The Lagrange inversion formula on non-Archimedean fields. Non-Analytical Form of Differential and Finite Difference Equations

The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n>1$ some results of [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of $\C^2$ we prove a quantitative estimate of a previous qualitative result of [MatteiMoussu] and we compare it with a result of [YoccozPerezMarco].


Introduction
Let k be a field of characteristic zero complete with respect to a non-trivial absolute value | | and let k ′ denote its residue field. When k = R or C, the classical Lagrange inversion formula (see [21] 1 , [10] chapter VIII, section 7 or [29] p. 286, for the 1-dimensional case, and to [16] for the multidimensional case) says that if G is an analytic function in a neighborhood of w ∈ k then there exists a unique solution h = H(u, w) of h = uG (h) + w , (1.1) provided that |u| is sufficiently small. The solution h = H(u, w) depends analytically on u and w and its Taylor series with respect to u is explicitly given by the formula: H (u, w) = w + n≥1 u n n! d n−1 dw n−1 (G (w)) n . (1.2) In sections 2, after recalling some elementary notions of theory of analytic functions on non-Archimedean fields, we give two generalizations of (1.1): in the ndimensional vector space k n , k non-Archimedean when G is an analytic function (Corollary 2.5), and for non-analytic G (Theorem 2.3). To deal with this second case we rewrite the Lagrange inversion formula by means of the tree formalism. We refer to [15] and references therein for a combinatorial proof of the Lagrange inversion formula using the tree formalism.
In sections 4 and 5 we will give some applications of the previous results in the setting of the formal Laurent series with applications to some dynamical systems problems. The idea of using trees in non-linear small divisors problems (in particular Hamiltonian) is due to H. Eliasson [12] who introduced trees in his study of the absolute convergence of Lindstedt series. The idea has been further developed by many authors (see, for example, [8,13,14] always in the context of Hamiltonian KAM theory, see also [1] which we take as reference for many definitions concerning trees). The fact that these formulas should be obtained by a suitable generalization of Lagrange's inversion formula was first remarked by Vittot [33].
When k is the field of formal Laurent series C((z)), we consider the vector space: C n ((z 1 , . . . , z n )); the non-analytic inversion problem can be applied to obtain the solution of semilinear differential or q-difference equations in an explicit (i.e. not recursive) form. Our results are formulated so as to include general first-order U -differential semilinear equations [11] and semilinear convolution equations. In particular we will study (section 4) the Siegel center problem [18,34] for analytic and non-analytic germs of (C n , 0), n ≥ 1, and (section 5) the Problem of linearization of analytic [3] and non-analytic vector fields of C n , n ≥ 1. The reader interested only in the Siegel center problem may find useful to assume Proposition 4.2 and to skip the reading of the whole of sections 2 and 3. The same is true for those interested in the linearization of vector fields, assuming Proposition 5.1 and reading the rest of section 5, even if they will find several useful definitions in section 4.
In [5] authors began the study of the Siegel center problem in some ultradifferentiable algebras of C ((z)), here we generalize these results to dimension n ≥ 1.
Consider two Classes of formal power series C 1 and C 2 of C n [[z 1 , . . . , z n ]] closed with respect to the composition and derivation. For example the Class of germs of analytic functions of (C n , 0) or Gevrey-s Classes, s > 0 (i.e. series F = α∈N n f α z α for which there exist c 1 , c 2 > 0 such that |f α | ≤ c 1 c |α| 2 (|α|!) s , for all α ∈ N n ). Let A ∈ GL(n, C) and F ∈ C 1 such that F(z) = Az + . . . , we say that F is linearizable in C 2 if there exists H ∈ C 2 , tangent to the identity, such that: When A is in the Poincaré domain (see § 4.2), the results of Poincaré [27] and Koenigs [20] assure that F is linearizable in C 2 . When A is in the Siegel domain (see § 4.2), the problem is harder, the only trivial case is C 2 = C n [[z 1 , . . . , z n ]] (formal linearization) for which one only needs to assume A to be non-resonant.
In the analytic case we recover the results of Bruno [3] and Rüssmann [28], whereas in the non-analytic case new arithmetical conditions are introduced (Theorem 4.6). Consider the general case where both C 1 and C 2 are different from the Class of germs of analytic function of (C n , 0), if one requires C 1 = C 2 , once again the Bruno condition is sufficient, otherwise if C 1 ⊂ C 2 one finds new arithmetical conditions, weaker than the Bruno one.
In section 5 we will consider the following differential equation: where t is the time variable and F is a formal power series in the n ≥ 1 variables z 1 , . . . , z n , with coefficients in C n , without constant term: F = α∈N n ,|α|≥1 F α z α , and we are interested in the behavior of the solutions near the singular point z = 0.
A basic but clever idea has been introduced by Poincaré (1879), which consists in reducing the system (1.3) with an appropriate change of variables, to a simpler form: the normal form. In [3] several results are presented in the analytic case (namely F is a convergent power series). Here we generalize such kind of results to the case of non-analytic F, with a diagonal, non-resonant linear part. More precisely considering the same Classes of formal power series as we did for the Siegel Center Problem , we take an element F ∈ C 1 with a diagonal, non-resonant linear part, Az, and we look for sufficient conditions on A to ensure the existence of a change of variables H ∈ C 2 (the linearization), such that in the new variables the vector field reduces to its linear part. We will show that the Bruno condition is sufficient to linearize in the same class of the given vector field, whereas in the general case, C 1 ⊂ C 2 , new arithmetical conditions, weaker than the Bruno one, are introduced (Theorem 5.2). Finally in the case of analytic vector field of C 2 , the use of the continued fraction and of a best description of the accumulation of small divisors (due to the Davie counting function [9]), allows us to improve (Theorem 5.5) the results of Theorem 5.2, giving rise to (we conjecture) an optimal estimate concerning the domain of analyticity of the linearization. This gives a quantitative estimate of some previous results of [25] and [26].
In our formulation we emphasize the strong similarities existing between this problem and the Siegel Center Problem, which becomes essentially the same problem; in fact once we reduced each problem to a Lagrange inversion formula (on some appropriate setting) we get the same functional equation to solve.

The Lagrange inversion formula on non-Archimedean fields
In this section we generalize the Lagrange inversion formula for analytic and non-analytic functions on complete, ultrametric fields of characteristic zero. In the first part we give for completeness some basic definitions and properties of non-Archimedean fields, referring to Appendix A and to [30,7,6] for a more detailed discussion. We end the section introducing some elementary facts concerning trees.
2.1. Statement of the Problem. Let (k, | |) be a non-Archimedean field 2 of characteristic zero, where | | is a ultrametric absolute value : |x + y| ≤ sup(|x|, |y|) for all x, y ∈ k. Moreover we assume that k is complete and the norm is non-trivial.
Let a be a real number such that 0 < a < 1, given any x ∈ k we define the real number v (x) by: |x| = a v(x) , the valuation 3 of x.
Since k is non-Archimedean one has the following elementary but fundamental result: x n converges if and only if x n → 0.
Let n ∈ N, we introduce the n-dimensional vector space k n and using the ultrametric absolute value, defined on k, we introduce a norm || · || : which results an ultrametric one and verifies a Schwartz-like inequality, for x, y ∈ k n then: |x · y| ≤ ||x|| ||y||, where x · y = n i=1 x i y i is a scalar product. This norm induces a topology, where the open balls are defined 4 by for x ∈ k n and r ∈ R + . We will denote the closed ball with B(x, r) = B 0 (x, r).
Let r > 0 and let us consider a function G : B 0 (0, r) ⊂ k n → k n×l , i.e. for x ∈ B 0 (0, r) and for all 1 ≤ i ≤ n, 1 ≤ j ≤ l: Given w ∈ k n , u ∈ k l and G as above, we consider the following problem: Solve with respect to h ∈ k n , the multidimensional non-analytic Lagrange inversion problem: where Λ is a k n -additive, k ′ -linear, non-expanding operator (i.e. ||Λw|| ≤ ||w|| for all w ∈ k n ). We will prove the existence of a solution of (2.3) using trees. We will now recall some elementary facts concerning trees; we refer to [17] for a more complete description.
2.2. The Tree formalism. A tree is a connected acyclic graph, composed by nodes and lines connecting together two or more nodes. Among trees we consider rooted trees, namely trees with an extra node, not included in the set of nodes of the tree, called the earth, and an extra line connecting the earth to the tree, the root line. We will call root the only node to which the earth is linked. The existence of the root introduces a partial ordering in the tree: given any two nodes 5 The order of a tree is the number of its nodes. The forest T N is the disjoint union of all trees 6 with the same order N . 3 From the properties of | | it follows that the valuation satisfies, for all x, y ∈ k : v (x) = +∞ if and only if , v (y)). 4 One could define [30] the open polydisks: P 0 (x, ρ) = {y ∈ k n : ∀i, 1 ≤ i ≤ n : |x i − y i | < ρ i }, for some x ∈ k n and ρ ∈ R n + . Clearly the induced topology is equivalent to the previously defined one. 5 To denote nodes we will use letters: u, v, w, . . . , with possible sub-indices. Lines will be denoted by ℓ, the line exiting from the node u will be denoted by ℓu. 6 Here we consider only semitopological trees (see [2]), we refer to [13] for the definition of topological trees.
The degree of a node, deg v, is the number of incident lines with the node. Let m v = deg v−1, that is the number of lines entering into the node v w.r.t. the partial ordering, if m v = 0 we will say that v is an end node; for the root v 1 , because the root line doesn't belong to the lines of the tree, we define m v1 = deg v 1 , in this way m v1 also represents the number of lines entering in the root. Let ϑ be a rooted tree, for any v ∈ ϑ we denote by L v the set of lines entering into v; if v is an end node we will set L v = ∅.
Given a rooted tree ϑ of order N , we can view it as the union of its root and the subtrees ϑ i obtained from ϑ by detaching the root. Let v 1 be the root of ϑ and t = m v1 , we define the standard decomposition of ϑ as: Using the definition of m v we can associate uniquely to a rooted tree of order N a vector of N N , whose components are just m v with v in the tree [33].
We can then rewrite the standard decomposition of ϑ as: ϑ = t, ϑ 1 , . . . , ϑ t where the subtrees satisfy: ϑ i ∈ T Ni with N 1 + . . . In the following we will also use labeled rooted trees. A labeled rooted tree of order N is an element of T N together with N labels: α 1 , . . . , α N . We can think that the label α i is attached to the i-th node of the standard decomposition of the tree. The label is nothing else that a function from the set of nodes of a tree to some set, usually a subset of Z m for some integer m. When needed we denote a labeled rooted tree of order N with the couple (ϑ, α), where ϑ ∈ T N and α = (α 1 , . . . , α N ) is the vector label.
2.3. The non-analytic Lagrange inversion formula. We are now able to extend equation (1.1), the classical analytic Lagrange inversion formula, to the setting of paragraph 2.1. We refer the reader to Appendix A for a brief introduction to the theory of analytic functions on k n (norms, Cauchy estimates, etc ... ).
Let N ∈ N * , U and V be open subsets of, respectively, k l and k n , and ϑ ∈ T N . We define the function Val Λ : T N ×U × V ∋ (ϑ, u, w) → Val Λ (ϑ) (u, w) ∈ k n as follows where ϑ = (t, ϑ 1 , . . . , ϑ t ) is the standard decomposition of the tree and Λ : k n → k n .
Remark 2.2. For t ≥ 1 and v (1) , . . . , v (t) ∈ k n we recall that n ) and D li are the l i -th partial derivatives of G at w (see Appendix A).
We can then state the following existence Theorem: Theorem 2.3 (Non-analytic case). Let n, l be positive integers. Let u ∈ k l , w ∈ k n and for Let Λ : k n → k n be a k n -additive, k ′ -linear and non-expanding operator. Assume that for (r 1 , . . . , r n ) ∈ R n + the G i 's are convergent in B (0, r i ), set 7 M i = ||G i || ri > 0, r = min i r i and M = max i M i . Then equation (2.3) has the unique solution , has been defined in (2.4). Moreover for any fixed ϑ the function Val Λ is continuous, series (2.5) converges on B 0 (0, r/M ) × B 0 (0, r) and the map (u, w) → H(u, w) ∈ B 0 (0, r) is continuous.
Remark 2.4. Since Λ is not k n -linear, Val Λ and H cannot be analytic. However the non-expanding condition implies that Λ is Lipschitz continuous from which the regularity properties of Val Λ and H follow.
If Λ is k n -linear then we have the following Corollary (particular case of the previous Theorem) with w ′ instead of Λw and G ′ · u ′ instead of Λ(G · u), which extends the analytic Lagrange inversion formula (1.1).
Corollary 2.5 (Analytic case). Let n, l be positive integers. Let u ′ ∈ k l , w ′ ∈ k n and for

Proofs.
This section is devoted to the proof of Theorem 2.3 and Corollary 2.5. Proof of Theorem 2.3. Using the fact that Λ is non-expanding the uniqueness of the solution can be proved easily. Let H 1 and H 2 be two solutions of (2.3), then but for all i = 1, . . . , n: By hypothesis ||u|| < r M , then µ < 1, from which we conclude that H 1 = H 2 . We now prove existence. Since G i are convergent, D α G i also are convergent and Proposition A.2 gives the following estimate which together with the non-expanding property of Λ allows us to prove that for all N ≥ 1 and all ϑ ∈ T N : , clearly H (j) → H as j → ∞ and it is easy to check that: which tends to 0 as j → ∞.
We give now the proof of Corollary 2.5. This one follows closely the one of Theorem 2.3 in particular the uniqueness statement, so we will outline only the main differences w.r.t to the previous proof.
Proof. The hypothesis on G ′ i gives an estimate similar to (3.2), then by induction on N it is easy to prove that for all N ≥ 1 and ϑ ∈ T N one has Remark 3.1. In the simplest case n = l = 1, namely u, w ∈ k and G ∈ k [[X]], the solution given by (2.6) coincides with the classical one of Lagrange (1.2). One can prove this fact either using the uniqueness of the Taylor development or by direct calculation showing that for all positive integer N ≥ 1 we have In the other cases formula (2.6) is the natural generalization of (1.2).
Remark 3.2. Series (2.6) is an analytic function of u, w, but it is not explicitly written as u-power series. We claim that introducing labeled rooted trees we can rewrite (2.6) explicitly as a u-power series.
4. The Non-analytic Siegel center problem.
In this part we show that the problem of the conjugation of a (formal) germ of a given function with its linear part near a fixed point (the so called Siegel Center Problem) can be solved applying Theorem 2.3 to the field of (formal) power series. The Siegel Center Problem is a particular case of first order semilinear q-difference equation, but our results apply to general first order semilinear q-difference and differential equations (see next section).

4.1.
Notations and Statement of the Problem. Let α = (α 1 , . . . , α n ) ∈ N n , λ = (λ 1 , . . . , λ n ) ∈ C n with λ i = λ j if i = j, we will use the compact notation λ α = λ α1 1 . . . λ αn n and |α| = α 1 + · · · + α n ; we will denote the diagonal n × n matrix with λ i at the (i, i)-th place with diag(λ 1 , . . . , λ n ). Let be the vector space of the formal power series in the n variables z 1 , . . . , z n with coefficients in We consider V endowed with the ultrametric absolute norm induced by the z- Let C be a Class (that we will define later, see paragraph 4.3) of formal power series, closed w.r.t. the (formal) derivation, the composition and where, roughly speaking, the (formal) Taylor series makes sense. One can think for example to the Class of germs of analytic diffeomorphisms of (C n , 0) or Gevrey-s Classes, in fact we will see that our classes will contain these special cases.
Let A ∈ GL(n, C) and assume A to be diagonal 8 with all the eigenvalues distinct. Let C 1 and C 2 be two classes as stated before, then the Siegel center problem can be formulated as follow [18,5]: Let F(z) = Az+f (z) ∈ C 1 , f ∈ C 1 ∩V 1 , find necessary and sufficient conditions on A to linearize in C 2 F, namely find H ∈ C 2 ∩ V 0 (the linearization) solving: for any g(z) ∈ V . We remark that the action of D λ on the monomial vz α , for any v ∈ C n and any α ∈ N n , is given by: Let A = diag(λ 1 , . . . , λ n ) ∈ GL(n, C), we say that A is resonant if there exist α ∈ N n , |α| ≥ 2 and j ∈ {1, . . . , n} such that: , it is nonexpanding (||D λ g|| ≤ ||g||), and clearly V -additive (D λ (f + g) = D λ f + D λ g) and C-linear. Then we claim that the Siegel center problem (4.1) is equivalent to solve the functional equation: where z is the identity formal power series, f ∈ C 1 ∩ V 1 and h ∈ C 2 ∩ V 1 . In fact from (4.1) we see that the linear part of H doesn't play any role, so we can choose H tangent to the identity (this normalization assures the uniqueness of the linearization): But then (4.1) can be rewritten as: and replacing the left hand side with the operator D λ h we obtain (4.5). Given which is a particular case of the non-analytic multidimensional Lagrange inversion formula (2.3) with u = 1, w = 0 and Λ = D −1 λ . The following Lemma assures that G f verifies the hypotheses of Theorem 2.3.
where (if we define f β = 0 for |β| = 0 and |β| = 1) series g β (f ) ∈ V are given by Here we used the compact notations α! = α 1 ! . . . α n ! and α+β β = (α+β)! α!β! , for α = (α 1 , . . . , α n ) ∈ N n and β ∈ N n . Moreover one has ||g 0 || = ||f ||, ||g β || = 2||f || for any |β| = 1 and ||g β || ≤ 4||f || for any |β| ≥ 2; (4.10) The proof is straightforward and we omit it. We can thus apply Theorem 2.3 to solve (4.7) with u = 1, w = 0, G = G f , Λ = D −1 λ , r = 1/2 and M = 1/4, and the unique solution of (4.7) is given by An explicit expression for the power series coefficients of h can be obtained introducing labeled rooted trees (see Remark 3.2). Let us now explain how to do this. Let N ≥ 1 and let ϑ be a rooted tree of order N ; to any v ∈ ϑ we associate a (node)-label α v ∈ N n s.t. |α v | ≥ 2 and to the line ℓ v (exiting w.r.t. the partial order from v) we associate a (line)-label β ℓv ∈ N n s.t. |β ℓv | = 1. We define β v = ℓ∈Lv β ℓ (so β v ∈ N n and |β v | = m v ) and the momentum flowing through the line ℓ v : ν ℓv = w∈ϑ:w≤v (α w − β w ). When v will be root of the tree, we will also use the symbol ν ϑ (the total momentum of the tree) instead of ν ℓv . It is trivial to show that the momentum function is increasing (w.r.t. the partial order of the tree), namely if v is the root of a rooted labeled tree and if v i is any of the immediate . . , n} and α ∈ N n , we finally define T N,α,j to be the forest of rooted labeled trees of order N with total momentum ν ϑ = α and β ℓv 1 = e j (being e j the vector with all zero entries but the j-th which is set equal to 1) for the root line ℓ v1 .
We are now able to prove the following For |α| ≥ 2 the j-th component of the coefficient h α is given by 9 : where the last product has to be set equal to 1 whenever v is an end node (L v = ∅). Remark 4.3. By definition β ℓw , for any w ∈ ϑ, has length 1, so it coincides with an element of the canonical base. Then for w ∈ ϑ and any choice of the labels, such that β ℓw = e i , the term ((Ω −1 ν ℓw f αw ) · β ℓw ) is nothing else that Proof. Let ϑ ∈ T N , for N ≥ 1 and α ∈ N n such that |α| ≥ N + 1. Let us define Val (ϑ) , (4.14) namely for a fixed tree, sum over all possible labels α vi and β ℓv i , with v i in the tree, in such a way that the total momentum is fixed to α and the root line has label β ℓv 1 = e j . It is clear that Actually in (4.15) h is ordered with increasing powers of z whereas in (4.16) with increasing order of trees. Convergence in V 1 for (4.16) is assured from the estimate ||h ϑ N +1 || ≤ 2 −(N +1) and uniform convergence assures that (4.15) and (4.16) coincide.
We claim that by induction on the order of the tree, we can prove that for all N ≥ 1 and all ϑ ∈ T N : In the Analytic case (both C 1 and C 2 are the ring of the germs of analytic diffeomorphisms of (C n , 0)) , let A be the derivative of F at the origin, then if A is non-resonant and it is in the Poincaré domain, the Analytic Siegel Center Problem has a solution [27,20] (see also [18] and references therein). Moreover if A is resonant and in the Poincaré domain, but F is formally linearizable, then F is analytically linearizable.
If A is in the Siegel domain the problem is harder, but we can nevertheless have a solution of the Analytic Siegel Center Problem, introducing some new condition on A. Let p ∈ N, p ≥ 2, and let us definẽ we remark that even if A is non-resonant, but in the Siegel domain, one has lim p→∞Ω (p) = 0; this is the so called small divisors problem, the main obstruction to the solution of equation (4.1). A non-resonant matrix A verifies a Bruno condition 10 if there exists an increasing sequence of natural numbers (p k ) k such Then if A satisfies a Bruno Condition, the germ is analytically linearizable [3,28].
For the 1-dimensional Analytic Siegel Center Problem, Yoccoz [34] proved that the Bruno condition is necessary and sufficient to linearize analytically any univalent germs with fixed linear part, in this case the Bruno condition reduces to the convergence of the series +∞ k=0 log q k+1 q k < +∞ , (4.21) where (q k ) k is the sequence of the convergent to ω ∈ R \ Q such that λ = e 2πiω .

4.3.
A new result: ultradifferentiable Classes. Let (M k ) k≥1 be a sequence of positive real numbers such that: We define the class C (M k ) ⊂ C n [[z 1 , . . . , z n ]] as the set of formal power series f = f α z α such that there exist A, B positive constant, such that: The hypotheses on the sequence (M k ) k assure that C (M k ) is stable w.r.t. the (formal) derivation, w.r.t. the composition of formal power series and for every tensor built with element of the class, its contraction 11 gives again an element of the class.
For example if f , g ∈ C (M k ) then also df (z)(g(z)) belongs to the same class.
Remark 4.5. Our classes include the Class of Gevrey-s power series as a special case: M k = (k!) s . Also the ring of convergent (analytic) power series are trivially included.
In [5] a similar problem was studied in the 1-dimensional case. Here we will extend the results contained there to the case of dimension n ≥ 1. The main result will be the following Theorem Theorem 4.6. Let (λ 1 , . . . , λ n ) ∈ C n , |λ i | = 1 for i = 1, . . . , n, and let A = diag(λ 1 , . . . , λ n ) be non-resonant, (M k ) k and (N k ) k be sequences verifying hypothe- If moreover F ∈ C (M k ) ∩V 0 and A verifies a Bruno condition (4.20), then also the linearization H belongs to C (M k ) ∩V 0 . 2. If F is a germ of analytic diffeomorphisms of (C n , 0) and there exists an increasing sequence of integer numbers (p k ) k such that A verifies: where κ(α) is the integer defined by: p κ(α) ≤ |α| < p κ(α)+1 , then the linearization H belongs to C (N k ) ∩V 0 . 3. If F ∈ C (M k ) ∩V 0 , the sequence (M k ) k is asymptotically bounded by the sequence (N k ) k (namely N k ≥ M k for all sufficiently large k) and there exists an increasing sequence of integer numbers (p k ) k such that A verifies: where κ(α) is the integer defined by: p κ(α) ≤ |α| < p κ(α)+1 , then the linearization H belongs to C (N k ) ∩V 0 .
The proof of Theorem 4.6 will be done in section 4.3.2, before we make some remarks and we prove some preliminary lemmata.  .23)) reduces to condition (2.10) (respectively condition (2.9)) of [5] except for the factor in front of the sum: here we have 2 instead of 1 in [5]. This is due to the better control of small denominators one can achieve using continued fractions and Davie's counting lemma [9] as explained in [5].
The proof follows closely the one of Lemma 10 p.218 of [3] and we don't prove it.
Let ℓ be a line of a rooted labeled tree, ϑ, let us introduce the notion of scale of the line ℓ. Let ν ℓ be the momentum of the line and let us defineν ℓ = ν ℓ − β ℓ . Let (p k ) k be an increasing sequence of positive integer, then for any k ≥ 0 we define: For short we will say that a line ℓ is on scale k if s ℓ (k) = 1. Let us define N k (ϑ) be the number of line on scale 1 in the rooted labeled tree ϑ. We can now prove the following Lemma, which roughly speaking says that the number of "bad" (too small) denominators is not too big, whereas Lemma 4.8 says that they do not occur so often.
where ⌊x⌋ denotes the integer part of the real number x. We recall that ν ϑ is the total momentum of the tree andν ϑ = ν ϑ − β ℓv 1 , being ℓ v1 the root line of ϑ.
Our proof follows the original one of Bruno but exploiting the tree formalism, the interested reader can find this proof in appendix B.

4.3.2.
Proof of Theorem 4.6. We are now able to prove the main Theorem, we will prove only point 3 which clearly implies point 1 (choosing M k = N k for all k) and point 2 (choosing M k = C k for all k and some constant C > 0).
For a fixed rooted labeled tree of order N ≥ 1 with total momentum equals to α ∈ N n , |α| ≥ 2, we consider the following term of equation (4.13): Recalling the definition of scale and the definition of number of lines on scale k we can bound (4.29) with where κ(α) is the integer defined by: p κ(α) ≤ |α| < p κ(α)+1 . Using hypothesis 3. of paragraph 4.3 on the sequence (M k ) k and the hypothesis f ∈ C (M k ) ∩V 1 , we will obtain for some positive constant A, B the bound: by definition of total momentum: v∈ϑ |α v | − (N − 1) = |ν ϑ |, which has been fixed to |α|, then Using finally the bound of the Counting Lemma 4.9 we get: for some positive constant C. Dividing (4.30) by |α| and passing to the limit superior we get the thesis. λ = e 2πiω , ω ∈ R \ Q. Let us call R (k) (ω) the radius of convergence of the unique linearization associated to F (k) . Then an easy adaptation of Theorem 4.6 case 1) with C (M k ) = z C{z}, allows us to prove: for some constant C k (depending on k but independent of ω). This can explain the 1/k-periodicity of R (k) (ω), as a function of ω, showed in Figures 5 and 7 of [23].

Linearization of non-analytic vector fields
The aim of this section is to extend the analytic results of Bruno about the linearization of an analytic vector field near a singular point, to the case of ultradifferentiable vector fields. We will show that this problem can be put in the framework of Theorem 2.3 and then obtaining an explicit (i.e. non-recursive) expression for the change of variables (the linearization) in which the vector field has a simpler form.
Our aim is also to point out the strong similarities of this problem with the Siegel Center Problem, previously studied. In particular when both problems are put in the framework of the multidimensional non-analytic Lagrange inversion formula on the field of formal Laurent series, they give rise to (essentially) the same problem. For this reason most results will only be stated without proofs, these being very close to the proofs of the previous section.

Notation and Statement of the Problem.
In this section we will use the same notations given at the beginning of section 4.3. Let A ∈ GL(n, C) and assume A to be diagonal. Let C 1 and C 2 be two classes of formal power series as defined before, then the problem of the linearization of vector fields can be formulated as follows: Let F(z) = Az + f (z) ∈ C 1 , f ∈ C 1 ∩ V 1 , and consider the following differential equation:ż where t denotes the time variable. Determine necessary and sufficient conditions on A to find a change of variables in C 2 ∩V 1 (called the linearization) which leaves the singularity (z = 0) fixed, doesn't change the linear part of F and allows to rewrite (5.1) in a simpler form 12 . Namely find h ∈ C 2 ∩ V 1 , such that z = w + h(w) and in the new variables w, equation (5.1) rewrites: Let ω = (ω 1 , . . . , ω n ) ∈ C n and A = diag(ω 1 , . . . , ω n ), we will say that A is resonant if there exist α ∈ Z n , with all positive component except at most one which can assume the value −1, |α| ≥ 2, and j ∈ {1, . . . , n} such that: ω ·α−ω j = 0, where ω·α = n i=1 ω i α i is the scalar product. Let α ∈ N n we introduce the diagonal matrix Ω ′ α = diag(ω · α − ω 1 , . . . , ω · α − ω n ). Let us introduce the operator 13 , it is non-expanding and clearly V -additive and C-linear. Then we claim that the linearization, h, is solution of the functional equation: where f ∈ C 1 ∩ V 1 , h ∈ C 2 ∩ V 1 and z denotes the identity formal power series.
Given f ∈ V 1 we consider the function G f (h) = f • (z + h), assume A to be non-resonant to invert the operator D ω , then we rewrite (5.4) as: which is a particular case of the non-analytic multidimensional Lagrange inversion formula (2.3). Apart from the different operator, this equation is the same of the Siegel Center Problem (4.5). Lemma 4.1 assures that G f verifies the hypotheses of Theorem 2.3 and thus we can apply it with: u = 1, w = 0, G = G f , Λ = D ′ −1 ω , r = 1/2 and M = 1/4. The unique solution of (5.5) is then given by: Once again we can give an explicit expression for the linearization h using rooted labeled trees. Introducing the same labels as for the Siegel center problem (see page 10) we can prove the following For |α| ≥ 2 the j-th component of the coefficient h α is given by: where the last product has to be set equal to 1 whenever v is an end node (L v = ∅). 12 Here we don't consider the most general case of looking for a change of coordinates which put (5.1) in normal form, namely containing only resonant terms:ẇ i = w i α·ω=0 g α,i w α . Our results will concern vector fields with non-resonant linear parts, so (5.2) will be the normal form. 13 An equivalent definition will be: D ′ ω g(w) = n i=1 (Aw) i ∂w i g(w) − Ag(w) = L A g, the Poisson bracket of the linear field Aw and g.
We don't prove this Proposition (whose proof is the same as the one of Proposition 4.2); moreover we point out that because both problems give rise to (essentially) the same multidimensional non-analytic Lagrange inversion formula, we can pass from one solution to the other with very small changes: Ω ′ −1 α instead of Ω −1 α . If both classes C 1 and C 2 are V 1 (formal case) then the linearization problem has solution if A is non-resonant. In the analytic case we distinguish again the Poincaré domain (the convex hull of the n complex points ω 1 , . . . , ω n doesn't contain the origin) and the Siegel domain (if they are not in the Poincaré domain). In the first case, under a non-resonance condition, Poincaré proved that the vector field is analytically linearizable, and then Dulac, in the resonant case, proved the conjugation to a normal form. In the Siegel non-resonant case Bruno proved analytic linearizability [3], under the Bruno condition which reads: for some increasing sequence of integer number (p k ) k , whereΩ(p) = min{|α · ω| : α ∈ N n , α · ω = 0, 0 < |α| < p, at most one component α i = −1.}.
We now extend this kind of results to the case of ultradifferentiable vector fields. Namely we consider two classes C (M k ) and C (N k ) , defined as in section 4.3 and we prove the following If moreover F ∈ C (M k ) ∩V 0 and A verifies a Bruno condition (5.8), then also the linearization h belongs to C (M k ) ∩V 1 . 2. If F is a germ of analytic diffeomorphisms of (C n , 0) and there exists an increasing sequence of integer numbers (p k ) k such that A verifies: where κ(α) is the integer defined by: p κ(α) ≤ |α| < p κ(α)+1 , then the linearization h belongs to C (N k ) ∩V 1 . 3. If F ∈ C (M k ) ∩V 0 , the sequence (M k ) k is asymptotically bounded by the sequence (N k ) k (namely N k ≥ M k for all sufficiently large k) and there exists an increasing sequence of integer numbers (p k ) k such that A verifies: where κ(α) is the integer defined by: p κ(α) ≤ |α| < p κ(α)+1 , then the linearization h belongs to C (N k ) ∩V 1 .
To prove this Theorem we will use again the majorant series method, the main step is to control the small denominators contributions. To do this, given an increasing sequence of positive integer (p k ) k we define a new counting function: for any ν ∈ Z n \0. By definition we trivially haveΦ (k) (ν) = 0 for all 0 < |ν| ≤ p k . Then we can prove the following Lemma (which will play the role of Lemma 4.8): Lemma 5.3. Let ν 1 ∈ Z n \0 such thatΦ (k) (ν 1 ) = 1, for some k. Then for all ν 2 ∈ Z n , such that 0 < |ν 2 | ≤ p k , we haveΦ (k) (ν 1 − ν 2 ) = 0.
We finally define a new notion of scale; let ℓ be a line of a rooted labeled tree, let ν ℓ be its momentum and let us recall thatν ℓ = ν ℓ − β ℓ , then: (5.12) we will say that a line ℓ is on scale k ifŝ ℓ (k) = 1. We can now prove the following Counting Lemma Lemma 5.4 (Bruno's Counting lemma 2 nd version). Let ϑ be a rooted labeled tree of order N ≥ 1, let k ≥ 1 be an integer and letN k (ϑ) be the number of line on scale k in the tree. Then the following bound holds: where ⌊x⌋ denotes the integer part of the real number x. We recall that ν ϑ is the momentum of the root line andν ϑ = ν ϑ − β ℓv 1 , being ℓ v1 the root line.
We don't prove it because its proof is the same as the one of Lemma 4.9 except for the use of Lemma 5.3 instead of Lemma 4.8.

5.2.
Proof of Theorem 5.2. Once again we will prove only point 3 which clearly contains points 1 and 2 as special cases.

5.3.
A result for some analytic vector fields of C 2 . For 2-dimensional analytic vector field the existence of the continued fraction and the convergents allows us to improve the previous Theorem, giving an optimal (we conjecture) estimate on the "size" of the analyticity domain of the linearization.
Mattei and Moussu [25] proved, using the holonomy construction, that linearization of germs implies linearization of the foliation 14 associated to the vector fields of (C 2 , 0). In [26] authors proved, using Hormander∂-techniques, the converse statement. More precisely they proved that the foliation associated to the analytic vector field: where ω > 0 and the suspension points mean terms of order bigger than 1, has the same analytical classification of the germs of (C, 0): f (z) = e 2πiω z + O(z 2 ). Using the results of [34] they obtain as corollaries that: if ω is a Bruno number then the foliation associated to (5.17) is analytically linearizable, whereas if ω is not a Bruno number then there exist analytic vector fields of the form (5.17) whose foliation are not analytically linearizable.
Here we push up this analogy between vector fields and germs, by proving that the linearizing function of the vector field is analytic in domain containing a ball of radius ρ which satisfies the same lower bound (in term of the Bruno function) as the radius of convergence of the linearizing function of the germ [5,34] does.
To do this we must introduce some normalization condition for the vector field; let ω > 0, we consider the family F ω of analytic vector fields F : D × D → C 2 of the form (5.18) with |f α,j | ≤ 1 for all |α| ≥ 2 and j = 1, 2.
For power series in several complex variables the analogue of the disk of convergence is the complete Reinhardt domain of center 0, R 0 , by studying the distance of the origin to the boundary of this domain we can obtain informations about its "size". Fixing the non linear part of the vector field: f = |α|≥2 f α z α , this distance is given by d F = inf (z1,z2)∈R0 (|z 1 | 2 + |z 2 | 2 ) 1/2 . The family F ω is compact w.r.t. the uniform convergence on compact subsets of D × D (use Weierstrass Theorem and Cauchy's estimates in C 2 , see for example [31]) so we can define d ω = inf F∈Fω d F .
Let ρ F > 0 and let us introduce P (0, ρ F ) = {(z 1 , z 2 ) ∈ C 2 : |z i | < ρ F , i = 1, 2}, the biggest polydisk of center 0 contained in R 0 , whose radius depends on the vector field F. Trivially ρ F and d F are related by a coefficient depending only on the dimension: √ 2ρ F = d F . We can then prove where B(ω) is the value of the Bruno function [24] on ω.
We don't prove this Theorem being its proof very close to the one of Theorem 5.2 case 1), we only stress that the use of the continued fraction allows us to give an "optimal" counting Lemma as done in [5,9], which essentially bounds the number of lines on scale k in a rooted labeled tree of order N and total momentum ν ϑ , with ν ϑ q k , being (q k ) k the denominators of the convergent to ω.
In the case of analytic germs of (C, 0) Yoccoz [34] proved that the same bound holds from above for the radius of convergence of the linearization; the sophisticate techniques used in [26] would lead to prove: for some constants C > 1 and C ′ , we conjecture that one can take C = 1. We are not able to prove this fact but can prove that the power series obtained replacing the coefficients of the linearization with their absolute values is divergent whenever ω is not a Bruno number (a similar result has been proved in [34] Appendix 2 and in [5] paragraph 2.4 for germs).
Remark 5.6 (Ultradifferentiable vector fields of C 2 ). In the more general case of ultradifferentiable vector fields of C 2 we can improve Theorem 5.2 showing that we can linearize the vector field under weaker conditions. Theorem 5.7. Let ω > 0 and let (p k /q k ) k be its convergents. Let F be a vector field of the form (5.18) (without additional hypotheses on the coefficients f α ), let (M n ) n and (N n ) n be two sequences verifying conditions 0)-3) of section 4.3. Then 1. If moreover F belongs to C (Mn) and ω is a Bruno number then also the linearization h belongs to C (M k ) ∩V 1 . 2. If F is a germ of analytic diffeomorphisms of (C 2 , 0) and ω verifies: where κ(α) is the integer defined by: q κ(α) ≤ |α| < q κ(α)+1 , then the linearization h belongs to C (N k ) ∩V 1 . 3. If F ∈ C (M k ) ∩V 0 , the sequence (M k ) k is asymptotically bounded by the sequence (N k ) k (namely N k ≥ M k for all sufficiently large k) and ω verifies: where κ(α) is the integer defined by: q κ(α) ≤ |α| < q κ(α)+1 , then the linearization h belongs to C (N k ) ∩V 1 .
The proof follows closely the one of Theorem 5.2 and the weaker arithmetical condition are obtained using the "optimal" counting function as done in the proof of Theorem 5.5 and in [5,9].
α}, for α ≥ 0, are ideals of A v . I ′ 0 is the maximal ideal of A v and it is an open set in the topology induced by the ultrametric absolute value defined on k: Let n and m be positive integers, we consider the set of the formal power series with coefficients in k n in the m variables X 1 , . . . , X m , S n,m = k n [[X 1 , . . . , X m ]], F ∈ S n,m : with F α = (F α,1 , . . . , F α,n ) ∈ k n for all α ∈ N m . We will be interested in composition problems, so it is natural to set m = n and to define the composition of two elements F, G, with v(G) ≥ 1, as We will set S n,n = S n and we will introduce some definitions and properties of S n , but it's clear that they also hold on S n,m with some small changes.
1. If F is convergent in B(0, r) then there exists M > 0 such that 2. If there exists M > 0 such that (A.2) holds for all α ∈ N n , F converges in B 0 (0, r) and uniformly in B(0, r ′ ) for all 0 < r ′ < r. 3. LetF : B 0 (0, r) → k n denote the continuous function defined as the sum of the series F ∈ S n convergent in B 0 (0, r). ThenF ≡ 0 ⇐⇒ F = 0.
We can therefore identify a convergent power series F with its associated functioñ F and vice versa. Let U be an open set of k n ,G : U → k n is said to be analytic in U if for all x ∈ U there is a formal power series G ∈ S n and a radius r > 0 such that: r), and for all y ∈ B 0 (0, r),G (x + y) = G (y).
With a slight abuse of notation we will omit the superscript˜to distinguish analytic functions from convergent power series. If F is a convergent power series on B(0, r) we denote ||F || r = sup ||G (x + y) − G (x) − Ly|| ||y|| = 0.
Clearly if the limit exists then the derivative is unique and it will be denoted by dG (x). Let δ i = (0, . . . , 1, . . . , 0) ∈ k n the vector with 1 at the i-th place, we call the i-th partial derivative of G at x. Higher order derivatives are defined analogously.
Let G = G α X α be an element of S n , then G α = D α G(x) α!
It is not difficult to prove that any power series G ∈ S n convergent in B 0 (0, r) defines an analytic function G in B 0 (0, r). However one should be aware of the fact that in general the local expansion of a function G analytic on U at a point x ∈ U such that B 0 (x, r) ⊂ U does not necessarily converge on all of B 0 (0, r). This is true if one assumes k to be algebraically closed.
B.1. Proof of Lemma 4.9. Let us recall briefly the object of the Lemma. We are considering rooted labeled trees ϑ, any line produces a divisor and we want to count the number of lines producing "small divisors", i.e. the number of lines on scale k for some integer k. The way these small divisors accumulate give rise to the arithmetical condition needed to prove the convergence of the series involved. We can then prove Lemma 4.9 (Bruno's Counting Lemma). The number of lines on scale k in a rooted labeled tree verifies the following bound: where ⌊x⌋ denotes the integer part of the real number x. We recall that ν ϑ is the total momentum of the tree andν ϑ = ν ϑ − β ℓv 1 , being ℓ v1 the root line of ϑ.