Jacky Cresson
email: jacky.cresson@univ-pau.fr

Jordy Palafox
email: jordy.palafox@univ-pau.fr

Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

Keywords: elsarticle.cls, L A T E X, Elsevier, template

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture has already been proved for quadratic and quartic systems. Using the correction of a vector eld to characterize isochronicity and explicit computations of this quantity for polynomial vector elds, we are able to describe a very large class of nonisochronous Hamiltonian systems of even arbitrarily large degree.

Introduction and main results

The Jarque-Villadelprat conjecture

In this paper, we study centers of planar polynomial Hamiltonian systems in the real case. In particular we focus on isochronous centers. Our main concern is the following conjecture stated by Jarque and Villadelprat in [START_REF] Jarque | Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four[END_REF]: 5

Let X be a real polynomial Hamiltonian vector eld of the form:

X(x, y) = -∂ y H(x, y)∂ x + ∂ x H(x, y)∂ y , (x, y) ∈ R 2 , (1)
where H(x, y) is a real polynomial in the variables x and y. The maximum degree of the polynomials ∂ x H and ∂ y H is the degree of the Hamiltonian vector eld.

Conjecture : Every center of a planar polynomial Hamiltonian system of even degree is nonisochronous.

The conjecture is known to be true for quadratic systems thanks to a result of Loud in [START_REF] Loud | Behaviour of the period of solutions of certain plane autonomous systems near centers[END_REF] and in the quartic case by a result of Jarque-Villadelprat in [START_REF] Jarque | Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four[END_REF]. The proof of Jarque and Villadelprat is based on a careful study of the bifurcations set and seems dicult to extend to an arbitrary degree. The conjecture is open for the other cases despite partial results in this direction obtain by B. Schuman in [START_REF] Schuman | Sur la forme normale de Birkho et les centres isochrones[END_REF][START_REF] Schuman | Une classe d'hamiltoniens polynomiaux isochrones[END_REF] using an explicit computation of the rst coecients of the Birkho normal form and Chen and al. [START_REF] Chen | Non-isochronicity of the center for polynomial Hamiltonian systems with even degree nonlinearities[END_REF] proving what they call a weak version of the conjecture, i.e. that any vector elds having only even components is nonisochronous.

Dierent strategies can be used to go further toward this conjecture. A rst class of methods can be called geometric and are related to some special features of Hamiltonian or isochronous centers. We can mention for example the work of L. Gavrilov [START_REF] Gavrilov | Isochronicity of plane polynomial Hamiltonian systems[END_REF] and P. Mardesic, C. Rousseau and B. Toni [START_REF] Mardesic | Linearisation of isochronous centers[END_REF]. Up to now, these methods are unable to reproduce some special results obtained by B. Schuman [START_REF] Schuman | Sur la forme normale de Birkho et les centres isochrones[END_REF][START_REF] Schuman | Une classe d'hamiltoniens polynomiaux isochrones[END_REF] for classes of polynomial vector elds of arbitrary degree. Another class of methods can be called analytic and are more or less all dealing with the computations of quantities which can be obtained algorithmically like period constants [START_REF] Francoise | Les conditions du centre pour un champ de vecteurs quadratiques[END_REF] and coecients of normal forms [START_REF] Cresson | Formes normales et problème du centre[END_REF] (see also [START_REF] Francoise | Géométrie analytique et systèmes dynamiques[END_REF]). However, such methods are usually assumed to be intractable when one is dealing with a vector eld of arbitrary large degree (see for example [START_REF] Jarque | Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four[END_REF] p.337). This is indeed the case when one has no informations on the algebraic structure of these coecients. Then one is reduced to compute Grobner bases or to use the elimination method. However, one is quickly limited by the computational complexity and the memory size need to perform these computations. Existing results are restricted to polynomials of order 5.

A natural problem is then to look for methods allowing us to bypass these technical limitations.

In the analytic setting, this can be done for (pre)normal forms or other local analytic objects using the mould formalism introduced by J. Ecalle in the 70 ( [START_REF] Ecalle | Les algèbres de fonctions résurgentes[END_REF], [START_REF] Ecalle | L'équation du pont et la classication analytique des objets locaux[END_REF], see also [START_REF] Cresson | Calcul Moulien[END_REF]) which allows an ecient algorithmic construction of these objects but also a very precise analysis of the coecients.

The idea is to separate in these coecients what is universal and what is not.

Main results

In this paper, using the formalism of moulds (see [START_REF] Ecalle | Les algèbres de fonctions résurgentes[END_REF], [START_REF] Ecalle | L'équation du pont et la classication analytique des objets locaux[END_REF]) and a particular object attached to a vector eld called the correction dened by Ecalle and Vallet in [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF]. In particular, we obtain a partial answer to the conjecture for arbitrarily large degree.

It is well known that isochronicity of a real center is equivalent to its linearisability (see [START_REF] Chavarriga | A survey of isochronous centers[END_REF], theorem 3.3, p.12). A main property of the correction is that it gives a very useful criterion for linearisability. Indeed, a vector eld is linearisable if and only if its correction is zero. As the correction possesses an algorithmic and explicit form which is easily calculable using mould calculus we are able to give more informations on the isochronous set. This strategy was already used by one of us in [START_REF] Cresson | Obstruction à la linéarisation des champs de vecteurs polynomiaux[END_REF].

In the following, we use the classical complex representation of real vector elds (see [START_REF] Llibre | Isochronicity and linearizability of planar polynomial Hamiltonian systems[END_REF]).

Let us denote by X lin = i(x∂ x -x∂ x) and X r = P r (x, x)∂ x + P r (x, x)∂ x with x ∈ C, P r is a homogeneous polynomial of degree r, P r (x, x) = r j=0 p r-j-1,j x r-j xj . We formulate our main results (Theorem 1 to Theorem 4) whose proofs are postpone in Section 4.

Theorem 1. Let X be a non trivial real Hamiltonian vector eld of even degree 2n given by:

X = X lin + 2n r=2 X r
If X satises one of the following conditions : a) there exists 1 ≤ r < n -1 such that p i,i = 0 for i = 1, ..., r -1 and Im(p r,r ) > 0, b) p i,i = 0 for i = 1, ..., n -1, then the vector eld is nonisochronous.

As a consequence we deduce that:

X = X lin + X 2 , X = X lin + X 2 + X 3 + X 4 with Im(p 1,1 ) ≥ 0, X = X lin + X 2 + X 3 + X 4 + X 5 + X 6 with p 1,1 > 0 or p 1,1 = 0 and Im(p 2,2 ) > 0, are nonisochronous.
As a corollary, we obtain the weak version of the Jarque-Villadelprat conjecture proved by X.

Chen and al. [3]:

Corollary 1 (weak Jarque-Villadelprat conjecture). Let X be a non trivial real Hamiltonian vector eld of even degree 2n given by

X = X lin + X 2 + X 4 + • • • + X 2n ,
then X is nonisochrnous.

The proof follows easily from Theorem 1 as for all i = 1, . . . , n -1, we have p i,i = 0 due to the fact that there exists no odd components. Theorem 2. Let X be a non trivial real Hamiltonian vector eld of the form:

X = X lin + X k + ... + X 2l , for k ≥ 2, 2l ≥ k and l ≤ k -1. Then X is nonisochronous.
Using this last theorem, without any conditions we have that:

X = X lin + X 2 , X = X lin + X 3 + X 4 , X = X lin + X 4 + X 5 + X 6 ,
or more funny X = X lin + 92 i=47 X i are nonisochronous. We see that Theorem 1 and Theorem 2 are complementary to each other.

Mixing the proofs of Theorem 1 and Theorem 2 we obtain: Theorem 3. Let X be a non trivial real polynomial Hamiltonian vector eld on the form:

X = X lin + X k + ... + X 2l + m n=1 2(cn-1) j=cn X j
where k ≥ 2, 2l ≥ k, l ≤ k -1, m ≥ 1 and the sequence c n is dened by :

c 1 = 4l and ∀n ≥ 2, c n = 4(c n-1 -1), (2)
Then X is nonisochronous.

A rst example of nonisochronous vector eld given by the last theorem is:

X = X lin + X 2 + X 4 + X 5 + X 6 .
Theorem 4. Let k ≥ 2 and l ≤ k -1, a real polynomial Hamiltonian vector eld denoted by X on one of these two forms:

i)X = X lin + X k + ... + X 2l + X 2l+1 + r+n m=r X m where r ≥ 2l + 2 and Im(p l,l ) > 0 or ii)X = X lin X k + ... + X 2l + X 4l-1 + r+n m=r X r
where X 2l is nontrivial, r ≥ 4l, with Im(p 2l-1,2l-1 ) > 0, are nonisochronous.

Using Theorem 3 and Theorem 4 we easily deduce the classical result that homogeneous perturbations of a linear center are non isochronous (see [START_REF] Schuman | Sur la forme normale de Birkho et les centres isochrones[END_REF]) as well its generalization (see [START_REF] Schuman | Une classe d'hamiltoniens polynomiaux isochrones[END_REF][START_REF] Cresson | Obstruction à la linéarisation des champs de vecteurs polynomiaux[END_REF]).

Plan of the paper

In Section 2, we give following J. Ecalle and B. Vallet [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF] the denition of the correction of a vector eld and remind some of its properties. We then look more specically to the correction of polynomial real vector elds. We derive explicit formula allowing us to analyse its structure.

In Section 3, we prove that the set of isochronous Hamiltonian centers is an ane variety which can be explicitly described. We also prove that this variety is invariant under a non trivial torus action.

In Section 4, we give the proofs of our main results and some technical Lemmas.

We then discuss several perspectives for this work.

Correction of vector elds and Hamiltonian systems

Correction of a vector eld

In this section, we remind the denition of the correction of a vector eld following the work of J. Ecalle and B. Vallet [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF]. In particular, we give the mould expansion of the correction, which plays a central role in our approach to study the linearisability. It must be noted that all these computations can be made in arbitrary dimension.

The correction of a vector eld

We denote by X an analytic vector eld on C ν at 0:

X = 1≤j≤ν X j (x)∂ xj
with X j (0) = 0 and X j (x) ∈ C{x}. We can write the vector eld X in its prepared form: Denition 1. A vector eld X is said to be in prepared form if it is given by

X = X lin + n∈A(X) B n (3)
where X lin is the linear part of X in diagonal form

X lin = ν j=1 λ j x j ∂ xj , (4)
the B n are homogeneous dierential operators of degree n = (n 1 , . . . , n ν ) ∈ Z ν , where all n i are integer except one which can be -1, satisfying for all monomial x m = x m1 1 . . . x mν m , where

x = (x 1 , . . . , x ν ) ∈ C ν , m = (m 1 , . . . , m ν ) ∈ N ν , the equality B n (x m ) = β n,m x n+m , β n,m ∈ C ν , (5)
and A(X) ∈ Z ν is the set of the degree of the operators B n in the decomposition of X.

From the point of view of Analysis, homogeneous dierential operators are more tractable. An operator B n is said to be homogeneous of degree n = (n 1 , n 2 ) if for all monomial x l y k we have

B n (x l y k ) = β l,k n • x n1+l y n2+k with β l,k n ∈ C.
In [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], J.Ecalle and B.Vallet introduce the correction of a vector eld following previous works of G.Gallavotti [START_REF] Gallavotti | wick ordering" of the perturbations in classical mechanics and invariance of the frequency spectrum[END_REF] and H.Eliasson [START_REF] Eliasson | Hamiltonian systems with linear normal form near an invariant torus[END_REF] in the Hamiltonian case.

Let us consider a vector eld in prepared form. The correction is dened as follows( [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], p.258):

Denition 2. Let X be a vector eld in prepared form with linear part X lin . The correction of X, denoted by Carr(X) is the vector eld solution of the following problem:

Find a vector eld Z such that

X -Z ∼ X lin , and [X lin , Z] = 0, (6)
where for two vector elds A and B, we denote A ∼ B if the vectors elds A and B are formally conjugate.

The second condition implies that Carr(X) is only made of resonant terms.

In [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], Ecalle and Vallet prove that the correction of a vector eld admits a mould expansion.

Precisely, let us denote by A * (X) the set of the words given by the letters in A(X) using the concatenation morphism conc on letters :

conc : A(X) p → A * (X) (n 1 , ..., n p ) → n 1 • n 2 • ... • n p .
for any integer p.

In the following, a word is denoted by

n 1 • n 2 • ... • n p or n 1 n 2 ...n p .
Remark 1. The length of the word n 1 ...n p is p. The word of length 0 is denoted by ∅.

Denition 3. The set A * (X) is composed by all the words of all lengths that is, if n ∈ A * (X) there exists an integer p ≥ 0 such that n = conc(n 1 , ..., n p ) where n j ∈ A(X) for j = 1, ..., p. We denote A p (X) the set of words of length p.

For all n = n 1 • ... • n r ∈ A * (X), we denote:

B n = B n1 • ... • B nr . (7)
The correction can be written as ( [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], Lemma 3.2 p.267):

Carr(X) = n∈A * (X) Carr n B n , (8)
where Carr n ∈ C for all n ∈ A * (X), or simply

Carr(X) = • Carr • B • (9)
following Ecalle's notations (see [START_REF] Ecalle | Les algèbres de fonctions résurgentes[END_REF] and [START_REF] Cresson | Calcul Moulien[END_REF]).

The mapping

Carr • : A(X) * -→ C, n -→ Carr n , (10)
is called a mould (see [START_REF] Ecalle | Les algèbres de fonctions résurgentes[END_REF][START_REF] Ecalle | L'équation du pont et la classication analytique des objets locaux[END_REF] and [START_REF] Cresson | Calcul Moulien[END_REF]).

The main point is that the mould Carr • can be computed algorithmically using a recursive formula on the length of words.

Precisely for all n ∈ A(X), let us denote by ω(n) the quantity:

ω(n) = n, λ ,
where the ., . is the usual scalar product on C n and λ = (λ 1 , . . . , λ ν ) are the eigenvalues of X lin .

We can extend ω to a morphism from (A * (X), conc) to (C, +), i.e. for all n = n 1 .n 2 . . . n r ∈ A * (X),

n i ∈ A(X), i = 1, . . . , r, we have ω(n) = ω(n 1 ) + • • • + ω(n r ). (11)
The quantity ω(n) is called the weight of the letter n.

We have the following theorem (formula 3.42 in [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF]):

Theorem 5 (Variance formula). The mould of the correction is given by the formula for any word

n = n 1 • ... • n r : ω(n 1 )Carr n1,n2,...,nr + Carr n1+n2,n3,...nr = n1bc=n Carr n1c Carr b . ( 12
)
The proof of this theorem is nontrivial. It follows from the variance formula for a vector eld discussed in ([14], Prop 3.1 p.270). The variance of a vector eld gives many dierent way to compute the mould of the correction.

The main consequence of the previous Theorem is the universal character of the mould of the correction. Precisely, following the denition of universality used in [START_REF] Cresson | Calcul Moulien[END_REF], we have :

Theorem 6 (Universality of the Correction's mould). There exists a one parameter family of complex functions

C r : D r ⊂ C r → C, r ∈ N such that for all X the correction's mould Carr • dened on A(X) * is given for all n ∈ A(X) * such that l(n) = r, r ∈ N by Carr n =    C r (ω(n 1 ), . . . , ω(n r )), if ω(n) = 0, 0 otherwise. ( 13
)
This property is fundamental concerning our problem as the computation of these coecients is done once and for all and does not depends on the value of the coecients entering the polynomials but on the alphabet generated by the vector eld. Up to our knowledge only the mould formalism is able to produce such kind of coecients allowing to write the correction (this is not the case for example dealing with the classical Lie framework).

As for prenormal forms (see [START_REF] Ecalle | Prenormalization, correction, and linearization of resonant vector elds or dieomorphisms[END_REF]), a specic role is played by resonant terms: Denition 4. A word n ∈ A * (X) is said to be resonant if ω(n) = 0.

In the following, we give explicit expressions for C 1 , C 2 and C 3 .

The mould of the correction

The following theorem concerns precisely the length 1,2 and 3 :

Theorem 7. The universal correction functions C r : C r → C of Theorem 6 are given for r = 1, 2, 3 by

C 1 (x) =    1 if z 1 = 0, 0 otherwise. C 2 (z 1 , z 2 ) =    - 1 z 1 if z 1 + z 2 = 0, z 1 = 0, 0 otherwise. ( 14
) C 3 (z 1 , z 2 , z 3 ) =      1 z 1 (z 1 + z 2 ) , if z 1 + z 2 + z 3 = 0, z 1 = 0, z 1 + z 2 = 0, 0 otherwise. ( 15
)
The proof is based on explicit computations which are summarized in the following Lemmas whose proof are given in Appendix.

We can remark that the values of the correction's mould depend on the weight of the letters.

Lemma 1. The mould Carr • veries :

1) Carr ∅ =0, 2)If n is non resonant, Carr n = 0, 3)If n = n 1 .
..n r is such that there exists j satisfying ω(n j ) = 0 then Carr n = 0.

Moreover, we have:

Lemma 2. 1) If ω(n) = 0, Carr n = 1, 2) If ω(n 1 • n 2 ) = 0 with ω(n 1 ) = -ω(n 2 ) = 0, we have Carr n1•n2 = -1 ω(n1) , 3) If ω(n 1 • n 2 • n 3 ) = 0 with ω(n j ) = 0 j = 1, 2, 3, we have Carr n1•n2•n3 = 1 ω(n1)(ω(n1)+ω(n2)) .
We refer to Appendix B.2 and Appendix B.3 for the proofs.

Some computations of the correction mould

Let us consider the quadratic case, i.e.

X = X lin + X 2 , (16)
where X lin is diagonal with eigenvalues (i, -i) and X 2 is a homogeneous polynomial vector eld of degree 2 given by

X 2 = (p 1,0 x 2 + p 0,1 xy + p-1, 2y 2 )∂ x + (q 2,-1 x 2 + q 0,1 xy + q 1,0 )∂ y . ( 17
)
We have: Lemma 3. The alphabet generated by X 2 is given by

A(X) = {n 1 = (1, 0), n -1 = (0, 1), n 3 = (2, -1), n -3 = (-1, 2)} . (18)
Proof. The alphabet generated by X 2 is related to the decomposition of X 2 in homogeneous dierential operators. A simple rearrangement of terms proves that

X 2 = B (1,0) + B (0,1) + B (-1,2) + B (2,-1) , (19)
where

B (1,0) = x(p 1,0 x∂ x + q 1,0 y∂ y ), B (0,1) = y(p 0,1 x∂ x + q 0,1 y∂ y ), B (-1,2) = p -1,2 y 2 ∂ x , B (2,-1) = q 2,-1 x 2 ∂ y . (20)
This concludes the proof.

All the letters in A(X) are non resonant. Indeed, we have

ω(n 1 ) = i, ω(n -1 ) = -i, ω(n 3 ) = 3i, ω(n -3 ) = -3i. ( 21
)
As a consequence, the correction mould is always zero in length 1.

In length 2 however, some resonant combinations are possible. We have Word n Carr n

n 1 • n -1 i n -1 • n 1 -i n 3 • n -3 i 3 n -3 • n 3 -i 3
200
In length 4, the correction mould is given by : Word n

Carr n

n -3 • n -3 • n 3 • n 3 -i 54 n -3 • n -1 • n 1 • n 3 -i 12 n -3 • n -1 • n 3 • n 1 i 12 n -3 • n 1 • n -1 • n 3 i 6 n -3 • n 1 • n 1 • n 1 i 6 n -3 • n 1 • n 3 • n -1 -i 6 n -3 • n 3 • n -3 • n 3 i 27 n -3 • n 3 • n -1 • n 1 0 n -3 • n 3 • n 1 • n -1 0 n -3 • n 3 • n 3 • n -3 0 n -1 • n -3 • n 1 • n 3 i 12 n -1 • n -3 • n 3 • n 1 -i 12 n -1 • n -1 • n -1 • n 3 i 6 n -1 • n -1 • n 1 • n 1 -i 2 n -1 • n -1 • n 3 • n -1 -i 2 n -1 • n 1 • n -3 • n 3 0 n -1 • n 1 • n -1 • n 1 i n -1 • n 1 • n 1 • n -1 0 n -1 • n 1 • n 3 • n -3 0 n -1 • n 3 • n -3 • n 1 -i 6 n -1 • n 3 • n -1 • n -1 i 2 n -1 • n 3 • n 1 • n -3 i 6
Word n

Carr n

n 1 • n -3 • n -1 • n 3 -i 6 n 1 • n -3 • n 1 • n 1 -i 2 n 1 • n -3 • n 3 • n -1 i 6 n 1 • n -1 • n -3 • n 3 0 n 1 • n -1 • n -1 • n 1 0 n 1 • n -1 • n 1 • n -1 -i n 1 • n -1 • n 3 • n -3 0 n 1 • n 1 • n -3 • n 1 i 2 n 1 • n 1 • n -1 • n -1 i 2 n 1 • n 1 • n 1 • n -3 -i 6 n 1 • n 3 • n -3 • n -1 i 12 n 1 • n 3 • n -1 • n -3 -i 12 n 3 • n -3 • n -3 • n 3 0 n 3 • n -3 • n -1 • n 1 0 n 3 • n -3 • n 1 • n -1 0 n 3 • n -3 • n 3 • n -3 -i 27 n 3 • n -1 • n -3 • n 1 i 6 n 3 • n -1 • n -1 • n -1 -i 6 n 3 • n -1 • n 1 • n -3 -i 6 n 3 • n 1 • n -3 • n -1 -i 12 n 3 • n 1 • n -1 • n -3 i 12 n 3 • n 3 • n -3 • n -3 i 54 2.2.
X = X lin + l r=2 X r , (22)
with

X r = P r (x, y)∂ x + Q r (x, y)∂ y , (23)
where P r and Q r are homogeneous polynomials of degree r such that:

P r (x, y) = r k=0 p k-1,r-k x k y r-k , Q r (x, y) = r l=0 q r-k,k-1 x r-k y k . ( 24
)
In the following, we describe explicitly the prepared form of X, the set A(X) and the operators

B n .
Lemma 4. For all r ≥ 2, we can associate an alphabet to the vector eld X r , written A(X r ) given by:

A(X r ) = {(r, -1), (-1, r), (k -1, r -k) with k = 1, ..., r}.
Moreover, we get from A(X r ) the set denoted B(X r ) of the homogeneous dierential operators given by the decomposition :

B (-1,r) = p -1,r y r ∂ x , B (r,-1) = q r,-1 x r ∂ y , B (k-1,r-k) = x k-1 y r-k (p k-1,r-k x∂ x + q k-1,r-k y∂ y )
with k ∈ {1, ..., r}.

210

Proof. We rst rewrite any vector eld X r , r = 2, . . . , l, in order to make apparent the order, in term of dierential operators, of all its components. We have :

P r (x, y)∂ x + Q r (x, y)∂ y = r k=1 p k-1,r-k x k-1 y r-k x∂ x + q r-k,k-1 x r-k y k-1 y∂ y + p -1,r y r ∂ x + q r,-1 x r ∂ y , = r k=1 (O (k-1,r-k) + Õ(r-k,k-1) ) + O (-1,r) + O (r,-1) , with O (k-1,r-k) = p k-1,r-k x k-1 y r-k x∂ x , Õ(r-k,k-1) = q r-k,k-1 x r-k y k-1 y∂ y , O (-1,r) = p -1,r y r ∂ x , O (r,-1) = q r,-1 x r ∂ y .
We want to know if there exists some operators of the same degree among the operators O (k-1,r-k) and Õ(r-k,k-1) . If it is the case, we gather them in a same operator of the form

B (n 1 ,n 2 ) = x n 1 y n 2 (p (n 1 ,n 2 ) x∂ x + q (n 1 ,n 2 ) y∂ y ).
For that, we have to solve the equation

k -1 = r -k (25)
where k, k ∈ {1, ..., r}. As k ∈ {1, ..., r}, r -k ∈ {0, ..., r -1} hence r -k + 1 ∈ {1, ..., r}. So there always exists solutions. This concludes the proof.

Example 1. We consider the vector eld X = X lin + X 2 + X 3 where

X 2 = p 1,0 x 2 + p 0,1 xy + p -1,2 y 2 ∂ x + q -1,2 x 2 + q 1,0 xy + q 0,1 yr ∂ y , X 3 = p 2,0 x 3 + p 1,1 x 2 y + p 0,2 xy 2 + p -1,3 y 3 ∂ x + q 3,-1 x 3 + q 2,0 x 2 y + q 1,1 xy 2 + q 0,2 y 3 ∂ y .
Hence we obtain the three following alphabets :

A(X 2 ) = {(2, -1), (1, 0), (0, 1), (-1, 2)}, A(X 3 ) = {(3, - 1
), (2, 0), (1, 1), (0, 2), (-1, 3)}.

For example, we also have the elements of the set B(X 3 ) :

B (3,-1) = q 3,-1 x 3 ∂ y , B (2,0) = x 2 (p 2,0 x∂ x + q 2,0 y∂ y ) , B (1,1) = xy (xp 1,1 x∂ x + q 1,1 y∂ y ) , B (0,2) = y 2 (p 0,2 x∂ x + q 0,2 y∂ y ) , B (-1,3) = p -1,3 y 3 ∂ x .
Denition 5. We dene the degree of a vector elds as the maximum of the degree of its dening polynomial.

In a same way, we dene the degree of the homogeneous dierential operators B n or of a Lie bracket of B n which appear in the decomposition of X. We denote by

deg(B n ) (resp. deg([B n ])) the degree of B n (resp. [B n ]).
Lemma 5. Let X be a vector elds of the form X = X lin + m r=2 X r then X admits the alphabet

A(X) = m ∪ r=2
A(X r ) and B(X), the set of homogeneous dierential operators of X, is given by

B(X) = m ∪ r=2 B(X r ).
Proof. For all n = (n 1 , n 2 ) ∈ A(X), we dene the application :

p : A(X) → N n = (n 1 , n 2 ) → n 1 + n 2 .
For every r ≥ 2, for all n ∈ A(X r ), we have p(n) = r -1, so ∀r, r , such that r = r , we have

A(X r ) ∩ A(X r ) = ∅ because p(A(X r )) = p(A(X r )). Moreover, as Der(C 2 ) = r≥1 Der r (C 2 ) and B(X r ) ⊂ Der r (C 2 ) , then B(X r ) ∩ B(X r ) = ∅ if r = r .
The elements of the alphabet A(X) are named letter .

Depth 230

As we have a one-to-one correspondence between (A * (X), conc) and (B * (X), •):

A * (X) → B * (X), n = n 1 • ... • n r → B n = B n1 • ... • B nr ,
the degree of [B n ] gives a natural notion of depth for the words dened by: Denition 6. We denote by p : A * (X) → N the mapping dened by:

p(n) = deg([B n ]) -1.
Lemma 6. The mapping p is a morphism from (A * (X), conc) in (N, +).

Proof. We prove it by induction on the length of the words. Let

n 1 , n 2 ∈ A(X), p(n 1 • n 2 ) = deg([B n1 B n2 ]) -1 = deg(B n1 ) + deg(B n2 ) -1 -1 = p(n 1 ) + p(n 2 ).
Let n 1 ∈ A(X) and n ∈ A * (X), so:

p(n 1 • n) = deg([B n1 , B n ]) -1 = deg(B n1 ) + deg(B n ) -1 -1 = p(n 1 ) + p(n).

The projection Theorem and rewriting of a mould expansion

Let M(X) be a mould series, i.e. an expression of the form 235

M(X) = n∈A * (X) M n B n , (26)
where M • is an alternal mould ( [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF]), i.e. M(X) is primitive( [START_REF] Serre | Lie algebras and Lie groups[END_REF], p.17) and then a vector eld. In this case, using the projection theorem ( [START_REF] Serre | Lie algebras and Lie groups[END_REF], p.28), the mould series M(X) can be expressed in the following form :

M(X) = r≥1 1 r n∈A * (X) l(n)=r M n [B n ], (27)
where

[B n ] = [B n1...nr ] = [...[[B n1 , B n2 ], B n3 ], ...], B nr-1 ], B nr ]. (28)
We can reorganize this sum using the depth as follows:

M(X) = d≥1 M d (X), (29)
where

M d (X) = n∈A * (X) p(n)=d M n B n . (30)
A useful consequence is that the equation M(X) = 0 is equivalent to M d (X) = 0 for all d ≥ 1.

Expression of the correction and criterion of linearisability

The main property of the correction is that it provides a useful and simple criterion of linearizability. Indeed, we have by denition of the correction (see [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], p.258) : Lemma 7. A vector eld X is linearizable if and only if Carr(X) = 0.

Using the above decomposition, we obtain an explicit criterion for linearizability writing Carr(X) as :

Carr(X) = p≥1      n∈A * (X) p(n)=p Carr n B n      = p≥1
Carr p (X).

Theorem 8. A vector elds X is linearizable if and only if Carr p (X) = 0 ∀p ≥ 1.

In the following, we derive some properties of the quantities Carr p (X).

Correction of real polynomial Hamiltonian vector elds 2.3.1. General properties

An interesting property of the correction is that we just have to consider the even depth, indeed:

Theorem 9. Let X be a real Hamiltonian vector elds as above. Its correction in odd depth is zero, i.e.

Carr 2p+1 (X) = 0,

for all integer p.

This theorem is a consequence of the following lemma :

Lemma 8. For a resonant word n, the related Lie bracket is on the form :

[B n ] = (xy) p(n) 2 (P n x∂ x + Q n y∂ y ).
Proof. For all resonant word n = n 1 • ... • n r , the related Lie bracket is :

[B n ] = x n 1 j y n 2 j (P n x∂ x + Q n y∂ y ).
where n j = (n 1 j , n 2 j ). As n is resonant, we have :

ω(n) = ω(n 1 ) + ... + ω(n r ) = i n 1 j - n 2 j = 0, then n 1 j = n 2 j = α ∈ N. We just have to remark that p(n) = p(n 1 ) + ... + p(n r ) = n 1 j + n 2 j = 2α, then α = p(n) 2 .
As a consequence, we can restrict our attention to the even components of the correction. For a given integer p, terms in Carr 2p can be decomposed with respect to the length of words. Precisely, we have

Carr p (X) = Carr p,1 (X) + • • • + Carr p,p (X), (32)
where Carr p,j (X) = n∈A * (X)

p(n)=p, l(n)=j Carr n B n , (33)
for j = 1, . . . , p.

The main point is that of course, this is a nite sum. Indeed, as each dierential operator entering in the denition are at least of depth one, we can not have more than a word of length p as for all j ∈ N * , n ∈ A * (X) such that l(n) = j, we have p(n) ≥ j.

Moreover, some of these terms are easily determined.

Lemma 9. Let p ∈ N * , we have

Carr 2p,2p (X) = Carr 2p,2p (X 2 ), Carr 2p,2 (X) = Carr 2p,2 (X p+1 ) + p r=2 Carr 2p,2 (X r , X 2p-r+2 ), Carr 2p,1 (X) = Carr 2p (X 2p+1 ) = B p,p . (34)
This Lemma has important implications on the following, as it gives the maximal degree of the homogeneous vector elds X r entering in the computation of a given correction term. In particular, for Carr 2p we have no terms coming from the X r with r ≥ 2p + 2.

The proof of Lemma 9 is based on the properties of the set of resonant words with respects to length and depth.

Proof. i) The rst equality easily follows from the fact that an element of depth 2p and length 2p

is necessarily made of elements of depth 1 corresponding to operators in X 2 .

ii) The second equality is a direct rewriting of the denition for a length 2 contribution to the correction term of depth 2p. We denoted by W(X r ) the set of weights coming from the component X r of X and given by :

W(X r ) = { n, λ , n ∈ A(X r )}. (35)
The rst term comes from the following decomposition lemma:

Lemma 10. For every r ≥ 2, W(X r ) can be decomposed in the following way :

W(X r ) = W + (X r ) ∪ W -(X r ) ∪ W 0 (X r ),
where

W + (X r ) is the set of positive weights coming from X r , W -(X r ) = -W + (X r ) and W 0 r (X)
is the set of the zero weight.

This decomposition shows the interaction between each homogeneous component X l intervening in Carr 2p,2 (X). In length 2, as any weight as its symmetric counterpart, we always have a contribution of X p+1 .

iii) The third equality follows directly from a computation: Lemma 11. A component X r , r ≥ 2, produces a resonant letter in A(X) if and only if r is odd. In this case, the letter is unique and given by n 0 = ( r-1 2 , r-1 2 ).

Proof. Let r ≥ 2 be xed. By Lemma 4 we know the set of letters produced by X r . The two letters (-1, r) and (r, -1) are never resonant. For the other ones given by (k -1, r -k), one must solve the equation of resonance

2k -r -1 = 0, (36)
for k = 1, ..., r. This equation has a unique solution given by

k = r + 1 2 , (37)
which is valid, as k must be an integer, only when r is odd.

This concludes the proof of Lemma 9.

Explicit computation and the fundamental Lemma

We now explicit the quantities Carr p (X) when X is a real Hamiltonian polynomial vector eld.

Let X a polynomial vector eld in C 2 of the form:

X = X lin + d r=2 (P r (x, y)∂ x + Q r (x, y)∂ y ) , (38)
where P r and Q r are homogeneous polynomials of degree r.

Lemma 12. The complex vector eld (38) corresponds to a real vector eld if for all r = 2, . . . , d, we have p i,j = q j,i , i = 0, . . . , r -1, j = r -i.

(

) 39
The proof follows easily from the fact that x = y and Q r (x, y) = P r (x, y) which gives Q r (x, y) = P r (y, x) for all r = 2, . . . , d.

Real Hamiltonian systems satisfy moreover the following conditions:

Lemma 13. The complex vector eld (38) corresponds to a real Hamiltonian vector elds if conditions (39) are satised and moreover if for all r = 2, . . . , d, we have

p i-1,r-i = - r -i + 1 i p r-i,i-1 , i = 1, . . . , r. (40)
We give some examples of relations between the coecients in X 2 and X 3 :

Example 2. For the vector eld X 2 dened above, we have :

p 1,0 = -1 2 p0,1 , p -1,2 = q2,-1 .
For the vector eld X 3 , we have :

p 2,0 = -1 3 p0,2 , p 1,1 = -p 1,1 , p -1,3 = q3,-1 .
Under the two previous conditions on the coecients we have:

Lemma 14 (Fundamental Lemma). Let X be a real Hamiltonian vector elds of the form X = i(x∂ x -y∂ y ) + 2r-1 j=r (P j (x, y)∂ x + P j (x, y)∂ y ) with x = y, then :

Carr 2(r-1) (X) = p r-1,r-1 + i   r k=[ r+1 2 ]+1 r(r + 1) (r -k + 1) 2 |p k-1,r-k | 2 + r r + 1 |p -1,r | 2   .
Proof. The computation of the correction in depth 2(k -1) requires to know which operators appear in it and the alphabet. In this depth, there are just the homogeneous dierential operators from the polynomial P r and the resonant letter from the polynomial P 2r-1 .

The alphabet given by P r is A(X r ) = {(r, -1), (-1, r), (k -1, r -k) with k = 1, ..., r}.

We also have the homogeneous dierential operators :

B (-1,r) = p -1,r y r ∂ x , B (r,-1) = q r,-1 x r ∂ y , B (k-1,r-k) = x k-1 y r-k (p k-1,r-k x∂ x + q k-1,r-k y∂ y )
with k ∈ {1, ..., r}.

Here the resonant words are :

((k -1, r -k), (r -k, k -1)), ((r -k, k -1), (k -1, r -k)),
((-1, r), (r -1)), ((r, -1), (-1, r)).

So the Lie brackets associated, using the real and Hamiltonian conditions, are:

[B (k-1,r-k) , B (r-k,k-1) ] = r(r + 1)(2k -r -1) (r -k + 1) 2 |p k-1,r-k | 2 (xy) r-1 (x∂ x -y∂ y ) = -[B (r-k,k-1) , B (k-1,r-k) ], [B (-1,r) , B (r,-1) ] = -r|p -1,r | 2 (xy) r-1 (x∂ x -y∂ y ) = -[B (r,-1) , B (-1,r) ].
The mould of the correction in this resonant word are:

Carr (k-1,r-k)•(r-k,k-1) = i 2k -r -1 , Carr (-1,r)•(r,-1) = -i r + 1
.

The only resonant letter in P 2k-1 is p r-1,r-1 with the operator B (r-1,r-1) = p r-1,r-1 (xy) r-1 (x∂ x -y∂ y ). Using the alternality of the mould Carr • and the skew-symmetric of the Lie brackets, we get the formula :

Carr 2(k-1) (X) = p r-1,r-1 + i   r k=[ r+1 2 ]+1 r(r + 1) (r -k + 1) 2 |p k-1,r-k | 2 + r r + 1 |p -1,r | 2   .

Examples of computations 315

We give examples of computations of the correction.

The quadratic case. We consider the real Hamiltonian vector eld :

X = X lin + X 2
We remind that the alphabet is given by A(X 2 ) = {(1, 0), (0, 1), (2, -1), (-1, 2)} (see Section 2.1.3) and is non resonant.

The rst none trivial correction term is then in depth 2. 320

The correction in depth 2 is given by:

Carr 2 (X) = Carr 2,2 (X) = n∈A * (X) p(n)=2 ω(n)=0 Carr n B n = n∈A * (X) p(n)=2 ω(n)=0 1 (n) Carr n B [n] = 1 2 Carr (1,0)•(0,1) B [(1,0)•(0,1)] + Carr (0,1)•(1,0) B [(0,1)•(1,0)] +Carr (2,-1)•(-1,2) B [(2,-1)•(-1,2)] + Carr (-1,2)•(2,-1) B [(-1,2)•(2,-1)]
.

By the Fundamental Lemma, we nally have :

Carr 2 (X) = i 6|p 1,0 | 2 + 2 3 |p -1,2 | 2 .
The cubic case. We consider the real Hamiltonian vector eld :

X = X lin + X 2 + X 3 with X 2 = (p 1,0 x 2 +p 0,1 xy +p-1, 2y 2 )∂ x +(q 2,-1 x 2 +q 0,1 xy +q 1,0 )∂ y and X 3 = (p 2,0 x 3 +p1, 1x 2 y + p 0,2 xy 2 + p -1,3 y 3 )∂ x + (q 3,-1 x 3 + q 2,0 x 2 y + q 1,1 xy 2 + q 0,2 y 3 )∂ y .
As above, the rst none trivial correction is in depth 2 :

Carr 2 (X) = Carr 2,1 (X 3 ) + Carr 2,2 (X 2 ).
The only operator which is of depth 2 from X 3 is given by its resonant letter (1, 1) and the operator B 1,1 = xy(p 1,1 x∂ x + q 1,1 y∂ y . So, using the previous result on the quadratic case, we have :

Carr 2 (X) = p 1,1 + i 6|p 1,0 | 2 + 2 3 |p -1,2 | 2 .
In depth 4, we have :

Carr 4 (X) = Carr 4,2 (X 3 , X 3 ) + Carr 4,3 (X 3 , X 2 , X 2 ) + +Carr 4,4 (X 2 , X 2 , X 2 , X 2 ).
Theorem 10 (Geometric structure). For all d ≥ 2, the set L d of isochronous centers is an ane variety over

Q in C N (d) .
The proof is based on a precise description of the algebraic form of the correction. For all n ∈ A * (X), let us denote by P (n) and Q(n) the coecients given by Lemma 8 and satisfying

[B n ] = (xy) p(n) 2 (P (n)x∂ x + Q(n)y∂ y ).
We have :

Theorem 11 (Algebraic structure). For all p ∈ N * the correction term Carr 2p (X) has the form

Carr 2p (X) = (xy) p Ca 2p (p)x∂ x + Ca 2p (p)y∂y , (41)
where

Ca 2p (p) = 2p i=1 1 i! Ca 2p,i (p), (42) with Ca 2p,i (p) :
= n∈A * (X) p(n)=2p, l(n)=i Carr n P (n), i = 1, . . . , 2p. (43)
The quantities Ca 2p,i (p), i = 1, . . . , 2p, are explicit polynomials of degree i in the coecients of P with coecients in Q if i is even and iQ otherwise. Moreover, these polynomials can be computed algorithmically using recursive formula.

The proof of this theorem is a consequence of two results. First, nested Lie brackets have a very special shapes which can be easily computed. Precisely, we have : Lemma 15. For all n ∈ A * (X), the coecients P (n) and Q(n) are polynomials in Z[C N (d) ] of degree l(n) and dened recursively on the length of n by

P (nn) = (| n | 1 -n 1 )p n P (n)+ | n | 2 q n P (n) -n 2 p n Q(n), Q(nn) = (| n | 2 -n 2 )q n Q(n)+ | n | 1 p n Q(n) -n 1 q n P (n). ( 44
)
where for n

= n 1 • ... • n r ∈ A * (X), we let |n| j = n j 1 + ... + n j r , j = 1, 2 with n i = (n 1 i , n 2 i ).
Second, the correction mould can also be computed by a recursive formula from which we deduce :

Lemma 16. For all n ∈ A * (X), the mould

Carr n belongs to Q if l(n) is odd and iQ if l(n)is even.
The proof of Theorem 11 easily follows.

Proof of Theorem 10

Using Theorem 11 and the characterization of isochronous centers given by Theorem 8, the set L d is dened by the zero set of an innite family of polynomials over C N (d) given by

L d = P ∈ C N (d) , Ca 2p (X) = 0, p ≥ 1 . (45)
We dene the ascending chain of ideals

I k generated by Ca 2 (p), . . . , Ca 2k (p) in C[p]
. By the Hilbert Basis Theorem (see [START_REF] Cox | Ideals, Varieties, and Algorithms, An introduction to computational algebraic geometry and commutative algebra[END_REF], Theorem 4, p.77), there exists an M (d) ∈ N * such that I M = I M +1 = . . . . We denote by I d the resulting ideal. As a consequence, the set L d can be obtained as (see [START_REF] Cox | Ideals, Varieties, and Algorithms, An introduction to computational algebraic geometry and commutative algebra[END_REF], Denition 8,p.81)

L d = V(I d ) = p ∈ C N (d) | f (p) = 0 for all f ∈ I d , (46)
and corresponds to the ane variety (see [START_REF] Cox | Ideals, Varieties, and Algorithms, An introduction to computational algebraic geometry and commutative algebra[END_REF], Proposition 9 p.81) dened by

L d = V(f 1 , . . . , f s(d) ) = p ∈ C N (d) | f i (p) = 0 for all i = 1, . . . , s(d) , (47)
where the nite family of polynomials f i , i = 1, . . . , s(d) is a generating set of I d . As the polynomials dening this variety have coecients in Q or iQ this concludes the proof.

Remark 2. Theorem 10 together with Theorem 11 gives explicit informations on the degree as well as on the growth of the rational coecients entering in the denition of the ane variety. A natural question is up to which extend these informations can be used to provide a natural upper bound on the number of generators for the ideal generating L d thanks to a constructive version of the Hilbert basis theorem. This will be explored in another work.

C * -invariance

The resonant character of the correction has an interesting consequence on the rational algebraic variety of isochronous center. Indeed, let us consider the following action of C * : Denition 7. Let λ ∈ C * , we denote by T λ the map

T λ : C N (d) → C N (d) p • → λ ω(•) p •
where • is an arbitrary letter.

We extend this action for all monomials p n = p n1 1 ...p nr r , with n is a word n 1 • ... • n r , we have:

T λ (p n ) = T λ (p n1 1 ...p nr r ), = λ ω(n1)+...+ω(nr) p n = λ ω(n) p n . X r and [X r , X r ] Depth X 3 2 [X 2 , X 2 ] 2 X 5 4 [X 4 , X 2 ] 4 [X 3 , X 3 ] 4 [X 3 , X 2 , X 2 ] 4 X 7 6 [X 6 , X 2 ] 6 [X 5 , X 3 ] 6 [X 4 , X 4 ] 6 [X 5 , X 2 , X 2 ] 6 [X 4 , X 3 , X 2 ] 6 [X 4 , X 2 , X 2 , X 2 ] 6 [X 3 , X 2 , X 2 , X 2 , X 2 ] 6 [X 2 , X 2 , X 2 , X 2 , X 2 , X 2 ] 6
The correction in depth 2 is given by : Carr 2 (X) = Carr 2,1 (X 3 ) + Carr 2,2 (X 2 ),

As the depth is a morphism we have the contribution of the Lie bracket of X 2 , we also have the contribution of the resonant letter of X 3 . By the Fundamental Lemma, the correction is given by

Carr 2 (X) = p 1,1 + i 6|p 1,0 | 2 + 2 3 |p -1,2 | .
By the linearisation criterion, we must have Carr 2 (X) = 0. This implies that 395

   Re(p 1,1 ) = 0, -Im(p 1,1 ) = 6|p 1,0 | 2 + 2 3 |p -1,2 |. ( 48
)
As X is real and Hamiltonian, the rst equation is always satised. The second one has only a non trivial solution if and only if Im(p 1,1 ) < 0. The situation when Im(p 1,1 ) ≥ 0 leads to two distinct cases. When Im(p 1,1 ) = 0, the Birkho sphere reduce to 0 and we obtain X 2 = 0. When Im(p 1,1 ) > 0, the equation can not be satised and the vector eld is then nonisochronous.

400

Assume that p 1,1 = 0 then X 2 = 0 and we are reduce to the case

X = X lin + X 3 + • • • + X r . (49)
As p 1,1 = 0, the rst non zero term of the correction is Carr 4 (X). By the fundamental Lemma, the term Carr 4 (X) has exactly the same algebraic structure than the preceding Carr 2 (X) = 0 case. The role of X 2 is played by X 3 and the role of the resonant term of X 3 is played by the resonant term of X 5 . Here again, we recover the same dichotomy between the case Im(p 2,2 ) = 0 and Im(p 2,2 ) > 0. In the rst case, we obtain that X 3 = 0 and we are leaded to the same situation as before. Otherwise if Im(p 2,2 ) > 0 the term Carr 4 (X) can not be zero and we have a nonisochronous center.

The preceding discussion is representative of the general strategy of proof. Let us assume that p j,j = 0 for j = 1, ..., r -1. Then, we prove by induction that X 2 = • • • = X r-1 = 0. In order to nish the proof, two cases must be discussed depending the value of r.

Case 1 : r < n -1. The component X 2r+1 is non trivial due to the condition Im(p r,r ) > 0.

By the Fundamental Lemma and the linearisability criterion, we must have

Carr 2r (X) = p r,r + i   r k=[ r+1 2 ]+1 r(r + 1) (r -k + 1) 2 |p k-1,r-k | 2 + r r -1 |p -1,r | 2   = 0 (50)
As Im(p r,r ) > 0, this equality can not be satised and X is nonisochronous.

Case 2 : r = n -1. In this case, we are reduced to an homogeneous perturbation of degree 2n and the correction is given by Carr

2n-2 (X) = r k=[ r+1 2 ]+1 r(r + 1) (r -k + 1) 2 |p k-1,r-k | 2 + r r -1 |p -1,r | 2 = 0
As X 2n is nontrivial, this equation can not be satised and X is nonisochrnous.

This concludes the proof of the Theorem.

Proof of Theorem 2

The proof follows the lines of those of Theorem 1. We have to distinguish two cases : k is even or odd.

Case 1: k is even. The vector eld X k does not contain resonant terms. As a consequence, its rst contribution to the correction appears in depth 2(k -1) corresponding to resonant Lie brackets of homogeneous dierential operators in X k of length two. As l ≤ k -1, this implies that 2l -1 < 2l ≤ 2(k -1). The even component between X k and X 2l will come into play in the correction only with a greater depth as 2(k -1) by Lie brackets of length at least two. In the same way, for odd components, the resonant term will intervene in the correction with a strictly smaller depth in length one and the other terms in depth greater than 2(k -1) by a Lie brackets of length at least two.

As a consequence, the correction term coming from an even k is given by :

Carr 2(k-1) (X) = i   r k=[ r+1 2 ]+1 r(r + 1) (r -k + 1) 2 |p k-1,r-k | 2 + r r -1 |p -1,r | 2   . (51)
In order to satisfy the linearisability criterion, we must have Carr 2(k-1) = 0. If the component X k is non trivial then the system is already non isochronous. Otherwise, we have X k = 0 and we are leaded to the same problem but with an odd component.

Case 2 : k is odd. In this case the vector eld X k contain a resonant homogeneous operator.

Let us write k = 2m + 1 then B m,m is of depth 2m and weight zero. We have

Carr k-1 (X) = Carr 2m (X) = B m,m . (52)
By the linearisability criterion, Carr k-1 (X) = 0 and the resonant term B m,m in X k is zero. The contribution of X k in length 2 follows the same argument as for the even case and we deduce that nally X k = 0.

As a consequence, we can prove by induction that in order to be linearisable the components X k , . . . , X 2l must be zero. But, by assumption, we have that X 2l is non trivial. As a consequence, the vector eld X is necessarily nonisochronous.

Proof of Theorem 3

The strategy of proof follows those of Theorem 2. The main observation is that there exists no interactions between each family of vector elds {X k , . . . , X 2l } and {X cn , . . . , X 2(cn-1) }, n = 1, . . . , m. Indeed, let us rst analyse the depth of all these objects. We have :

X r Depth X k k -1 X k+1 k . . . . . . X 2l-1 2l -2 X 2l 2l -1 0 0 X 4l 4l -1 . . . . . . X 8l-2 8l -3 ... ... X cm c m -1 . . . . . . X 2(cm-1) 2(c m -1) -1
Following the same lines as for Theorem 2, we see that the arguments based on the contributions of a given component belonging to {X k , . . . , X 2l } are valid. In other words, we easily proved that in order to be linearisable, then one must have X k = • • • = X 2l-2 = 0. The last argument concerning X 2l is also satises because the rst contribution of X 2l to the correction is of length two and depth 4l -2 which is not disturbed by terms of the remaining family {X cn , . . . , X 2(cn-1) }, n = 1, . . . , m as the minimal contribution of these terms to the correction is in depth 4l -1.

As a consequence, a vector elds of this type will be linearisable if

X k = • • • = X 2l = 0.
By the same argument, we see that there exists no interaction between the family {X c1 , . . . . . . , X 2(c1-1) } and the remaining one {X cn , . . . , X 2(cn-1) }, n = 2, . . . , m. We deduce that a linearisable vector eld of this type must satisfy

X c1 = • • • = X 2(c1-1)
.

By induction, we easily deduce that a vector eld of this type is linearisable if and only if all the components vanish. As by assumption we are considering a non trivial vector eld, we are leaded to a contradiction and the vector eld is necessarily nonisochronous.

Conclusion and perspectives

5.1. Toward a complete proof of the Jarque-Villadelprat conjecture

Our main results give a very strong support to the Jarque-Villadelprat conjecture. The remaining cases always deal with the role of the resonant term in the deformation of the Birkho 's spheres. However, the phenomenon which is working for a quartic perturbation, which was precisely studied by Jarque and Villadelprat using geometrical methods, must applies in the same way for arbitrary degrees. Indeed, as we have seem in our derivation of the main results, the algebraic structure of the correction during the cancellation process does not change and can be closely investigated. We have then some directions in order to solve completely the Jarque-Villadelprat conjecture :

• Can we prove using an extension of our method the quartic case ?

• Can we extend the geometrical method of Jarque-Villadelprat in the cases which are let open by our work ?

• Can we prove the remaining cases using other methods ?

We believe that a better understanding of the algebraic structure of the correction will be of importance in order to go further.

Eective Hilbert basis and the isochronous centers ane variety

A second aspect of our work is the explicit and algorithmic description of the isochronous center ane variety. As already said, we have informations about the growth of the degree and the coecients entering in the description of this variety. A natural question is then to look for eective version of the Hilbert basis theorem in order to get some informations about the minimal number of generators of the ideal. The isochronous centers seem to be more tractable than the usual center. However, it is clear that any advance in this direction will have consequences on the local 16th Hilbert problem. Indeed, the same kind of combinatoric and tools can be used to obtain analogous information for centers of polynomial vector elds (see [START_REF] Cresson | Formes normales et problème du centre[END_REF]).

Isochronicity for complex Hamiltonian systems

In [START_REF] Llibre | Isochronicity and linearizability of planar polynomial Hamiltonian systems[END_REF], the authors study isochronicity of complex Hamiltonian systems when the linear part has for spectrum (1, -1). Our method and results extend naturally to this case and give an explicit and algorithmic description of the isochronous centers ane variety. This will be the subject of a forthcoming work.

As above we will use the mould T ram • in length 2, Carr n1•n2 = T ram n1•n2 = ω(n2)-ω(n1) ω(n1)ω(n2) . As ω(n 1 ) = -ω(n 2 ), we nally have :

Carr n1•n2 = 1 ω(n 1 )
.

As we can see the result diers by a multiplication by -1. It is due to the fact in [START_REF] Ecalle | Prenormalization, correction, and linearization of resonant vector elds or dieomorphisms[END_REF] and [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF],

the nested Lie brackets are taken in this form Lemma 19. If n ∈ A(X) * is such that l(n) = 3, ω(n) = 0, we have the following possibilities:

1. If all the letters have zero weight, i.e. ω(n i ) = 0 for all i = 1, 2, 3, then Carr n = 0. 3. If all the letters have a non zero weight, i.e. ω(n i ) = 0, i = 1, 2, 3, then Carr n1•n2•n3 = 1 ω(n 1 )(ω(n 1 ) + ω(n 2 )) .

(B.17)

Proof. 1) This case follows from Lemma ??.

2) Let us take for example a = n 2 such that ω(n 2 ) = 0, then ω(n 1 n 3 ) = 0 and ω(n 3 ) = -ω(n 1 ).

As a consequence, using (B.16) and the fact that Carr n2 = 1, we obtain (B.20)

2 .

 2 If there exists a letter a ∈ {n 1 , n 2 , n 3 } such that ω(a) = 0 and ω(b) = 0 for b ∈ {n 1 , n 2 , n 3 } \ {a}, then Carr n = 0.

ω(n 1 )

 1 Carr n1•n2•n3 + Carr (n1+n2)•n3 = Carr n1•n3 + Carr n1 Carr n2•n3 . (B.18)Let us assume that n 1 is such that ω(n 1 ) = 0, then Carr n1 = 0 and the previous equality reduces toω(n 1 )Carr n1•n2•n3 = Carr n1•n3 -Carr (n1+n2)•n3 . (B.19)Using the expression of the mould Carr • in length 2, we haveCarr n1•n3 = -1/ω(n 1 ), Carr (n1+n2)•n3 = -1/ω(n 1 + n 2 ) = -1/ω(n 1 ).

  [B n1•...•nr ] = [B nr , [B nr-1 , [...[B n2 , B n1 ]..]] whereas in[START_REF] Cresson | About the Trimmed and the Poincaré-Dulac Normal Form of Dieomorphisms[END_REF] and[START_REF] Cresson | Calcul Moulien[END_REF] we consider [B n1•...•nr ] = [...[B n1 , B n2 ], ...], B nr-1 ], B nr ].And we have the relation:[B nr , [B nr-1 , [...[B n2 , B n1 ]..]] = (-1) r+1 [B n1 , B n2 ], ...], B nr-1 ], B nr ].Appendix B.3.3. The mould Carr • in length 3Using the variance formula we obtain for a word n = n 1 n 2 n 3 of length 3 the following equality:ω(n 1 )Carr n1•n2•n3 + Carr (n1+n2)•n3 = Carr n1•n3 Carr n2 + Carr n1 Carr n2•n3 .

(B.

[START_REF] Francoise | Les conditions du centre pour un champ de vecteurs quadratiques[END_REF]

One can prove the following lemma:

The dierent Carr i,j (X) are given by : Carr 4,2 (X 3 , X 3 ) = i(12|p 2,0

Carr 4,3 (X 3 , X 2 , X 2 ) = -i(120Im(p 2,0 p2 1,0 ) + 26 3

Im(p -1,3 p -1,2 p1,0 ) + 40Im(p 2,0 p -1,2 p 1,0 ))

|p -1,2 | 4 + 40Re(p -1,2 p 3 0,1 ) .

The quartic case. We consider the real Hamiltonian vector eld :

with X 2 and X 3 as above and X 4 = (p 3,0 x 4 + p 2,1 x 3 y + p 1,2 x 2 y 2 + p 0,3 xy 3 + p -1,4 y 4 )∂ x + (q 4,-1 x 4 + q 3,0 x 3 y + q 2,1 x 2 y 2 + q 1,2 xy 3 + q 0,3 y 4 )∂ y .

The correction in depth 2 is the same as the cubic case.

In depth 4, we have :

where Carr 4,2 (X 4 , X 2 ) is given by : Carr 4,2 (X 4 , X 2 ) = i (12Re(p 2,1 p1,0 ) + 8Re(p 3,0 p -1,2 )) .

Maple program. For the interested readers, we can send some Maple program to compute the correction of a polynomial vector elds. The program can be found in [START_REF] Palafox | Calcul moulien, arborication, symétries et applications[END_REF].

The isochronous center ane variety

In this part, we prove that the set of isochronous center is a rational ane variety which is invariant under a non trivial C * action. This ane variety is moreover explicitly described.

We also give estimates on the growth of the degree of each rational polynomials entering in this description as well as the growth of the rational coecients.

Ane variety of isochronous center

We consider real vector elds written in complex form as X = X lin + P (x, y)∂ x + Q(x, y)∂ y where P and Q are polynomials with coecients in C such that P (x, y) = Q(y, x). We denote by N (d) the number of independent coecients dening P and by p any element of this set. By the reality condition, the coecient of Q can be deduced from those of P . We then identify the set of complex polynomials of a given degree d with C Lemma 17. For all resonant word n, we have :

Proof. By denition of a resonant word, we have ω(n) = 0. So :

Finally we can generalise this lemma in the following corollary :

Corollary 2. For all λ ∈ C * , the algebraic variety L d of the isochronous centers is invariant under the action of T λ .

Proof. To prove this corollary we just have to remind that only the resonant word contribute to the linearisability. We can conclude by the above lemma.

4. Proof of the main results

Proof of Theorem 1

Let X be a real Hamiltonian vector eld of even degree 2n of the form :

For each X r , we can associate its depth as follows :

By Theorem 7, we are only interested by the even depth. As a consequence, we look for all possible combinations of arbitrary length which give rise under Lie bracket to an even depth vector eld.

In the following, we denote by [X k1 , . . . , X kr ] the set of operators that one obtain by nested Lie brackets of homogeneous dierential operators B ni coming from X ki , i = 1, . . . , r. As an example, we have :

Appendix A. Notations

• A(X) alphabet associated to a vector eld X;

• A * (X) set of words given by the alphabet A(X);

• B(X) set of homogeneous dierential operator associated to a vector elds X;

• Carr(X) the correction associated to a vector elds X;

• Carr • the mould of the correction;

• n element of A(X);

• n element of A * (X);

• B n element of B(X);

• B n element of (B(X), •);

• ω the weight application;

• p the depth application;

• ping the application which inverts the two components of a letter;

• ret the application on word which inverts the order of the letters;

Appendix B. Properties of the Correction's mould

The proofs of Lemma 1 and 2 use the denition of the mould of the correction using the variance formula [START_REF] Ecalle | The nilpotent and distinguished form of resonant vector elds or dieomorphisms[END_REF] and also an alternative one based on the knowledge of the mould of a given prenormal form that we remind in the next Section.

Appendix B.1. An alternative denition of the correction

We rst remind the composition of two moulds M • and N • (see [START_REF] Cresson | Calcul Moulien[END_REF], Denition III.44 p.347):

where * ω1•...•ω k means the sum on all the decomposition of the word n in k words. Moreover, w j is a letter obtained by the word ω j summing all its letter if the alphabet is provided with a law of semi-group.

The neutral element for the composition of moulds is denoted by I • and is dened by

To prove the dierent results about the correction, we use another equivalent denition of the mould of the correction related to the choice of a given prenormal form (see [START_REF] Ecalle | Correction an linearization of resonant vector elds and dieomorphisms[END_REF], p.267 Lemma 3.2) an

where n = n 1 • ... • n r and M • is a mould associated to a given prenormal form.

In the following, we often use the mould T ram • associated to the trimmed prenormal form (see [START_REF] Ecalle | Prenormalization, correction, and linearization of resonant vector elds or dieomorphisms[END_REF], [START_REF] Cresson | Calcul Moulien[END_REF] and [START_REF] Morin | Calcul moulien et théorie des formes normales classiques et renormalisés[END_REF]) and dened by

where Sam • is a mould dened as follows: Let P r be the family of Poincaré functions P = {P r } r∈N * dened by G. Morin in ( [START_REF] Morin | Calcul moulien et théorie des formes normales classiques et renormalisés[END_REF], §.7.9.1 p.114-115) by

and for r ≥ 2 by

where S r,i = (C * ) i-1 × {0} × (C * ) r-i .

The mould Sam • is dened by (see [START_REF] Morin | Calcul moulien et théorie des formes normales classiques et renormalisés[END_REF], Lemma 7.9.9., p.115):

Moreover we have :

where k ∈ N.

as I ∅ = M ∅ = 0, we deduce that Carr ∅ = 0.

2) Let n = n 1 • ... • n r be a non resonant word of length r, i.e. ω(n) = 0, then we have :

where M n = M n = 0 and I n = I n = 1 hence 1 -Carr n = 1 and Carr n = 0.

If the length of n is greater than 2, the variance formula gives:

By induction, we assume the result is true in length r -1. As n is non resonant, for any given decomposition in k sub-words n = w 1 . . . w k , there exists at least one of these sub-words which is non resonant, let says w j of length l(w j ) < l(n) such that I wj -Carr wj = 0 as long as l(w j ) ≥ 2.

If this sub-word w j is a letter, then the decomposition is made of at least two terms, i.e.

k ≥ 2, and all the remaining sub-words are either resonant or non resonant. and the quantity

However, the mould M • is zero over non resonant words as it corresponds to a prenormal form, so that -M • ) w1 •...•wj •...• w k = 0

and Carr n = 0.

3) We prove this result by induction on the length.

as n is resonant, n 2 is also resonant. So we have :

Now we assume that the result is satised for all words of length ≤ r -1, r ≥ 3. We con-

and at least one letter, let says n j is resonant.

We have to study the following equality,

There exists an integer l such that n j appears in the decomposition of one w l for 1 ≤ l ≤ k.

Either (w l ) = 1, so w l = n j and I w l -Carr w l = 1 -1 = 0, or (w l ) ≥ 2, so by the induction hypothesis Carr w l = 0 and I w l = 0 by denition. As a consequence, we obtain that Carr n = 0.

Let n a letter such that ω(n) = 0. By the above denition,

, where M • is a prenormal form. We can take for example the mould T ram • (see [START_REF] Cresson | Calcul Moulien[END_REF]), which is the mould of the Poincaré-Dulac normal form. So Carr n = T ram n = 1.

Appendix B.3.2. The mould

, ω(n) = 0, we have the following possibilities:

1. If all the letters have zero weight, i.e. ω(n i ) = 0 for all i = 1, 2, 3, then Carr n = 0.

2. If all the letters have a non zero weight then Carr n = 1/ω(n 1 ).

Proof. If ω(n 1 n 2 ) = 0, we have ω(n 1 ) = -ω(n 2 ) as ω is a morphism. Using the variance formula, we have :

As ω(n i ) = 0, we have Carr n1 = Carr n2 = 0 so that ω(n 1 )Carr n1•n2 + Carr n1+n2 = 0.

We then obtain ω(n 1 )Carr n1•n2 = -Carr n1+n2 = -1.

(B.14)

As a consequence, we have

.

We can also prove this result using the alternative denition :

We deduce easily that ω(n 1 )Carr n1•n2•n3 = 0,

and as a consequence Carr n1•n2•n3 = 0.

The same computations prove that Carr n = 0 if ω(n 3 ) = 0 and ω(n 1 ) = 0.

The remaining case, i.e. ω(n 1 ) = 0 and ω(n 2 ) = 0, can be deduced from the alternality of the mould Carr • which induces the following equality (see [START_REF] Cresson | Calcul Moulien[END_REF], §.4.1 p.331):

The second and third terms are zero thanks to our previous computations. As a consequence, we obtain Carr n1n2n3 = 0 if ω(n) = 0, ω(n 1 ) = 0 and ω(n 2 ) = 0.

3) From now on, let us assume that all the ω(n i ) are non zero. Then using formula (B.16) and the fact that Carr n1 = 0 and Carr n2 = 0, we obtain

Using the expression of Carr (n1+n2)•n3 we deduce that

As in length 2, we can give an alternative proof using the mould T ram • to compute the correction : P n,m = (m 1 -n 1 )p n p m + m 2 q n p m -n 2 p n q m , Q n,m = (m 2 -n 2 )q n q m + m 1 p n q m -n 1 q n p m .

We easily prove by induction that all the Lie bracket in any length are of the above form.


Benoît Delpont

Agnès Katell Beauvais

Jacquin Piques

Valérie Alavoine

Philippe Rault

Christelle Blanc-Labarre

G.-V Osseby

Marie Hervieu-Bègue

Maurice Giroud
email: maurice.giroud@chu-dijon.fr

Yannick Béjot

Agnès Jacquin-Piques

Guy-Victor Osseby

Clinical features of pain in amyotrophic lateral sclerosis: A clinical challenge

Keywords: Amyotrophic lateral sclerosis, Pain, Analgesic drugs, Quality of life

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

Introduction

Amyotrophic lateral sclerosis (ALS) was first described by Jean-Martin Charcot [START_REF] Couratier | Epidemiology of amyotrophic lateral sclerosis : a review of literature[END_REF], and is a rare but severe neurodegenerative disease, usually leading to death within 2 to 4 years [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Kiernan | Amyotrophic lateral sclerosis[END_REF]. Without treatment, involvement of the first-and second-order motor neurons induces palsy of the four limbs, dysphagia and dysarthria, with respiratory failure as the cause of the fatal outcome. Besides these motor issues, other, non-motor symptoms may be a major problem, such as cognitive decline, semantic memory involvement [START_REF] Hervieu-Bègue | Semantic memory assessment in 15 patients with amyotrophic lateral stenosis[END_REF], frontotemporal dementia [START_REF] Canosa | 18F-FDG-PET correlates of cognitive impairment in ALS[END_REF], malnutrition and pain, although the latter is often underestimated by practitioners compared with fatal motor and respiratory failure [START_REF] Handy | Pain in amyotrophic lateral sclerosis : a neglected aspect of disease[END_REF]. In addition, the treatment of pain in ALS is not well managed [START_REF] Andersen | EFNS guidelines on the clinical management of amyotrophic lateral sclerosis (MALS) revised report of an EFNS task force[END_REF] because its underlying mechanisms and assessment are still as yet unclear [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Brettschneider | Drug therapy for pain in amyotrophic lateral sclerosis or motor neuron disease[END_REF]. Yet, despite this, every practitioner in charge of patients with ALS is familiar with the negative effects of pain on quality of life for both patients and their caregivers [START_REF] Wallace | The evaluation of pain in amhyotrophic lateral sclerosis : a case controlled observational study[END_REF]. To provide a clinical tool for practitioners, the present article reviews what is currently known of the clinical characteristics of pain in ALS, its underlying mechanisms and different treatments according to the mechanisms involved.

Clinical features of pain in ALS

These features are multiple and differ among patients because they depend on the patient's age, duration of disease, and clinical form of the disease and its underlying primary or secondary mechanisms. These differences explain the variable prevalence of cases, ranging from 15% [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF] to 85% [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF][START_REF] Rivera | Prevalence and characteristics of pain in early and late stages of ALS[END_REF][START_REF] Stephens | Pain in amyotrophic lateral sclerosis : patient and physician perspectives and practices[END_REF]. Indeed, a common observation is that pain is less frequent in bulbar forms of ALS [START_REF] Stephens | Pain in amyotrophic lateral sclerosis : patient and physician perspectives and practices[END_REF][START_REF] Chio | Pain in amyotrophic lateral sclerosis : a population-based controlled study[END_REF].

Primary causes of pain

These are induced as a direct result of the neuropathological damage due to the disease.

Cramps and spasticity

Cramps are the most frequent type of pain in ALS, with rates between 25% [START_REF] Caress | Natural history of muscle cramps in amyotrophic lateral sclerosis[END_REF] to 63% [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF]. They usually affect the calf and other distal muscles, but also the abdomen, during the night. They are short-lasting and may be induced by movement and cold temperatures [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF], but they are also often experienced for several months prior to the onset of motor weakness [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF]. Cramps and fasciculations are due to hyperexcitability of motor units due to axonal sprouting [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF], and are usually associated with muscle denervation as a consequence of involvement of secondorder motor neurons at the level of the anterior horn of the spinal cord [START_REF] Caress | Natural history of muscle cramps in amyotrophic lateral sclerosis[END_REF], as shown by needle electromyography (EMG) tests, with rare giant motor unit potentials. Spasticity can produce paroxysmal, spontaneous, violent and hyperalgic contractures of hypertonic muscles, thereby suggesting involvement of first-order motor neurons. It is estimated to affect close to 6% of patients [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF].

Neuropathic pain

It is surprising to describe neuropathic pain in a disorder of the motor system that usually spares the somatosensory system [START_REF] Soriani | Care management in amyotrophic lateral sclerosis[END_REF]. In fact, only 9% of patients with ALS report neuropathic pain [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF]. It may be spontaneous, presenting as burning feet, tingling, paroxysmal shooting pain and/or numbness, or evoked, with allodynia, hyperalgesia and flashback algesia [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF]. These sensations may be localized to the distal limbs (hands, feet) and either focal (abdomen) or diffuse. This type of pain suggests involvement of the sensory tracts [START_REF] Treede | Neuropathic pain : redefinition and a grading system for clinical and research purposes[END_REF] affecting the posterior columns of the spine in ALS, as shown by familial cases [START_REF] Wakabayashi | Sporadic motor neuron disease with severe sensory neuronopathy[END_REF][START_REF] Isaacs | Amyotrophic lateral sclerosis with sensory neuropathy : part of a multisystem disorder ?[END_REF][START_REF] Pugdahl | Generalised sensory system abnormalities in amyotrophic lateral sclerosis : a European multicenter study[END_REF], sensory evoked potentials [START_REF] Iglesias | Electrophysiological and spinal imaging evidences for sensory dysfunction in amyotrophic lateral sclerosis[END_REF], magnetic resonance imaging (MRI) of the spinal cord [START_REF] Foerster | Multimodal MRI as a diagnostic biomarker for amyotrophie lateral sclerosis[END_REF] and some animal models [START_REF] Guo | Sensory involvement in the SOD1-G93A mouse model of amyotrophic lateral sclerosis[END_REF]. A recent study [START_REF] Isak | Involvement of distal sensory nerves in amyotrophic lateral sclerosis[END_REF] found that sensory nerve conduction was altered in 44% of ALS patients, and up to 50% of patients had small fiber neuropathy [START_REF] Weiss | Small-fiber neuropathy in patients with ALS[END_REF]. The use of skin biopsies may be useful in ALS patients with neuropathic pain to analyze their intraepidermal nerve fibers [START_REF] Lauria | Skin biopsy as a diagnostic tool in peripheral neuropathy[END_REF]. Yet, despite these data, it is still difficult to clarify the relationship between neuropathic pain in ALS and the lesions observed in the sensory system. However, clinical experience has demonstrated positive responses to antiepileptic drugs, but a lack of efficacy of anti-inflammatory drugs for this type of ALS pain [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Soriani | Care management in amyotrophic lateral sclerosis[END_REF][START_REF] Tard | Clinical features of amyotrophic lateral sclerosis and their pronostic value[END_REF][START_REF] Moisset | Is there pain with neuropathic characteristics in patients with amyotrophic lateral sclerosis ? A cross-sectional study[END_REF].

Secondary pain

This is induced by complications of the disease, and increase in intensity as the disease progresses [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF][START_REF] Soriani | Care management in amyotrophic lateral sclerosis[END_REF][START_REF] Tard | Clinical features of amyotrophic lateral sclerosis and their pronostic value[END_REF]. Nociceptive pain is the main cause, induced by the involvement of non-neural tissue [START_REF] Smart | Clinical indicators of "nociceptive", "peripheral neuropathic" and "central" mechanisms of musculoskeletal pain. A Delphi survey of expert clinicians[END_REF]. In ALS, this type of pain develops from the association of muscle atrophy, long-term reduced mobility, and involvement of smooth tissue, bone, tendon, ligament and joint contractures [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF]. The most frequent joint pain is due to shoulder joint contractures, resulting from muscle weakness, loss of protective muscle sheaths due to atrophy and microtrauma induced by caregivers when helping patients to stand [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Treede | Neuropathic pain : redefinition and a grading system for clinical and research purposes[END_REF]. Decubitus ulcers (bed sores) are not classic symptoms in ALS despite long periods of immobility. When present, pain appears mainly during their care. Non-invasive ventilation (NIV) using, for example, face or nasal masks, is another secondary pain induced by lesions on the skin around the nose and mouth, reported in ALS patients after beginning NIV. Close to 80% of ALS patients report experiencing this type of pain [START_REF] Hirano | Ventilator dependence and expressions of need : a study of patients with amyotrophic lateral sclerosis in Japan[END_REF][START_REF] Dangers | Relieving dyspnoea by non-invasive ventilation decreases pain thresholds in amyotrophic lateal sclerosis[END_REF], which is usually treated by non-steroidal anti-inflammatory drugs (NSAIDs) [START_REF] Woolf | Central sensitization : implications for the diagnosis and treatment of pain[END_REF].

Unspecified pain

In some patients, it is not possible to identify either neuropathic or nociceptive types of pain except through the association of both organic and psychic pain [START_REF] Treede | Neuropathic pain : redefinition and a grading system for clinical and research purposes[END_REF][START_REF] Moisset | Is there pain with neuropathic characteristics in patients with amyotrophic lateral sclerosis ? A cross-sectional study[END_REF]. This type of pain is known as 'central sensitization' [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF], for which four mechanisms are currently suggested: (i) spontaneous increase in neuronal activation resulting in pain hypersensitivity [START_REF] Woolf | Central sensitization : implications for the diagnosis and treatment of pain[END_REF]; (ii) increased response of cerebral neurons to light stimuli [START_REF] Meyer | Peripheral neural mechanisms of nociception[END_REF]; (iii) hypoxemia, which can stimulate nociceptive inhibitory control [START_REF] Morelot-Panzini | Dyspnea as a noxious sensation : inspiratory threshold loading may trigger diffuse noxious inhibitory controls in humans[END_REF]; and (iv) greater susceptibility to pain by a decrease in pain threshold [START_REF] Dangers | Relieving dyspnoea by non-invasive ventilation decreases pain thresholds in amyotrophic lateal sclerosis[END_REF]. There are also new data [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF] to explain how sensory light stimuli are able to create and amplify responses in nonstimulated nociceptive neurons [START_REF] Woolf | Central sensitization : implications for the diagnosis and treatment of pain[END_REF].

Natural history of pain in ALS

It is important to bear in mind that pain may precede the onset of motor symptoms, by 2 years [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF] in 25% of cases [START_REF] Wallace | The evaluation of pain in amhyotrophic lateral sclerosis : a case controlled observational study[END_REF] and in up to 50% in other patients [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF][START_REF] Vucic | Sensory and automatic nervous system dysfunction in amyotrophic lateral sclerosis[END_REF]. Cramps are the classic initial pain [START_REF] Carvalho | Cramps, muscle pain, and fasciculations : not always benign ?[END_REF] along with low back pain. Similar results are reported even with the use of analgesic drugs 2 years prior to ALS onset [START_REF] D'ovidio | Amyotrophic lateral sclerosis incidence and previous prescriptions of drugs for the nervous system[END_REF].

Usually, there are also no differences in the frequency [START_REF] Wallace | The evaluation of pain in amhyotrophic lateral sclerosis : a case controlled observational study[END_REF][START_REF] Rivera | Prevalence and characteristics of pain in early and late stages of ALS[END_REF] and severity [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF][START_REF] Pizzimenti | Depression, pain and quality of life in patient with amyotrophic lateral sclerosis a cross-sectional study[END_REF] of pain between early and late stages of the disease [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF]. However, recent studies [START_REF] Adelman | Disparities in perceptions of distress and burden in ALS patients and family caregivers[END_REF] have reported the presence of all-day-long moderate-to-severe pain in one patient out of two during the last month of life. In addition, the natural history of pain in ALS associates pain with comorbidities such as depression [START_REF] Pizzimenti | Depression, pain and quality of life in patient with amyotrophic lateral sclerosis a cross-sectional study[END_REF][START_REF] Lepine | The epidemiology of pain in depression[END_REF]. Indeed, pain is more frequent in ALS patients with than without depression [START_REF] Atassi | Depression in amyotrophic lateral sclerosis[END_REF], whereas the same excess of depression is found in patients with ALS and pain [START_REF] Stephens | The role of mental health and self-efficacy in the pain experience of patients with amyotrophic lateral sclerosis[END_REF]. This double association explains the poor quality of life of these patients. Other comorbidities, such as anxiety, apathy [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Caga | Apathy is associated with poor prognosis in amyotrophic lateral sclerosis[END_REF], and impaired memory [START_REF] Hervieu-Bègue | Semantic memory assessment in 15 patients with amyotrophic lateral stenosis[END_REF] and cognitive functions [START_REF] Couratier | ALS and frontotemporal dementia belong to a common disease spectrum[END_REF], may also arise during the course of the disease.

Pain treatment in ALS

Given the importance of pain in terms of quality of life for patients with ALS, physicians need to be able to identify and reduce the pain quickly or even prevent it (Tables 1 and2). Such strategies are mostly based on clinical experience, with pharmacological treatments being the more efficacious approach for neuropathic and secondary pain. For neuropathic pain [START_REF] Soriani | Care management in amyotrophic lateral sclerosis[END_REF][START_REF] Chio | Amyotrophic lateral sclerosis care in Italy : a nationwide study in neurological centers[END_REF][START_REF] Borasio | Standards of palliative care for patients with amyotrophic lateral sclerosis : results of a European survey[END_REF], gabapentin, pregabalin and tricyclic antidepressants are recommended. For cramps, quinine, benzodiazepine, magnesium and carbamazepine have proved significantly effective [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF]. For severe spasticity, oral or intrathecal baclofen may be efficacious, as well as benzodiazepine and dantrolene [START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF][START_REF] Soriani | Care management in amyotrophic lateral sclerosis[END_REF][START_REF] Chio | Amyotrophic lateral sclerosis care in Italy : a nationwide study in neurological centers[END_REF]. Articular pain such as in the shoulder joints can be relieved by anti-inflammatory drugs and acetaminophen/paracetamol or by intra-articular injections of analgesics or corticosteroids. Finally, in all cases, physiotherapy, massage, daily stretching, therapeutic ultrasound, laser therapy, transcutaneous electrical nerve stimulation (TENS) and acupuncture may also help patients with ALS and pain [START_REF] Chio | Pain in amyotrophic lateral sclerosis[END_REF][START_REF] Hanisch | Characteristics of pain in amyotrophic lateral sclerosis[END_REF].

Conclusion

Pain is a major clinical complaint reported by ALS patients, and physicians need to be trained to identify the different causes, including neuropathic pain, cramps, spasticity and nociceptive pain. In addition, pain may be reported by patients before the first motor symptoms of ALS; it then increases throughout the course of disease and, in most cases, is accompanied by depression, thereby worsening patients' quality of life. Thus, pain is a major challenge in the management of ALS that now requires good clinical practices such as those developed in France, thanks to the creation of specific ALS centers by the Ministry of Health.

Disclosure of interest

The authors declare that they have no financial or personal conflicts of interests concerning this article.

Table 1 -Pharmacological treatment of pain in patients with amyotrophic lateral sclerosis (ALS).

Types of pain