF] and is essentially based on the repetitive patterns occurring in automatic sequences.

The sequence a being p-automatic, the p-kernel is finite. We let e denote the number of states of the minimal automaton generating a (in direct reading) and s the cardinality of the p-kernel. Let consider a prefix P of σ ∞ (a) of length e + 1.

Observe that e is greater than or equal to the cardinality of the internal alphabet of a, that is, A m . It follows, from the pigeonhole principle, that there exists a letter b ∈ A m occurring at least twice in P . This means that there exist two (possibly empty) words U ′ and V ′ and a letter b (both defined over A m ) such that For any n ≥ 1, we set

P := U ′ bV ′ b. Now, if we set U := U ′ , V := bV ′ , |U | := k, |V | := ℓ, ω := 1 + 1/ℓ, then U V ω is a prefix of σ ∞ (a). Let n ∈ N, U n := ϕ(σ n (U )) and V n = ϕ(σ n (V )). Since a = ϕ(σ ∞ (a)) then, for any n ∈ N, U n V ω n is
(5) Q n (T ) = T kp n -1 (T ℓp n -1).
Let c n denote the infinite word U n V ∞ n . There exists P n (T ) ∈ F q [T ] such that

f cn (T ) = P n (T ) Q n (T ) .
More precisely, by Lemma 3.3, the polynomial P n (T ) may be defined by the following formula (6) P n (T ) = P Un (T )(T ℓp n -1) + P Vn (T ).

Since a and c n have the common prefix U n V ω n it follows by Lemma 3.4 that ( 7)

f a - P n Q n ≤ c 2 |Q n | k+ωℓ k+ℓ
, where c 2 = |T | k+l k+ωl . Furthermore, if a and c n have the common prefix U n V ω n and the (k + ωℓ)p n + 1-th letters are different, then, by Lemma 3.4, Inequality (7) becomes an equality. On the other hand, we also have the following result, which is an easy consequence of Lemma 3.1. Lemma 3.5. Let s be the cardinality of the p-kernel of the sequence a := (a i ) i≥0 . We have

f a - P n Q n ≥ 1 |Q n | p s .
Proof. Using Lemma 3.1 we obtain

|U n V ω n | < p s |U n V n |.
This implies that a and c n cannot have the same first p s |U n V n | digits. Hence

f a - P n Q n ≥ 1 |T | (|Un|+|Vn|)p s = 1 (|T ||Q n |) p s ≥ c 1 |Q n | p s .
where

c 1 := 1/|T | p s .
This shows that (P n /Q n ) n≥1 satisfies the assumptions of Lemma 3.2 with θ = p, ρ = p s -1 and δ = (ω-1)ℓ k+ℓ . With this notation, we obtain the following theorem.

Theorem 3.1. Let f a (T ) = i≥0 a i T -i ∈ F q [[T -1 ]
] be an algebraic Laurent series over F q (T ). Let k, l, ω, s be the parameters of f a defined above. Then, the irrationality measure of f a , µ(f a ), satisfies the following inequality:

(8) k + ωℓ k + ℓ ≤ µ(f a ) ≤ p s+1 (k + ℓ) (ω -1)ℓ . Remark 3.1.
If, for every n, a and c n have the same first (k + ωℓ)p n digits, while the (k + ωℓ)p n + 1-th digits are different, then we have

f a - P n Q n = c 2 |Q n | 1+δ .
Thus, Inequality [START_REF] Dyson | The approximation to algebraic numbers by rationals[END_REF] does not depend on s (the cardinality of p-kernel) anymore. More precisely, in this case, P n and Q n satisfy Lemma 3.2 with θ = p, ρ = δ = (ω-1)ℓ k+ℓ and we have k

+ ωℓ k + ℓ ≤ µ(f a ) ≤ p(k + ωℓ) (ω -1)ℓ .
Moreover, if there exists N such that, for any n ≥ N , (P n , Q n ) = 1, then

k + ωℓ k + ℓ ≤ µ(f a ) ≤ max k + ωℓ k + ℓ , 1 + p(k + ℓ) (ω -1)ℓ . If U = ε, that is k = 0, then ω ≤ µ(f a ) ≤ p ω ω -1 . Furthermore, if ω -1 ≥ √ p and (P n , Q n ) = 1, then µ(f a ) = ω.
All this explains why, in many cases, the general upper bound we obtained in Theorem 1.2 can be significatively improved.

Proof of Theorem 1.2. By construction, ω = 1 + 1/ℓ and k + ℓ ≤ d. By Theorem 3.1, it immediately follows that µ(f a ) ≤ p s+1 e.

3.4.

An equivalent condition for coprimality of P n and Q n . We have seen in Remark 3.1 that, in the case where the numerator P n and the denominator Q n of our rational approximations are relatively prime, the bound for the irrationality exponent obtained in Theorem 1.2 can be significantly improved. This serves as a motivation for this section, which is devoted to the coprimality of P n and Q n . First, let us recall the following result, which is an easy consequence of the fact that the greatest common divisor of two polynomials, defined over a field K, also belongs to K.

Lemma 3.6. Let P, Q ∈ F q [T ]. Then (P, Q) = 1 over F q [T ] if and only if (P, Q) = 1 over F p [T ].
We recall that F p is the classical notation for an algebraic closure of F p . Let n ∈ N * and Q n (T ) = T kp n -1 (T ℓp n -1) ∈ F q [T ]. Since we work in characteristic p, we have that

Q n (T ) = T kp n -1 (T ℓ -1) p n .
Now, let P be an arbitrary polynomial with coefficients in F q . Then (P, Q n ) = 1 if and only if (P (T ), T ) = 1 and (P (T ), T ℓ -1) = 1. In other words, (P, Q n ) = 1, if and only if P (0) = 0 and P (a) = 0 for all a ∈ F p such that a ℓ = 1.

Therefore, we easily obtain the following lemma, which will simplify the study of the coprimality of polynomials P n and Q n , by using some properties of P Un and P Vn . (We recall that U n = ϕ(σ n (U )) and V n = ϕ(σ n (V )), where U and V are introduced in Section 3.3.) Lemma 3.7. Let n ∈ N * and P n , Q n defined in Eq. ( 6) and [START_REF] Baum | Continued fractions of algebraic power series in characteristic 2[END_REF]. Then, (P n , Q n ) = 1 over F q (T ) if and only if we have (i) P Un (0) = P Vn (0), (ii) for any a ∈ F p , such that a ℓ = 1, P Vn (a) = 0.

Matrices associated with morphisms

The purpose of this section is to give an approach which will allow one to compute the polynomials P Un (T ) and P Vn (T ), described in the previous section. In particular, we show that, if α ∈ F p , the sequences (P Un (α)) n≥1 and (P Vn (α)) n≥1 are ultimately periodic. Lemma 3.7 implies that we have to test the coprimality of P n and Q n only for a finite number of index n.

Let U = a 0 a 1 • • • a k-1 be a finite word on A m and let i ∈ A m . We let P U (i) denote the set of positions of i in the word U ; or, simply, P i if there is no doubt about U . Définition 4.1. We associate with U the row vector v U (T ) = (β U,j (T )) 0≤j≤m-1 with coefficients in F p [T ] where, for any j ∈ A m , β U,j is defined as: [START_REF] Eilenberg | Automata, languages and machines[END_REF] β U,j (T ) = l∈P j T l , if j occurs in U , 0, otherwise.

Example 4.1. We consider U = 1020310 ∈ A * 5 . Then P 0 = {0, 3, 5}, P 1 = {1, 6}, P 2 = {4}, P 3 = {2} and P 4 = ∅. The vector associated with U is

v U (T ) = (1 + T 3 + T 5 , T + T 6 , T 4 , T 2 , 0).
We also recall that P U (T ) = T 6 + 2T 4 + 3T 2 + T (see Definition 3.1) and we observe that

P U (T ) = v U (T )       0 1 2 3 4       .
M σ (T ) =   T 2 + 1 T 0 T T 2 + 1 T 3 T 4 + T 3 T + 1 T 2   .
It is not difficult to see that such matrices satisfy some interesting general properties as claimed in the following remarks.

Remark 4.1. The matrix M σ (1) is the reduction modulo p of the so-called incidence matrix associated with the morphism σ. This matrix satisfy some very nice properties and has been the subject of considerable study (see for instance [START_REF] Quefféllec | Substitution Dynamical Systems. Spectral Analysis[END_REF]). Remark 4.2. If σ 1 and σ 2 are two p-morphisms over A m then we have that

M σ 1 •σ 2 (T ) = M σ 2 (T p )M σ 1 (T ).
Now, our main goal is to prove that, if α ∈ F p , the sequences (P Un (α)) n≥1 and (P Vn (α)) n≥1 are ultimately periodic. This will be the subject of Proposition 4.2. In order to prove this, we will need the following auxiliary results.

Lemma 4.1. Let σ : A m → A * m be a p-morphism and U = a 0 • • • a k-1 ∈ A * m . For any n ∈ N we denote U n = σ n (U ) = σ n (a 0 ) • • • σ n (a k-1
). We have

P Un (T ) = v U (T p n )R n (T ),
where, for any n ∈ N,

R n (T ) =      P σ n (0) (T ) P σ n (1) (T )
. . .

P σ n (m-1) (T )      . Proof. Since U n = σ n (a 0 ) • • • σ n (a k-1
), we infer from the fact that σ is a p-morphism, that

P σ n (U ) (T ) = P σ n (a 0 ) (T )T (k-1)•p n + P σ n (a 1 ) (T )T (k-2)•p n + • • • + P σ n (a k-1 ) (T ).
Hence there exists a vector S n (T ) = (s 0 (T

p n ), s 1 (T p n ), . . . , s m-1 (T p n ))
, where s i (T ), 0 ≤ i ≤ m -1, are some polynomials with coefficients 0 or 1, such that

P σ n (U ) (T ) = S n (T )      P σ n (0) (T ) P σ n (1) (T ) . . . P σ n (m-1) (T )     
and S n (T ) = S 0 (T p n ). For n = 0, the equality above becomes

P U (T ) = S 0 (T )      0 1 . . . m -1     
.

By Definitions 4.1 and 3.1, we deduce that S 0 (T ) = v U (T ). This ends the proof. Hence,

P σ n (U ) (T ) =P σ n (1) (T )T 4•3 n + P σ n (0) (T )T 3•3 n + + P σ n (1) (T )T 2•3 n + P σ n (0) (T )T 3 n + P σ n (0) (T ) =(T 3•3 n + T 3 n + 1, T 4•3 n + T 2•3 n ) P σ n (0) (T ) P σ n (1) (T ) =v U (T 3 n )R n (T ).
Lemma 4.2. Let n ∈ N and let σ be a p-morphism over A m . We have

R n+1 (T ) = M σ (T p n )R n (T ),
where M σ (T ) is the matrix associated with σ, as in Definition 4.2.

Proof. Let σ be defined as follows

           σ(0) = a (0) 0 a (0) 1 • • • a (0) p-1 σ(1) = a (1) 0 a (1) 1 • • • a (1) p-1 . . . . . . . . . σ(m -1) = a (m-1) 0 a (m-1) 1 • • • a (m-1) p-1 ,
where a

(j)

i ∈ A m , for any i ∈ {0, 1, . . . , p -1} and j ∈ {0, 1, . . . , m -1}. Then, for any j ∈ {0, 1, . . . , m -1} and n ∈ N * , we have

σ n+1 (j) = σ n (σ(j)) = σ n (a (j) 0 a (j) 1 • • • a (j) p-1 ) = σ n (a (j) 0 )σ n (a (j) 1 ) • • • σ n (a (j) p-1 ). Hence P σ n+1 (j) (T ) = P σ n (a (j) 0 ) (T )T (p-1)p n + • • • + P σ n (a j p-1 ) (T ). It follows by Lemma 4.1 that      P σ n+1 (0) (T ) P σ n+1 (1) (T ) . . . P σ n+1 (m-1) (T )      = (β σ(i),j (T p n )) 0≤i,j≤m-1      P σ n (0) (T ) P σ n (1) (T ) . . . P σ n (m-1) (T )      , that is, R n+1 (T ) = M σ (T p n )R n (T )
, which ends the proof.

Remark 4.3. In particular, if n = 0 in the previous lemma, we obtain the following equalities

     P σ(0) (T ) P σ(1) (T ) . . . P σ(m-1) (T )      = (β σ(i),j (T )) 0≤i,j≤m-1      0 1 . . . m -1      = M σ (T )R 0 (T ),
and this being true for any p-morphism σ defined over A m . Therefore, we observe that given a matrix M of this form, we can find only one morphism whose matrix is M . Notice also that, if ϕ is a coding defined over A m , we have a similar identity      P ϕ(σ(0)) (T ) P ϕ(σ( 1)) (T ) . . .

P ϕ(σ(m-1)) (T )      = (β ϕ(σ(i)),j (T )) 0≤i,j≤m-1      ϕ(0) ϕ(1) . . . ϕ(m -1)      ,
The following corollaries immediately yield.

Corollary 4.1. Let n ∈ N * and let σ be a p-morphism defined on A m . We have

R n (T ) = M σ (T p n-1 )M σ (T p n-2 ) • • • M σ (T )      0 1 . . . m -1     
, where M σ (T ) is the matrix associated with σ, as in Definition 4.2.

Corollary 4.2. Let σ be a p-morphism defined on A m . Then for any n ∈ N * we have

M σ n (T ) = M σ (T p n-1 )M σ (T p n-2 ) • • • M σ (T )
where M σ (T ) is the matrix associated with σ, as in Definition 4.2. Let p be a prime, q a power of p and σ : A m → A * m be a pmorphism. Let α ∈ F r , where r = p t , t ∈ N * . Then for any positive integer k we have [START_REF] Gill | An analogue for Algebraic Functions of the Thue-Siegel Theorem[END_REF] M σ kt (α) = (M σ t (α)) k .

Proof. We argue by induction on k. Obviously, this is true for k = 1. We suppose that Eq. ( 10) is satisfied for k and we prove it for k + 1. Using Corollary 4.2 and the fact that α r = α we obtain that

M σ (k+1)t (α) = M σ (α p kt+t-1 ) • • • M σ (α p t ) M σ (α p t-1 ) • • • M σ (α) = M σ (α p kt-1 ) • • • M σ (α) M σ (α p t-1 ) • • • M σ (α) = M σ kt (α)M σ t (α) = (M σ t (α)) k+1 .
Proposition 4.2. Let p be a prime, q a power of p and

U = a k-1 • • • a 0 ∈ A * m . Let σ : A m → A *
m be a p-morphism and ϕ : A m → F q a coding. Let α ∈ F r , where r = p t , r ∈ N * . Then the sequence (P ϕ(σ n (U )) (α)) n≥0 is ultimately periodic.

Proof. First, notice that, as in Lemma 4.1, we have

P ϕ(σ n (U )) (T ) = v U (T p n )      P ϕ(σ n (0)) (T ) P ϕ(σ n (1)) (T )
. . .

P ϕ(σ n (m-1)) (T )      . By Remark 4.3, we have that      P ϕ(σ n (0)) (T ) P ϕ(σ n (1)) (T ) . . . P ϕ(σ n (m-1)) (T )      = M σ n (T )      ϕ(0) ϕ (1)
. . .

ϕ(m -1)      . Hence P ϕ(σ n (U ) (α) = v U (α p n )M σ n (α)      ϕ(0) ϕ (1)
. . .

ϕ(m -1)      .
Clearly, the sequence (v U (α p n )) n≥0 is periodic with a period less than or equal to t since v U (α p n+t ) = v U (α p n ), for any n ∈ N * . We now prove that the sequence (M σ n (α)) n≥0 is ultimately periodic. Since α ∈ F p t , we have α p t = α and thus, by Corollary 4.2, for any k and n ∈ N, we obtain M σ n+kt (α) = M σ n (α)M σ kt (α). Therefore, by Proposition 4.1 we have, for any k ∈ N,

M σ n+kt (α) = M σ n (α)M σ kt (α) = M σ n (α)(M σ t (α)) k .
Since M σ t (α) is a m × m matrix with coefficients in a finite field, there exist two positive integers m 0 and n 0 , m 0 = n 0 (suppose that m 0 < n 0 ) such that M σ t (α) m 0 = M σ t (α) n 0 . This implies that

M σ n+m 0 t (α) = M σ n (α)(M σ t (α)) m 0 = M σ n (α)(M σ t (α)) n 0 = M σ n+n 0 t (α)
and thus, the sequence (M σ n (α)) n≥0 is ultimately periodic, with pre-period at most m 0 t and period at most (n 0m 0 )t. Since (v U (α p n )) n≥0 is periodic with period at most t, it follows that (P ϕ(σ n (U )) (α)) n≥0 is ultimately periodic (with pre-period at most m 0 t and period at most (n 0m 0 )t). This ends the proof. Remark 4.4. All the properties (we have proved here) of these matrix associated with morphisms are still true when replacing p-morphisms by p s -morphisms, for any s ∈ N * .

Examples

In Theorem 1.2 we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. In many cases, the sequence of rational approximations (P n /Q n ) n≥0 we construct turns out to satisfy the conditions (i) and (ii) of Lemma 3.7. This naturally gives rise to a much better estimate, as hinted in Remark 3.1. In this section, we illustrate this claim with few examples of algebraic Laurent series for which the irrationality exponent is exactly computed or at least well estimated.

Example 5.1. Let us consider the following equation over F 2 (T ) (11)

X 4 + X + T T 4 + 1 = 0.
This equation is related to the Mahler algebraic Laurent series, previously mentioned. Let

E 1 = {α ∈ F 2 ((T -1
)), |α| < 1}. We first notice that Eq. ( 11) has a unique solution f in E 1 . This can be obtained by showing that the application

h : E 1 -→ E 1 X -→ X + T T 4 + 1
is well defined and is a contracting map from E 1 to E 1 . Then the fixed point theorem in a complete metric space implies that the equation h(X) = X, which is equivalent to Eq. ( 11), has a unique solution in E 1 . Let f (T ) := i≥0 a i T -i denote this solution (with a i = 0, since f belongs to E 1 ).

The second step is to find the morphisms that generate the sequence of coefficients of f , as in Cobham's theorem. Notice that there is a general method that allows one to obtain these morphisms when we know the algebraic equation. In this example, we try to describe the important steps of this method and we will give further details later.

By replacing f in Eq. ( 11) and using the fact that f 4 (T ) = i≥0 a i T -4i we easily obtain the following relations between the coefficients of f a i+1 + a i + a 4i+4 + a 4i = 0, ( 12)

a 1 = 0, a 2 = 0, a 3 = 1, (13)
a i+4 + a i = 0, if i ≡ 0[4]. ( 14
)
By Eq. ( 13) and ( 14), we get that a 4i+1 = 0, a 4i+2 = 0 and a 4i+3 = 1, for any i ≥ 0. From Eq. ( 12) we deduce that

a 16i+4 = a 16i+8 = a 16i + a 4i , a 16i+12 = a 16i+8 + 1 = a 16i + a 4i + 1.
This implies that the 4-kernel of a := (a i ) i≥0 is the following set K 4 (a) = {(a i ) i≥0 , (a 4i ) i≥0 , (a 16i ) i≥0 , (0), (1)}.

Consequently, the 4-automaton generating a is q 0 /0 q 1 /1 0, 1, 2 0, 3 3 1, 2 Once we have the automaton, there is a general approach to obtain the morphisms that generate an automatic sequence. More precisely, the proof of Cobham's theorem precisely describes this process. The reader may consult the original article of Cobham [START_REF] Cobham | Uniform tag sequences[END_REF] or the monograph [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF] (Theorem 6.3.2, page 175). Following this approach, we obtain that a = σ ∞ (0), where σ is defined as follows

σ(0) = 0001 σ(1) = 1001.
It is now possible to apply our approach described in the first part of the paper. We will prove the following result. Notice that, Mahler's theorem implies only that µ(f ) ≤ 4 and Osgood's Theorem or Lasjaunias and de Mathan's Theorem cannot be applied since this Laurent series is clearly hyperquadratic.

Proof. We are going to introduce an infinite sequence of rational fractions (P n /Q n ) n≥1 converging to f . Since a begins with 0001, we denote V = 0 and for any n ≥ 1, V n = σ n (V ). Hence a begins with V n V n V n for any nonnegative integer n.

Since |V n | = 4 n , we set Q n (T ) = T 4 n -1. In Section 3.3, we showed that there exists a polynomial P n (T ) ∈ F 2 [T ] such that:

P n (T ) Q n (T ) = f V ∞ n (T ).
The Laurent series expansion of P n /Q n begins with σ n (0001)σ n (0001)σ n (0001)σ n (0), and we deduce that it begins with σ n (0001)σ n (0001)σ n (0001)0 while the sequence a begins with

σ n (0001)σ n (0001)σ n (0001)1.
Hence, the first 3 • 4 n th digits of the Laurent series expansion of P n /Q n and f are the same, while the following coefficients are different. According to Remark 3.1, we deduce that: 3 ≤ µ(f ) ≤ 6, or, if for any n ≥ 1 we have (P n , Q n ) = 1 then

µ(f ) = 3.
It thus remains to prove that (P n , Q n ) = 1 for every positive integer n. Let n ≥ 1. By Lemma 3.7 we have to prove that P n (1) = 0, i.e., P σ n (0) (1) = 0. Remark 4.3 implies that

P σ n (0) (T ) P σ n (1) (T ) = M σ n (T ) 0 1 .
Hence, P σ n (0) (1) = 0 if and only if

1 0 M σ n (1) 0 1 = 0,
that is, if and only if

1 0 M n σ (1) 0 1 = 0.
Indeed, by Corollary 4.3, we have

M σ n (1) = M n σ (1). Since M σ (1) = 1 1 0 0 ,
we deduce that M σ n (1) = M σ (1) and then,

1 0 1 1 0 0 0 1 = 0.
Consequently, we obtain that (P n , Q n ) = 1. This ends the proof.

Remark 5.1. Let f (T ) = i≥0 a i T -i be a Laurent series with coefficients in a finite field F q , where q is a power of p, and suppose that there is P (T ) ∈ F q [T ] such that P (f ) = 0. As mentioned before, there is a general approach that allows one to find the automaton that generates the infinite sequence a := (a i ) i≥0 . This consists of the following steps. First, there exists a polynomial Q with coefficients in F q , of the form Q(X) = i≥0 B i (T )X p i , B 0 (T ) = 0 such that Q(f ) = 0. This is known as an Ore's polynomial and the existence is due to Ore's theorem (for a proof see, for example, [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF], Lemma 12.2.3, page 355). Hence, the first step is to find such a Ore's polynomial vanishing f (this is possible by passing P to the power of p as many times as we need). The second step is to find some recurrent relations between the terms a i in order to find the kernel of a. This is possible thanks to the Frobenius morphism. The third step is the construction of the automaton generating a. Notice that a sequence is k-automatic if and only if its k-kernel is finite. The proof of this well-known result is explicit and we refer the reader to [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF] (Theorem 6.6.2, page 18). The last step is to find the morphisms generating a, as described in Cobham's theorem. In order to do this, the reader may refer to the proof of Cobham's theorem, which is explicit as well.

The following examples present different computations of irrationality exponents of Laurent power series over finite fields. We do not give the algebraic equations of them because the computation is quite long, but, as in the previous example (where we find the morphisms if we know the equation), there is a general approach that allows one to compute the equation of a Laurent series when we know the automaton generating the sequence of its coefficients. Indeed, by knowing the morphisms we can find the automaton (see the proof of Cobham's theorem), then knowing the automaton allows to find the kernel (see [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF], Theorem 6.6.2, page 185) and also the relations between the coefficients. These relations allow one to find a polynomial that vanishes the Laurent series (the reader may consult the proof of Christol's theorem in [START_REF] Christol | Ensembles presque periodiques k-reconnaissables[END_REF] or [START_REF] Allouche | Automatic Sequences: Theory, Applications, Generalizations[END_REF]-page 356, but also [START_REF] Kedlaya | The algebraic closure of the power series field in positive characteristic[END_REF]-where a generalisation of Christol's theorem is given). More precisely the polynomial that we compute from these relations is also an Ore's polynomial. Finally, we have to factor this polynomial and to check which is the irreducible factor vanishing our algebraic Laurent series.

Example 5.2. We now consider the Laurent series

f a (T ) = i≥0 a i T -i ∈ F 2 [[T -1 ]],
where the sequence a := (a i ) i≥0 is the image under the coding ϕ of the fixed point of the 8-uniform morphism σ, ϕ and σ being defined as follows Proof. We are going to introduce an infinite sequence of rational fractions (P n /Q n ) n≥0 converging to f a . Since a begins with 00000, we denote V = 0 and, for any n ≥ 1, V n = σ n (0). Hence, a begins with V 5 n for any n ≥ 1. Now, we set Q n (T ) = T 8 n -1. In Section 3.3 we showed that there exists a polynomial P n (T ) ∈ F 2 [T ] such that:

ϕ(0) = 1 ϕ(1) = 0 ϕ(2)
P n (T ) Q n (T ) = f V ∞ n (T ).
The Laurent series expansion of P n /Q n begins with

σ n (0)σ n (0)σ n (0)σ n (0)σ n (0)σ n (0)
and we deduce that it begins with

σ n (0)σ n (0)σ n (0)σ n (0)σ n (0)0,
while the sequence a begins with

σ n (0)σ n (0)σ n (0)σ n (0)σ n (0)1.
Hence, the first 5 • 8 n th digits of the Laurent series expansion of P n /Q n and f a are the same, while the following coefficients are different. Using Remark 3.1, we deduce that 5 ≤ µ(f a ) ≤ 10.

Moreover, if for any n ≥ 1 we have (P n , Q n ) = 1, then we obtain µ(f a ) = 5.

We now prove that (P n , Q n ) = 1 for every n ≥ 1. By Lemma 3.7 we have to prove that P n (1) = 0, i.e., P ϕ(σ n (0)) (1) = 0. By Remark 4.3 we have that

  P ϕ(σ n (0)) (T ) P ϕ(σ n (1)) (T ) P ϕ(σ n (2)) (T )   = M σ n (T )   ϕ(0) ϕ(1) ϕ(2)   and by Corollary 4.3 M σ n (1) = M n σ (1)
. The matrix associated with σ, when T = 1, is equal to the incidence matrix of σ:

M σ (1) =   1 1 0 1 0 1 0 1 1   .
Hence, by an easy computation, we obtain that, for any j ≥ 1,

M 2j+1 σ (1) =   1 1 0 1 0 1 0 1 1   and M 2j σ (1) =   0 1 1 1 0 1 1 1 0  
and thus, for every n ≥ 1, we have

1 0 0 M n σ (1)   ϕ(0) ϕ(1) ϕ(2)   = 1 = 0.
Consequently, for every n ≥ 1, P ϕ(σ n (0)) (1) = 0 and thus (P n , Q n ) = 1.

Example 5.3. We now consider the Laurent series

f a (T ) = i≥0 a i T -i ∈ F 3 [[T -1 ]],
where the sequence a := (a i ) i≥0 is the fixed point beginning with zero of the following 3-uniform morphism:

σ(0) = 010 σ(1) = 102 σ(2) = 122.
Thus a = 010102010 • • • .

Proposition 5.3. The irrationality exponent of the Laurent series f a satisfies 2.66 ≤ µ(f a ) ≤ 2.81.

In this case, we are not able to compute the exact value of the irrationality exponent but the lower bound we found shows that the degree of f a is greater than or equal to 3. Hence our upper bound obviously improves on the one that could be deduced from the Liouville-Mahler's theorem.

Proof. We are going to introduce an infinite sequence of rational fractions (P n /Q n ) n≥0 converging to f a . Since a begins with 010102, we denote V := 010102 and for any n ≥ 1, V n := σ n (V ). Hence a begins with σ n (010102)σ n (010102)σ n (0101) for any n ≥ 1. Now, we set Q n (T ) = T 2•3 n -1. There exists a polynomial P n (T ) ∈ F 3 [T ] such that:

P n (T ) Q n (T ) = f V ∞ n (T )
. The Laurent series expansion of P n /Q n begins with σ n (010102)σ n (010102)σ n (0101)σ n (0) and we deduce that it begins with σ n (010102)σ n (010102)σ n (0101)0, while the sequence a begins with σ n (010102)σ n (010102)σ n (0101)1.

Hence, the first 16 • 3 n th digits of the Laurent series expansion of P n /Q n and f a are the same, while the following coefficients are different. By Remark 3.1, we deduce that 2.66 ≤ µ(f a ) ≤ 4.81.

Furthermore, if for every n ≥ 1 we have (P n , Q n ) = 1, then 2.66 ≤ µ(f a ) ≤ 2.81.

Let n ≥ 1. We now prove that (P n , Q n ) = 1. By Lemma 3.7, since

Q n (T ) = (T -1) 3 n (T + 1) 3 n , we have to prove that, for all n ≥ 1, P n (1) = 0 and P n (-1) = 0. By definition of (P n (T )) n≥0 (see Eq. ( 6)) we have P n (1) = P Vn (1) and P n (-1) = P Vn (-1).

Since V n = σ n (010102) = σ n+1 (01) then, P Vn (T ) = P σ n+1 (0) (T )T 3 n+1 + P σ n+1 (1) (T ). The matrix associated with σ is

M σ (T ) =   T 2 + 1 T 0 T T 2 1 0 T 2 T + 1   .
Hence,

M σ (1) =   2 1 0 1 1 1 0 1 2   and M σ (-1) =   2 -1 0 -1 1 1 0 1 0   .
Notice that M 2 σ (±1) = M 4 σ (±1) and, by an easy computation, one can obtain that the sequence (P Vn (1)) n≥0 is 2-periodic and (P Vn (-1)) n≥0 is 1-periodic; more precisely, (P Vn (1)) n≥0 = (12) ∞ and (P Vn (-1)) n≥0 = (1) ∞ . This proves that P Vn (1) and P Vn (-1) never vanish, which ends the proof.

Example 5.4. We now consider the Laurent series

f a (T ) = i≥0 a i T -i ∈ F 5 [[T -1 ]],
where the sequence a := (a i ) i≥0 is the fixed point beginning with zero of the following 5-uniform morphism: Notice that, after Mahler's theorem, the degree of algebricity of f a is greater than or equal than 4.

Proof. We are going to introduce an infinite sequence of rational fractions (P n /Q n ) n≥0 converging to f a . We can remark that the sequence a begins with 00043000430004300144, and thus, more generally, a begins with σ n (00043)σ n (00043)σ n (00043)σ n (00)σ n (1) for every n ≥ 1. Now, we set Q n (T ) = T 5 n -1. There exists a polynomial P n (T ) ∈ F 5 [T ] such that:

P n (T ) Q n (T ) = f σ n (00043) ∞ (T ).

The Laurent series expansion of P n /Q n begins with σ n (00043)σ n (00043)σ n (00043)σ n (00043) = = σ n (00043)σ n (00043)σ n (00043)σ n (00)σ n (0)σ n (43)

and we deduce that it begins with σ n (00043)σ n (00043)σ n (00043)σ n (00)0, while the sequence a begins with σ n (00043)σ n (00043)σ n (00043)σ n (00)1.

Hence, the first 17 • 5 n th digits of the Laurent series expansion of P n /Q n and f a are the same, while the following coefficients are different. According to Remark 3.1, we deduce that 3.4 ≤ µ(f a ) ≤ 7.08.

Moreover, if for every n ≥ 1 we have (P n , Q n ) = 1, then we obtain µ(f a ) = 3.4.

Let n ≥ 1. We are going to prove that (P n , Q n ) = 1, for any n ≥ 1. By Lemma 3.7, it is necessary and sufficient to prove that P n (1) = 0, i.e., P σ n (0) (1) = 0, which is equivalent to:

Lemma 3 . 4 .

 34 Let a = (a i ) i≥0 and b = (b i ) i≥0 be two infinite sequences over a finite alphabet, satisfying a i = b i for 0 ≤ i ≤ L -1, where L ∈ N * . Then, we have|f af b | ≤ 1 |T | L .the equality being obtained when a L = b L .

  a prefix of a. Notice also that |U n | = |U |p n and |V n | = |V |p n and the sequence (|V n |) n≥1 is increasing.

Définition 4 . 2 .Example 4 . 2 .

 4242 Let σ : A m → A * m be a morphism. We associate with σ the m × m matrix M σ (T ) with coefficients in F p [T ] defined by M σ (T ) = (β σ(i),j (T )) 0≤i,j≤m-1 . Let σ : A 3 → A * 3 , σ(0) = 010, σ(1) = 2101 and σ(2) = 00211. Then

Example 4 . 3 .

 43 Let σ be a 3-morphism over A 2 and U = 10100. Then, for any n ∈ N, σ n (U ) = σ n (1)σ n (0)σ n (1)σ n (0)σ n (0) and v U (T ) = (T 3 + T + 1, T 4 + T 2 ).

Corollary 4 . 3 .

 43 Let a ∈ F p . Then for any n ∈ N we have M σ n (a) = M n σ (a).

Proposition 4 . 1 .

 41

Figure 2 .

 2 Figure 2. A 4-automaton recognizing a

Proposition 5 . 1 .

 51 One has µ(f ) = 3.

Proposition 5 . 2 .

 52 One has µ(f a ) = 5.

P

  Vn (1) = P σ n+1 (0) (1) + P σ n+1 (1) (1)P Vn (-1) = -P σ n+1 (0) (-1) + P σ n+1 (1) (-1). Remark 4.3 implies that   P σ n (0) (T ) P σ n (1) (T ) P σ n (2) (T )   = M σ n (T )If we now replace T = 1 (respectively T = -1), we obtain that P n (1) = 0 (respectively P n (-1) = 0) if and only if 1

4 )

 4 = 00144. Thus a = 0004300043 • • • . Proposition 5.4. One has µ(f a ) = 3.4.


Nicolas Schmidt
email: schmidt@ceremade.dauphine.fr

Gabriel Peyré

Yves Frégnac

Dissipative Wave Model Fitting Using Localized Sources

published or not. The documents may come   L'archive ouverte pluridisciplinaire

1 Introduction Biological motivation. This work studies a particular inverse problem in 2-D wave imaging, motivated by the problem of neural sources separation in the visual brain. Optical imaging with voltage sensitive dye (VSDOI) offers the possibility to image with a high temporal and spatial resolution the activity of the early visual area of the cortex [START_REF] Grinvald | VSDI : A new era in functional imaging of cortical dynamics[END_REF]. A challenging problem is to separate the different activity patterns emerging over the cortex. In some restricted experimental setups, it has been observed that the activity is characterized by waves that propagate over the 2-D cortical surface at approximately constant speed [START_REF] Jancke | Imaging cortical correlates of illusion in early visual cortex[END_REF], [START_REF] Bringuier | Horizontal propagation of visual activity in the synaptic integration field of area 17 neurons[END_REF], [START_REF] Roland | Cortical feedback depolarization waves: A mechanism of top-down influence on early visual areas[END_REF]. In a previous work [START_REF] Schmidt | Separation of traveling waves in cortical networks using optical imaging[END_REF]we proposed to model this activity using a simple wave equation with spatially localized sources. Estimating the parameters of this model (e.g. speed and diffusivity) is however challenging because of the very large noise level that contaminates the observations. In this paper, we propose a fast method to achieve a robust estimation of these parameters. Previous works. A large amount of work has been devoted in seismic imaging to estimate spatially varying speeds when sources are fully known and the propagating waves are observed on the boundary of the domain, see for instance [START_REF] Bleistein | Multi-dimensional Seismic Imaging, Migration, and Inversion[END_REF] and the reference therein. This leads to a non-convex inverse problem, that is challenging to solve because of its high dimensionality. Specific methods can be developed in the case where the sources are sparse and localized, see for instance [START_REF] Harris | Nonlinear multifrequency wave equation inversion[END_REF]. Another class of methods assumes some degree of randomness on the (unknown) sources, and use a statistical analysis on the recorded signals to estimate the parameters (e.g. travel times or speeds), see for instance [START_REF] Snieder | Retrieving the green's function of the diffusion equation from the response to a random forcing[END_REF], [START_REF] De Hoop | Estimating a green's function from "field-field" correlations in a random medium[END_REF], [START_REF] Garnier | Passive sensor imaging using cross correlations of noisy signals in a scattering medium[END_REF], [START_REF] Derode | Recovering the green's function from field-field correlations in an open scattering medium[END_REF]. In this paper, we consider a different setup, where only the spatial support of the sources are known. Contribution. This paper introduces a novel imaging paradigm, that might be relevant for some applications in computational neuroscience. We stress the importance of having some prior knowledge about the spatial localization of the sources, and imposing a small number of parameters (e.g. constant coefficients). We also provide a numerical scheme that makes use of a representation of the propagating wave over the Laplace domain. This representation uses a reduced number of green kernels localized on the boundary of the source support. This leads to a fast scheme that does not need repeated simulations of the space/time wave propagations.

2 Wave Generation Forward Model Linear PDE model. Biological considerations leaded us to formulate a simple linear model for the propagation of the information in the early areas of the visual cortex [START_REF] Schmidt | Separation of traveling waves in cortical networks using optical imaging[END_REF]. The measured observation X = u + w is assumed to be composed of a signal u caused by the visual excitation and an an additive noise w caused by the neural activity and the imaging device. The signal is assumed to propagate in time according to a linear wave equation parameterized by (unknown) coefficients α 0 = (a 0 , b 0 , c 0 ) ∈ R 3 and an (unknown) source f 0 (t, x). It thus solves

a 0 ∂ 2 u ∂t 2 + b 0 ∂u ∂t + c 0 u -∆u = f 0 (1)
on R 2 × (0, +∞), with boundary conditions ∂ k u ∂t k = 0 at t = 0 for k = 0, 1. In the following, we denote as u = T α 0 (f 0 ) the propagating operator that maps the sources to the signal. For positive values of α 0 , this model leads to propagating and dissipative waves that might be useful to model certain neural activities. In this paper we restrict our attention to this second order model, although our method could be used for more general linear PDEs. Source constraint. In accordance with some biological observations [START_REF] Schmidt | Separation of traveling waves in cortical networks using optical imaging[END_REF], we consider a spatial constraint on the support of the sources

C x = {f \ ∀ x / ∈ Ω, ∀ t > 0, f (x, t) = 0}
where Ω ⊂ R 2 is a known compact set. Figure 1 shows an example of a typical propagation that is achieved with this model using sources supported inside two small disks. Inverse problem. The coefficient estimation problem requires to estimate α = (a, b, c) ∈ R 3 in order for the observations X to be well approximated by a propagating wave X ≈ T α (f ) for some source f . An efficient method should provide an estimate α close to the true value α 0 even in the presence of a large noise w contaminating the observations. The following theorem asserts that the true value α 0 can be uncovered in the absence of noise.

Theorem 2.1. If f 0 ∈ C x , then T α 0 (f 0 ) = T α (f ) implies (α, f ) = (α 0 , f 0 ).
This suggests to jointly estimate α and f by minimizing the deviation between X and T α (f ). This is however a difficult task because of the high dimensionality of f and the presence of a large noise which might lead to a poor solution. We propose here to avoid estimating the source f by exploiting the constraint C x . Once α has been computed, it is easy to recover the sources f by solving the linear equation X = T α (f ), possibly with some regularization to remove the noise.

Green Representation of the Wave Model

This section details our main mathematical analysis of the model. The resulting estimation procedure is described in the next section. Laplace transform. To reduce the computational complexity of the method, we remove the temporal dimension by performing a Laplace transform, given some β > 0

Y (x) = +∞ 0 e -βt X(t, x)dt (2)
where we have omitted the dependency on β for the sake of readability. The following proposition shows that the original PDE ( 1) is projected to a 2-D Helmholtz equation over the Laplace domain.

q 0 (β)Y -∆Y = 0 in R 2 \Ω ( 3
)
where the shape parameter is q 0 (β) = a 0 β 2 + b 0 β + c 0 .

Inspired by this proposition, our strategy is thus to estimate the value of the shape parameter q(β) for different values of β in order to recover the value of α 0 . Green kernel extrapolation. For each value of the shape parameter q ∈ R, the Green solution (or fundamental solution) G x,q at x ∈ R 2 of the Helmoltz equation ( 3) on the whole domain R 2 is given by

G x,q (y) = K 0 ( √ q ||x -y||)
where K 0 is the modified Bessel function of the second kind. At a point y ∈ ∂Ω of the boundary of Ω, we denote as ∂ n G x,q (y) the derivative of G x,q in the direction normal to ∂Ω.

Given a set of values A = (A(x)) x∈∂Ω and derivative values B = (B(x)) x∈∂Ω defined on this boundary, one obtains a function on the whole spatial domain by the following Green extrapolation

H q (A, B)(x) = - 1 2π ∂Ω A(y)∂ n G x,q (y) -B(y)G x,q (y) dy
where the integral is performed along the 1-D contour of the boundary. Dissipative wave reproducing formula. Our main result is the following theorem, that expresses the fact that in the noiseless setting, the Laplace transform of the observations can be extrapolated to the whole domain given only its values on the boundary ∂Ω. We denote receptively as A Y and B Y the restriction to ∂Ω of Y and its derivative normal to ∂Ω.

Theorem 3.1. If w = 0, one has for all x / ∈ Ω Y (x) = H q 0 (β) (A Y , B Y )(x)
where q 0 (β) is defined in proposition 3.1.

Wave Model Fitting Algorithm

Fitting procedure. Our parameter estimation method proceeds in two steps:

Step 1: for each β ∈ Σ, compute an estimate of the shape parameter by solving

q(β) = argmin q E β (q) (4)
where

E β (q) = min A,B ||Y -H q (A, B)|| 2 L 2 (Ω c ) . ( 5
)
Step 2: estimate the parameter α 0 by computing the optimal α solving min α=(a,b,c)∈R 3 β∈Σ

aβ 2 + bβ + c -q(β) 2 . ( 6
)
Theorem 3.1 asserts that in the noiseless case, w = 0, one has q(β) = q 0 (β), and proposition (3.1) thus implies that α = α 0 so that the estimation procedure succeeds. Equality does not hold in the general case, but the L 2 fit (5) of a single parameter from data on the domain Ω c = R 2 \Ω guarantees a strong robustness to noise. Discrete algorithm. In a numerical scenario, the data X is sampled at discrete locations in space and time, so that the L 2 regression ( 5) is performed on the discrete grid points lying outside Ω.

In order to be able to compute accurately the Laplace transform Y of X we assume that the propagating wave T α 0 (f 0 ) is vanishing for t > t 0 , and the integral (2) is estimated numerically from the available time samples. The set Σ of tested β values is an uniform discretization of an interval [0, β max ] at |Σ| locations.

The resolution (4) corresponds to a minimization of a non-convex function of a single variable. Each estimation of this function requires a linear regression [START_REF] Grinvald | VSDI : A new era in functional imaging of cortical dynamics[END_REF] where the variables A and B are sampled along the 1-D contours ∂Ω. The number of variables in this regression is thus small so that we decided to perform a brute force exhaustive search for q(β). More advanced optimization schemes could be used as well to speed up the process. The final polynomial fit (6) only requires the resolution of a 3 × 3 linear system.

Numerical Results

We benched the efficiency of our method on a synthetic model that is intended to be biologically plausible in term of both the spatial and temporal dynamics of the source. Signal model. We generate a smooth source f 0 as a space-time colored Gaussian noise. We restrict the support of the source by multiplying it with a smooth windowing function in order to obtain a compact support in Ω × [0, t 0 ] where t 0 = 40, Ω is an union of two disks of radius 0.08, centered at (0.5, 0.2) and (0.5, 0.7).

The observation X = u + w with u = T α 0 (f 0 ) is generated by using α 0 = (a 0 , b 0 , c 0 ) with

a 0 = 1 s 2 0 , b 0 = 2 ρ 0 s 2 0 , c 0 = ρ 0 s 0 2
where s 0 represent the biological speed of the propagating wave, and ρ 0 accounts for the dissipation of the propagation. The noise w is a Gaussian white noise of standard deviation σ ||u|| ∞ , where σ 0 controls the strength of the noise. The wave propagation u is generated by solving numerically the (1) with semi-implicit finite difference scheme in time, spectral discretization of the spatial Laplacian, and absorbing boundary conditions using a perfect matched layer (PML) scheme. In the numerical experiments, we use s 0 = 1 and ρ 0 = 1/10 for a square domain of observations [0, 1] 2 discretized using 150×150 points on an uniform 2-D grid. Figure 2 shows a typical signal used for the numerical experiments. Numerical evaluation. We measure the efficiency of our fitting procedure according to three noise level corresponding to σ = 0 (no noise), σ = 0.03 (low noise) and σ = 0.1 (strong noise). We perform the fitting procedure described in Section 4 using β max = 0.5 and |Σ| = 25. q(β) is computed for our shape parameter by solving (4) and then the polynomial fit β → aβ 2 + bβ + c is obtained by solving [START_REF] Harris | Nonlinear multifrequency wave equation inversion[END_REF]. After having computed the values for (a, b, c), we estimate the relevant biological parameters (speed and dissipation) as s = 1 √ a and ρ = 2 c b .q(β) is computed for 200 realizations of the noise w. Then we compute the relative error q r (β) = q(β)-q 0 (β) q 0 (β) for each realization. Figure 3 shows the expected value E β with respect to realizations of the noise w (resp. standard deviation Std β ) of q r (β) for 0 β 2β max .

Table 5 shows statistics of speed s and dissipation ρ parameters. We observed in this table that estimation is not perfect in the noiseless case (σ = 0) because of discretization error for both the signal generation and the Laplace transform. Our method is able to estimate the value of s 0 even in the presence of a reasonable noise. In the presence of a strong noise (σ = 0.1), our method is more accurate for the estimation of the speed s 0 than for the dissipation parameter ρ 0 .

Conclusion and Perspectives

In this paper we have proposed a fast and robust method to estimate constant parameters in a linear PDE model using only a constraint on the (known) support of the sources. This method might be useful to calibrate biological models in computational neuroscience. We are currently working on the application of our method to real data from VSDOI experiments. This method is also quite general and might be extended to other settings, such as 3-D propagations.

Figure 1 :

 1 Example of a 2-D wave u propagating according to our model.

Proposition 3 . 1 .

 31 If w = 0 and f 0 ∈ C x , Y satisfies

Figure 2 :

 2 Example of synthetic signal X considered in our numerical experiment, here σ = 0.1 (seeSection 5).

Figure 3 :σ

 3 Figure 3: Display of the expected value E β (blue and red solid lines) of q r (β). The blue and red regions corresponds to confidence intervals defined as the set of (β, q) with |q -E β | 2Std β .Table 1: Statistics of speed s and dissipation ρ σ Expected value Std. Deviation

Acknowledgement. We thank Laurent Demanet for his PML implementation of the wave equation solver. This work is supported by ANR grant NatImages (ANR-08-EMER-009) and FACETS (FET Bio-I3: 015879).


Belinda Klemmensen Moessner

Merete Skamling

Tina Riis Jørgensen

Jørgen Georgsen

Peer Court Pedersen

Brehm Christensen

Mössner Bk

Riis Jørgensen

Pedersen

Christensen Pb

M.D. Belinda K Mössner

observed after 11 years of regional vaccination among Danish drug users

Keywords: hepatitis c, hiv, prevalence Hepatitis C, HIV, Prevalence

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The primary aim of this study was to estimate the 2007 prevalences of viral hepatitis and HIV, and to compare these estimates with previous findings in the same geographical region. The HBV vaccination status among drug users was also determined. A survey investigating the prevalence of chronic hepatitis and HIV infection was performed among drug users attending treatment centers in the county of Funen in 2007. Clients present during these visits were eligible to participate in the survey, after written informed consent was obtained. A standardized questionnaire about hepatitis and vaccination status was answered by the participant.

Afterwards, blood samples for serologic analysis were drawn and separated within eight hours. The samples were either analyzed immediately or kept frozen at -80 °C until analyzed. All samples were analyzed with the AxSYMTM assays (Abbott Laboratories, North Chicago, IL, USA) according to the manufacturer's instructions. Samples were tested for hepatitis B surface antigen (HBsAg), antibodies to hepatitis B surface antigen (anti-HBs), antibodies to hepatitis B core antigen (anti-HBc), antibodies to hepatitis C virus (anti-HCV), and antibodies to human immunodeficiency virus (anti-HIV). If HBsAg was present, samples were tested for hepatitis E envelope antigen (HBeAg), antibodies to hepatitis E envelope antigen (anti-HBe) and antibodies type IgM to hepatitis B core antigen (anti-HBc IgM). Quantitative anti-HBs analysis was performed with Axsym AUSABTM ,that is calibrated according to the WHO standard. Samples with anti-HBs > 10 IU/L were classified as reactive.

HBsAg reactive samples were confirmed by the AxSYM HBsAg Microparticle Enzyme Immuno Assay (MEIA). Anti-HIV and anti-HCV reactive samples were confirmed with Line Immuno Assay (LIA) tests. Quantitative HCV-RNA and HBV-DNA tests were performed with in-house real-time PCR assays at the Department of Clinical Immunology, Aalborg Hospital. The sensitivities of these assays were 29 IU/mL for HCV-RNA and 30 IU/mL for HBV DNA [START_REF] Krarup | A quantitative HCV-PCR test for routine diagnostics[END_REF][START_REF] Krarup | Benign course of long-standing hepatitis B virus infection among Greenland Inuit?[END_REF]. If serology results were not obtained during the survey, results from routine serology tests within one year of the survey date were used. All previous positive HIV tests were included. Based on the treatment center register of all clients non-participants did not differ significantly by age or gender. In 2007, 235 (73%) of the clients were male, with a median age of 40 years (interquartile range (IQR) 34-47). The prevalence of injecting drug users (IDUs) was 75%; median time since first injection was 12 years (IQR 7-19 years). 137 clients with known IDU status (62% of available clients) from 1996 were included in the present study.

Overall prevalence

The prevalence of hepatitis B and C infection fell significantly during the time between the two surveys for the entire surveyed population (Table I). Anti-HBc decreased from 70% to 50.2% (P < 0.0001), and anti-HCV decreased from 82.8% to 66.8% (P < 0.0001). The HIV prevalence remained low and unchanged (1.1%). The overall change in prevalences was primarily due to an increase (7% to 25%) in the proportion of non-injectors, as this group displayed a much lower prevalence of bloodborne viral infections.

Prevalence changes among IDUs

The prevalence of HBsAg among IDUs decreased from 7% in 1996 to 1.3% in 2007 (P = 0.005), and the prevalence of anti-HBc IgM antibody fell from 29% to 1.4% (P < 0.0001). The total anti-HBc prevalence remained stable (Table II). The 1996 survey was performed in the late part of an HBV outbreak with many recent infections. The exclusion of anti-HBc IgM positives still resulted in a significant HBsAg decline between the two surveys, from 6% to 1.3% (P = 0.024).

Correspondingly, the HBsAg prevalence among antiHBc positives fell from 8% (7/90) in 1996 to 1% (1/151) in 2007 (P = 0.002). During the interim between the two surveys, the time until half of the population became infected after initiating injecting drug use rose from three to 13 years for HBV, but the HCV time lag remained stable at 1-2 years (Fig. 1).

Prevalence changes in non-IDUs

Among the non-IDU population, no significant changes in prevalence were observed; however, only 9 non-IDUs were included in the 1996 survey (Table II).

Hepatitis B vaccination

Among the participants in the 2007 survey, 24% (76/320) had been vaccinated against hepatitis B in the treatment centers prior to the survey. Among those vaccinated, 47% (34) displayed protective levels of anti-HBs, 24% (18) were negative for all hepatitis B serologic markers, and 28% (21) were anti-HBc positive (with/ without anti-HBs), indicating previous HBV-infection.

Among participants not registered as vaccinated, 39% (90/232) tested negative for all hepatitis B serologic markers, and hepatitis B vaccination was indicated for these participants. 32% (75/232) were "anti-HBc only" (anti-HBc positive and anti-HBs negative).

In the 1996 survey only 0.7% (1/137) was vaccinated. (P < 0.001).

HIV and hepatitis C

Three of the IDUs were identified as anti-HIV positive (1.3%) during the 2007 survey.

Additionally, 1.1% (5/439) of the non-participants had previously tested positive for anti-HIV, and thus the overall prevalence of anti-HIV was 1.1%. This rate was unchanged compared to 1.0% Analysis of two surveys performed 11 years apart revealed that IDUs continued to be at high risk for viral hepatitis, and at a much higher risk than non-injectors. The most important difference between the two surveys was a significant decline in HBsAg among IDUs without a corresponding decline in anti-HBc (table II). This decline in the current infection proportion (HBsAg/anti-HBc, the sAg/core proportion) from 7.7% (7/90) in 1996 to 0.7% (1/151) in 2007 has to our knowledge not been reported previously.

The SAg/core proportion represents an unknown mixture of chronic and acute infection. However, the SAg/core proportion has been reported as rather stable across HBV surveys in low endemic countries and among IDUs. An international review of 20 studies of 7,626 IDUs in 1994 (prior to routine vaccinations among drug users) reported an overall 9.3% prevalence of HBsAg, with an average SAg/core proportion of 13% [START_REF] Levine | Epidemiology of hepatitis B virus infections among injecting drug users: seroprevalence, risk factors, and viral interactions[END_REF]. A 1997 survey in Copenhagen, where no HBV epidemic was observed among drug users, reported a SAg/core proportion of 8.4%, and in a Danish prison survey in 1996, 14% was recorded [START_REF] Christensen | Prevalence and incidence of bloodborne viral infections among Danish prisoners[END_REF][START_REF] Fuglsang | Prevalence of HIV and hepatitis B and C among drug addicts in the city of Copenhagen[END_REF]. This study's 1996 SAg/core proportion was therefore in accordance with previous reports, but the 2007 SAg/core proportion was significantly lower (table I

and II).

There are several possible explanations for this observation. The difference could reflect a high proportion of recent infection in 1996, when an outbreak of hepatitis B had taken place in the years just prior to sampling. This hypothesis is supported by the high proportion of anti-HBc IgM positives found in 1996. The low SAg/core proportion in 2007 could reflect a situation in which the 1996 acute infections had been cleared, leaving the patients only anti-HBc positive. However, the prevalence found in 1996 was not higher than reported elsewhere, and when clients participating in both surveys were excluded, similar results were obtained (table II). The cross-sectional study design hindered assessment of whether the low 2007 SAg/core proportion was due to a decrease in the number of chronically-infected individuals, or whether the acute infections had a shorter HBsAg-positive phase. As there were only 3 HBsAg positives in the 2007 survey, both scenarios must have diminished, consequently lowering the risk of further transmission. These results are supported by the observation that the time from exposure (drug injection) needed to infect half of the population with hepatitis B increased from 3 to 13 years between the two surveys (Fig. 1). A lower overall prevalence of anti-HBc in 2007 among IDUs was also found, although this figure was not statistically significant (table II).

The HBsAg decline coincided with an increase in HBV vaccination coverage from 0.7% to 24% in the period between the two surveys. The vaccination coverage reported between the surveys was probably a conservative estimate, as some drug users may have been vaccinated outside the treatment centers included in this study. In addition to treatment center vaccination, all drug users admitted to Danish prisons have been offered hepatitis B vaccination since 2000. Depending on the uptake among prisoners, this initiative could have covered more than half of the drug-using population within a six-year period [Christensen et al., 2006b]. The protective levels of anti-HBs (without detectable anti-HBc) among 4% of the 2007 clients who claimed unvaccinated status support the underreporting hypothesis. Documentation on the rate of full vaccination course completion was unavailable. However, even persons who are vaccinated without developing a durable anti-HBs response may have enough immune memory to protect against the development of chronic infection upon transmission. This scenario was reported in follow-up studies of childhood HBV vaccination in which anti-HBc sero-conversion was observed in a number of cases without development of chronic hepatitis B [START_REF] Bialek | Persistence of protection against hepatitis B virus infection among adolescents vaccinated with recombinant hepatitis B vaccine beginning at birth: a 15-year follow-up study[END_REF][START_REF] But | Twenty-two years follow-up of a prospective randomized trial of hepatitis B vaccines without booster dose in children: Final report[END_REF][START_REF] Dentinger | Persistence of antibody to hepatitis B and protection from disease among Alaska natives immunized at birth[END_REF][START_REF] Harpaz | Elimination of new chronic hepatitis B virus infections: results of the Alaska immunization program[END_REF][START_REF] Mcmahon | Antibody levels and protection after hepatitis B vaccination: results of a 15-year follow-up[END_REF][START_REF] Van Der Sande | Long-term protection against carriage of hepatitis B virus after infant vaccination[END_REF].

Decreasing the carrier pool by implementing vaccination programs leads to an increased immunized population proportion. Individuals who have cleared their infections may also contribute to some degree of herd immunity among the individuals who were not previously infected or vaccinated.

In conclusion, for the first time, a decline in hepatitis B infection among drug users in Denmark has been demonstrated, and this decline coincides with an increase in HBV vaccination coverage. Due to the cross-sectional design of this study, a definitive causal relationship between vaccination coverage and declining prevalence of HBV infection could not be established. Nonetheless, the results of the present study suggest that every effort should be taken to enhance vaccination coverage among drug users in the future.

F

  The treatment centers' vaccination program for hepatitis B consisted of 20µg HBsAg (Engerix B®)(GlaxoSmithKline Biologicals, Rixensart, Belgium) administered in the deltoid muscle at days 0, 7, 21, and at 3 months to anti-HBc-negative susceptible clients. The treatment was supplemented with a hepatitis A vaccination consisting of inactivated hepatitis A virus (Havrix®) (GlaxoSmithKline Biologicals, Rixensart, Belgium) given as a two-dose schedule if the client was anti-HAV negative. Anti-HBs tests were performed one month after the last dose of vaccine, and titers >=10 IU/L were considered protective. Additional vaccines were offered if protection was not achieved. Vaccination status was ascertained by interview and extracted from client records. Data was entered into an Access (Microsoft office 2003, Seattle, WA, USA) database and proofread for clerical errors. Statistical analysis was performed with STATA version 10 (Statacorp LP, College Station, TX, USA); nonparametric tests (Fischer's exact test and Wilcoxon rank sum test) were used as appropriate. This study was approved by the research ethics committee of Region Southern Denmark (Project-ID VF-20060078), and by the Danish Data Protection Agency.

  clients who attended the treatment centers in 2007, 374 (49%) were included in the present study. The most frequent reason for exclusion was client absence at the time of the study visit. Additionally, 54 clients who participated in both surveys were excluded from the 2007 survey.

Figure 1 .

 1 Figure 1. Relationship between duration of intravenous drug use and sero-positivity.

Table I .

 I Antibody prevalences of Funen county drug users attending a treatment center by year

	BKMössner\ Prevalence of viral hepatitis and HIV\Manuscript\Version2\Jan2010	1
		2007 (n = 320)*	1996 (n= 221)
	n pos/N total, %	n/N	%	n/N	%	p-value
	Anti-HAV total	91/316	28.8	36/157	22.9	0.188
	Anti-HBs	121/312	38.8	77/191	40.3	0.778
	HBsAg	3/318	0.9	20/205	9.8	<0.0001
	Anti-HBc IgM Anti-HBc Anti-HBc only	o r F o r F o r 2/150 F 156/311 o r 79/156	1.3 50.2 50.6	36/119 119/170 43/118	30.3 70 36.4	<0.0001 <0.0001 0.02
	HBeAg Anti-Hbe	P e P e P e 1/3 P e 9/31	33.3 29	8/33 42/155	24.2 27.1	1 0.828
	sAg/core proportion Anti-HCV	e r e r e r 3/156 2 e r 209/313 66.8	20/119 164/198	16.8 82.8	<0.0001 <0.0001
	HCV-RNA	116/290	R R R R 40	76/136	56.3	0.002
	Anti-HIV *	3/318 †	e v i e e v i e e v i e e 0.9 2/203 v e i	1	1
				w w w w

Excluding 54 persons who participated in 1996. †In addition, 1.1% (5/439) of the non-participants had previously tested positive for anti-HIV; the overall prevalence of anti-HIV was therefore 1.1%.

	BKMössner\ Prevalence of viral hepatitis and HIV\Manuscript\Version2\Jan2010	2
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table II .

 II Changes in serologic markers according to drug use in 1996 and 2007

		Non-IDU				IDU
		1996 n = 9	2007 n = 79		1996 n = 128	2007 n = 241
		(7%)		(25%)			(93%)	(75%)
	n pos/N total, %	n/N	%	n/N	%	p-value	n/N	%	n/N	%	p-value
	Anti-HAV total	2/9	22	24/78	31	0.719 30/127	24	67/238	28	0.385
	HBsAg	0/9	0	0/79	0	1	9/128	7	3/239	1.3	0.005
	Anti-HBs	0/9	0	24/78	31	0.058 57/128	45	97/234	41	0.580
	Anti-HBc IgM	0/1	0	0/5	0	1	26/90	29	2/145	1.4	<0.0001
	Anti-HBc	1/9	11	5/78	6	0.491 90/126	71	151/233	65	0.239
	Anti-HBc only	1/1	100	3/5	60	1	31/90	34.4	76/151	50.3	0.022
	HBeAg	0/1	0	0/0	0	1	4/22	18	1/3	33	0.504
	Anti-Hbe	0/9	0	1/2	50	0.182 34/126	27	8/29	28	1
	sAg/core proportion*	0/1	0	0/5	0	1	9/90	10	3/151	2	<0.0001
	Anti-HCV	1/9	11	9/79	11	1	115/128	90	200/234	85	0.257
	HCV-RNA	1/9	11	4/77	5	0.433 70/119	59	112/213	53	0.301
	Anti-HIV	0/8	0	0/79	0	1	0/127	0	3/239	1.3	0.554

*The sAg/core proportion is a calculated value (n HBsAg/n Anti-HBc)

Acknowledgements:

We thank Inge Birkemose for support and comments. This study was supported by grants from "Fonden til Laegevidenskabens fremme, AP Møller" and "Aage Thuesen Bruun og hustrus legat."

Peer Brehm Christensen has been a member of the advisory board for Roche a/s. No other conflicts of interest are associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.