Content-Type: multipart/mixed; boundary="-------------0110121122597" This is a multi-part message in MIME format. ---------------0110121122597 Content-Type: text/plain; name="01-370.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-370.keywords" Lagrange's formula, non--Archimedean fields, Siegel center problem, Bruno condition, Linearization of vector fields ---------------0110121122597 Content-Type: application/x-tex; name="carletti.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="carletti.tex" %&LaTeX \documentclass{amsart} \usepackage{amsmath,amsfonts} \usepackage{amscd,amsthm} \theoremstyle{plain} %% This is the default \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{notation}[theorem]{Notation} \renewcommand{\thenotation}{} \newcommand{\val}[2]{\mathop{ {\it Val}\left( #1 \right)\left( #2 \right) }} \newcommand{\vval}[2]{\mathop{ {\it Val}\left( #1,#2 \right)}} \newcommand{\vall}[3]{\mathop{ {\it Val}_{ #3 }\!\left( #1 \right)\left( #2 \right) }} \newcommand{\forest}[1]{\mathop{ \mathcal{T}_{#1}}} \newcommand{\R}{\mathop{\mathbb{R}}} \newcommand{\Q}{\mathop{\mathbb{Q}}} \newcommand{\Z}{\mathop{\mathbb{Z}}} \newcommand{\N}{\mathop{\mathbb{N}}} \newcommand{\C}{\mathop{\mathbb{C}}} \newcommand{\bigo}[1]{\mathop{\mathcal{O}(#1)}} \newcommand{\class}[1]{\mathop{\mathcal{C}_{(#1)}}} \renewcommand{\v}[2]{\mathop{ v_{ #2 }\left( #1 \right) }} \newcommand{\diag}[1]{\mathop{\it diag(#1_1,\dots,#1_n)}} \renewcommand{\subjclassname}{\textup{2000} Mathematics Subject Classification} \numberwithin{equation}{section} \begin{document} \title[Lagrange's inversion formula. Non--Analytic Linearization Problems.]{The Lagrange inversion formula on non--Archimedean fields. Non--Analytical Form of Differential and Finite Difference Equations.} \author{Timoteo Carletti} %\date{\today,file {\bf c.fin.21092001.tex}} \date{\today} \address[Timoteo Carletti]{Dipartimento di Matematica "U. Dini", viale Morgagni 67/A, 50134 Firenze, Italy} \email[Timoteo Carletti]{carletti@math.unifi.it} \subjclass{Primary 37F50, 34A25; Secondary 05C05, 32A05} \keywords{Lagrange's formula, non--Archimedean fields, Siegel center problem, Bruno condition, Linearization of vector fields} \begin{abstract} The classical formula of Lagrange for the inversion of analytic functions 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. In particular we will be interested in studying the Siegel center problem (Linearization of germs of functions near a fixed point) and the problem of conjugation of a vector fields with its linear part near a critical point. In addition to the analytic Siegel Center problem, 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 the Bruno condition is sufficient for the linearization to belong to the same Class of the germ. If one allows the linearization to be less regular than the germ new conditions are introduced, weaker than the Bruno condition. This generalizes to dimension $n\geq 1$ some results of~\cite{CarlettiMarmi}. A similar statement holds in the case of the linearization of ultradifferentiable vector fields. Moreover our formulation of the non--analytic Lagrange inversion formula by mean of the tree formalism, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. In the case of analytic vector fields of $\C^2$ we prove a quantitative estimate of a previous qualitative result of~\cite{YoccozPerezMarco}. \end{abstract} \maketitle \section{introduction} \indent 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 $k=\mathbb{R}$ or $\mathbb{C}$, the classical Lagrange inversion formula (see~\cite{Lagrange}\footnote{J. H. Lambert was the first interested in determining the roots $x$ of the equation $x^m+px=q$ developing it in an infinite series~\cite{Cajori}. His results stimulated J. Lagrange, who first generalized the method to solve the equation $a-x+\phi (x)=0$ for an analytic function $\phi$, and then he applied the idea to the Kepler problem: solving the elliptic motion of a point mass planet about a fixed point, according to the law of the inverse square. To do this Lagrange studied the possibility of inverting the fundamental relation between the mean anomaly, $M$, and the eccentric anomaly, $E$: $E=e\sin E + M$, being $e$ the eccentricity of the orbit.}, \cite{Dieudonne} chapter VIII, section 7 or~\cite{Sansone} p. 286, for the $1$--dimensional case, and to~\cite{Good} for the multidimensional case) says that if $G$ is an analytic function in a neighborhood of $w\in k$ then there exists a unique solution $h=H(u,w)$ of % $$h =u G \left( h \right) + w \, , \label{inversion}$$ % 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: % $$H\left(u,w\right)=w+\sum_{n \geq 1}\frac{u^n}{n!} \frac{d^{n-1}}{dw^{n-1}}(G\left( w \right))^n \, . \label{eq:lagrangeorig}$$ % \indent In sections~\ref{sec:anallag}, after recalling some elementary notions of theory of analytic functions on non--Archimedean fields, we give two generalizations of~\eqref{inversion}: in the $n$--dimensional vector space $k^n$, $k$ {\em non--Archimedean} when $G$ is an analytic function (Corollary~\ref{existence}), and for non--analytic $G$ (Theorem~\ref{nonanex}). To deal with this second case we rewrite the Lagrange inversion formula by means of the tree formalism. We refer to~\cite{Gessel} and references therein for a combinatorial proof of the Lagrange inversion formula using the tree formalism. \indent In sections~\ref{sec:someappl} and~\ref{sec:nonanvf} 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~\cite{Eliasson} 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, ~\cite{ChierchiaFalcolini,Gallavotti1,Gallavotti2} always in the context of Hamiltonian KAM theory, see also~\cite{BerrettiGentile1} which we take as reference for many definitions concerning trees). The fact that these formulas should be obtainable by a suitable generalization of Lagrange's inversion formula was first remarked by Vittot~\cite{Vittot}. When $k$ is the field of formal Laurent series ${\mathbb C}((z))$, we consider the vector space: $\mathbb{C}^n\left(\left(z_1,\ldots,z_n\right)\right)$; 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\cite{Du} and semilinear convolution equations. In particular we will study (section~\ref{sec:someappl}) the Siegel center problem~\cite{Herman,Yoccoz} for analytic and non--analytic germs of $(\mathbb{C}^n,0)$, $n \geq 1$, and (section~\ref{sec:nonanvf}) the Problem of linearization of analytic~\cite{Bruno} and non--analytic vector fields of $\C^n$, $n \geq 1$. The reader interested only in the Siegel center problem may find useful to assume Proposition~\ref{thecoefficients} and to skip the reading of the whole of sections~\ref{sec:anallag} and~\ref{sec:proofs}. The same is true for those interested in the linearization of vector fields, assuming Proposition~\ref{thecoefficientsvf} and reading the rest of section~\ref{sec:nonanvf}, even if they will find several useful definitions in section~\ref{sec:someappl}. In~\cite{CarlettiMarmi} authors began the study of the Siegel center problem in some ultradifferentiable algebras of $\mathbb{C}\left(\left(z\right)\right)$, here we generalize these results to dimension $n\geq 1$. Consider two Classes of formal power series $\mathcal{C}_1$ and $\mathcal{C}_2$ of $\C^n\left[\left[z_1,\dots,z_n\right]\right]$ closed with respect to the composition. For example the Class of germs of analytic functions of $(\C^n,0)$ or Gevrey--$s$ Classes, $s>0$ (i.e. series ${\bf F}=\sum_{\alpha\in\N^n}{\bf f}_{\alpha}z^{\alpha}$ such that there exist $c_1,c_2>0$ such that $|{\bf f}_{\alpha}|\leq c_1 c_2^{|\alpha|}(|\alpha|!)^s$, for all $\alpha\in\N^n$). Let $A\in GL(n,\C)$ and ${\bf F}\in \mathcal{C}_1$ such that ${\bf F}({\bf z})=A{\bf z}+\dots$, we say that ${\bf F}$ is {\em linearizable} in $\mathcal{C}_2$ if there exists ${\bf H}\in \mathcal{C}_2$, tangent to the identity, such that: % \begin{equation*} {\bf F}\circ {\bf H}({\bf z})={\bf H}(A{\bf z}) \end{equation*} % When $A$ is in the Poincar\'e domain, the results of Poincar\'e~\cite{Poincare1} and Koenigs~\cite{Koenigs} assure that ${\bf F}$ is linearizable in $\mathcal{C}_2$. When $A$ is in the Siegel domain, the problem is harder, the only trivial case is $\mathcal{C}_2=\C^n\left[\left[z_1,\dots,z_n\right]\right]$ ({\em formal linearization}) for which one only needs to assume $A$ to be {\em non--resonant}. In the analytic case we recover the results of Bruno~\cite{Bruno} and R\"ussmann~\cite{Russmann}, whereas in the non--analytic case new arithmetical conditions are introduced (Theorem~\ref{maintheorem}). Consider the general case where both $\mathcal{C}_1$ and $\mathcal{C}_2$ are different from the Class of germs of analytic function of $(\C^n,0)$, if one requires $\mathcal{C}_1=\mathcal{C}_2$, once again the Bruno condition is sufficient, otherwise if $\mathcal{C}_1\subset\mathcal{C}_2$ one finds new arithmetical conditions, weaker than the Bruno one. In section~\ref{sec:nonanvf} we will consider the following differential equation: % $$\dot {\bf z}=\frac{d{\bf z}}{dt}={\bf F}({\bf z}) \, , \label{leq:diffeqa}$$ % where $t$ is the time variable and ${\bf F}$ is a formal power series in the $n\geq 1$ variables $z_1,\dots , z_n$, with coefficients in $\C^n$, without constant term: ${\bf F}=\sum_{\alpha \in \N^n,|\alpha|\geq 1}{\bf F}_{\alpha}z^{\alpha}$, and we are interested in the behavior of the solutions near the singular point ${\bf z}=0$. A basic but clever idea has been introduced by Poincar\'e (1879), which consists in reducing the system~\eqref{leq:diffeqa} with an appropriate change of variables, to a simpler form: the {\em normal form}. In~\cite{Bruno} several results are presented in the analytic case (namely ${\bf F}$ is a convergent power series). Here we generalize such kind of results to the case of non--analytic ${\bf F}$, 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 ${\bf F}\in \mathcal{C}_1$ with a diagonal, non--resonant linear part, $A{\bf z}$, and we look for sufficient conditions on $A$ to ensure the existence of a change of variables ${\bf H}\in \mathcal{C}_2$ (the {\em 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, $\mathcal{C}_1\subset \mathcal{C}_2$, new arithmetical conditions, weaker than the Bruno one, are introduced (Theorem~\ref{the:mainvf}). Finally in the case of analytic vector field of $\mathbb{C}^2$, the use of the continued fraction and of a best description of the accumulation of small divisors (due to the Davie counting function~\cite{Davie}), allows us to improve (Theorem~\ref{the:mainvfn2}) the results of Theorem~\ref{the:mainvf}, 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~\cite{MatteiMoussu} and~\cite{YoccozPerezMarco}. 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 an inversion Lagrange formula (on some appropriate setting) we get the same functional equation to solve. {\it Acknwoledgements.} We are grateful to J.-C. Yoccoz for pointing out the difference between the analytic and non--analytic inversion problems and the relation of the latter with small divisors problems. We also thank A. Albouy for some details about the history of the Lagrange inversion formula, in particular for reference~\cite{Cajori}. \section{The Lagrange inversion formula on non--Archimedean fields} \label{sec:anallag} 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~\ref{ultrametricappendix} and to~\cite{Serre,Christol,Chenciner} for a more detailed discussion. We end the section introducing some elementary facts concerning trees. \subsection{Statement of the Problem} \label{subsect:ultramatric} \indent Let $(k,|\;|)$ be a non--Archimedean field of characteristic zero, where $|\;|$ is a ultrametric absolute value : $|x+y|\le \sup (|x|,|y|)$ for all $x,y\in k$. Moreover we assume that $k$ is complete and the norm is {\em non--trivial}. Let $a$ be a real number such that $00$ and let us consider a function $G: B_0(0,r)\subset k^n \rightarrow k^{n \times l}$, i.e. for $x \in B_0(0,r)$ and $\forall \; 1 \leq i \leq n ,1 \leq j \leq l$: % \begin{equation*} G \left( x \right) = \left( G_{ij}\left( x \right) \right)_{ij}, \quad G_{ij}\left( x \right) \in k. \end{equation*} % Given $w\in k^n$, $u \in k^l$ and $G$ as above, we consider the following problem: % \begin{quotation} Solve with respect to $h\in k^n$, the {\em multidimensional non--analytic Lagrange inversion problem}: % $$h = \Lambda \left[w + G\left( h \right) \cdot u\right] ,\label{multilagrange}$$ % where $\Lambda$ is a $k^n$--additive, $k^\prime$--linear, non--expanding operator (i.e. $\lvert \lvert \Lambda w \rvert \rvert \leq \lvert \lvert w \rvert \rvert$ for all $w\in k^n$). \end{quotation} % We will prove the existence of a solution of~\eqref{multilagrange} using \emph{trees}. We will now recall some elementary facts concerning trees; we refer to~\cite{Harary} for a more complete description. \subsection{The Tree formalism} \label{sssec:treeform} A tree is a connected acyclic graph, composed by {\em nodes}, and {\em lines} connecting together two or more nodes. Among trees we consider {\em rooted trees}, namely trees with an extra node, not included in the set of nodes of the tree, called the {\em earth}, and an extra line connecting the earth to the tree, the {\em root line}. We will call {\em root} the only node to which the earth is linked. The existence of the root introduces a {\em partial ordering} in the tree: given any two nodes\footnote{To denote nodes we will use letters: $u,v,w,\ldots$, with possible sub-indices. Lines will be denoted by $\ell$, the line exiting from the node $u$ will be denoted by $\ell_u$.} $v,v^{\prime}$, then $v\geq v^{\prime }$ if the (only) path connecting the root $v_1$ with $v^{\prime}$, contains $v$. The {\em order} of a tree is the number of its nodes. The {\em forest} $\mathcal{T}_N$ is the disjoint union of all trees~\footnote{Here we consider only \emph{semitopological} trees (see~\cite{BerrettiGentile2}), we refer to~\cite{Gallavotti1} for the definition of \emph{topological trees}.} with the same order $N$. The {\em 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 {\em end node}; for the root $v_{1}$, because the root line doesn't belong to the lines of the tree, we define $m_{v_1}=\deg v_1$, in this way $m_{v_1}$ also represents the number of lines entering in the root. Let $\vartheta$ be a rooted tree, for any $v \in \vartheta$ we denote by $L_v$ the {\em set of lines entering into $v$}; if $v$ is an end node we will set $L_v=\emptyset$. Given a rooted tree $\vartheta$ of order $N$, we can view it as the union of its root and the subtrees $\vartheta^i$ obtained from $\vartheta$ by detaching the root. Let $v_1$ be the root of $\vartheta$ and $t=m_{v_1}$, we define the {\em standard decomposition} of $\vartheta$ as: $\vartheta = \{ v_1 \} \cup \vartheta^1 \cup \dots \cup \vartheta^t$, where $\vartheta^i \in \mathcal{T}_{N_i}$ with $N_1+ \ldots +N_t=N-1$. Using the definition of $m_v$ we can associate uniquely to a rooted tree of order $N$ a vector of $\mathbb{N}^N$, whose components are just $m_v$ with $v$ in the tree~\cite{Vittot}. Thus $\mathcal{T}_N =\{ (m_1,\ldots,m_N) \in \mathbb{N}^N : \sum_{i=1}^N m_i=N-1, \quad \sum_{i=j}^N m_i \leq N-j \quad \forall j=1, \cdots ,N \}$. We can then rewrite the standard decomposition of $\vartheta$ as: $\vartheta = \left( t,\vartheta^1, \ldots , \vartheta^t \right)$ where the subtrees satisfy: $\vartheta^i \in \mathcal{T}_{N_i}$ with $N_1+ \ldots +N_t=N-1$. \begin{center} \begin{figure}[ht] \setlength{\unitlength}{4144sp}% % \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(1898,1343)(128,-519) \put(2026, 29){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}$\Leftrightarrow \quad (3,1,0,0,0)$}}} \put(406,-241){\makebox(0,0)[lb]{\smash{\SetFigFont{5}{6.0}{\rmdefault}{\mddefault}{\itdefault}1}}} \put(361,-196){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(901,254){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}$\ell$}}} \put(946,209){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(991,164){\makebox(0,0)[lb]{\smash{\SetFigFont{5}{6.0}{\rmdefault}{\mddefault}{\itdefault}2}}} \put(1171,-376){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1216,-421){\makebox(0,0)[lb]{\smash{\SetFigFont{6}{7.2}{\rmdefault}{\mddefault}{\itdefault}5}}} \put(1261,434){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}$\ell$}}} \put(1306,389){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1351,344){\makebox(0,0)[lb]{\smash{\SetFigFont{5}{6.0}{\rmdefault}{\mddefault}{\itdefault}3}}} \put(1171,749){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1216,704){\makebox(0,0)[lb]{\smash{\SetFigFont{6}{7.2}{\rmdefault}{\mddefault}{\itdefault}2}}} \put(1621,749){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1666,704){\makebox(0,0)[lb]{\smash{\SetFigFont{6}{7.2}{\rmdefault}{\mddefault}{\itdefault}3}}} \put(541,209){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(586,164){\makebox(0,0)[lb]{\smash{\SetFigFont{6}{7.2}{\rmdefault}{\mddefault}{\itdefault}1}}} \put(766,-376){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}$\ell$}}} \put(811,-421){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(856,-466){\makebox(0,0)[lb]{\smash{\SetFigFont{5}{6.0}{\rmdefault}{\mddefault}{\itdefault}5}}} \put(1396,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1441,119){\makebox(0,0)[lb]{\smash{\SetFigFont{6}{7.2}{\rmdefault}{\mddefault}{\itdefault}4}}} \put(1036,-61){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}$\ell$}}} \put(1081,-106){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\itdefault}v}}} \put(1126,-151){\makebox(0,0)[lb]{\smash{\SetFigFont{5}{6.0}{\rmdefault}{\mddefault}{\itdefault}4}}} \thinlines \put(1126,614){\circle*{90}} \put(1576,614){\circle*{90}} \put(181, 74){\circle{90}} \put(631, 74){\circle*{90}} \put(1126,-466){\circle*{90}} \put(1351, 74){\circle*{90}} \put(631, 74){\line( 1, 1){517.500}} \put(1126,614){\line( 1, 0){450}} \put(631, 74){\line(-1, 0){405}} \put(629, 71){\line( 1,-1){517}} \put(631, 74){\line( 1, 0){720}} \put(136,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}e}}} \put(316,-151){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\itdefault}$\ell$}}} \end{picture} \caption{A rooted tree of order $5$, $v_i$, $i=1,\ldots,5$, are the nodes whereas $\ell_{v_i}$, $i=1,\ldots,5$, are the lines. The earth is denoted by the letter $e$, $\ell_{v_1}$ is the root line and $v_3,v_4,v_5$ are end nodes. On the right we show the standard decomposition of this tree.} \label{figtree5} \end{figure} \end{center} \indent In the following we will also use {\em labeled rooted trees}. A labeled rooted tree of order $N$ is an element of $\mathcal{T}_N$ together with $N$ labels: $\alpha_1,\ldots, \alpha_N$. We can think that the label $\alpha_i$ 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 $\mathbb{Z}^m$ for some integer $m$. When needed we denote a labeled rooted tree of order $N$ with the couple $(\vartheta,\mathbf{\alpha})$, where $\vartheta \in \mathcal{T}_N$ and $\mathbf{\alpha}=(\alpha_1,\ldots,\alpha_N)$ is the vector label. \subsection{The non--analytic Lagrange inversion formula} We are now able to extend equation~\eqref{inversion}, the classical analytic Lagrange inversion formula, to the setting of paragraph~\ref{subsect:ultramatric}. We refer the reader to Appendix~\ref{ultrametricappendix} for a brief introduction to the theory of analytic functions on $k^n$ (norms, Cauchy estimates, etc ... ). Let $N\in \mathbb{N}^*$, $U$ and $V$ be open subsets of, respectively, $k^l$ and $k^n$, and $\vartheta \in \mathcal{T}_N$. We define the function ${\it V\!al}_{\Lambda}:\forest{N} \times U \times V \ni (\vartheta,u,w)\mapsto \vall{\vartheta}{u,w}{\Lambda}\in k^n$ as follows % $$\label{defvallambda} \vall{\vartheta}{u,w}{\Lambda}=\begin{cases} \Lambda \left( G\left( \Lambda w \right) \cdot u \right)& \text{if \vartheta\in\mathcal{T}_1}\\ \frac{1}{t!}\Lambda \left[ d^{t}G\left( \Lambda w \right) \left( \vall{\vartheta^1}{u, w}{\Lambda}, \ldots , \vall{\vartheta^t}{u, w}{\Lambda} \right) \cdot u \right] & \text{otherwise} \end{cases}$$ % where $\vartheta=(t,\vartheta^1,\ldots,\vartheta^t)$ is the standard decomposition of the tree and $\Lambda :k^n\rightarrow k^n$. \begin{remark} \label{rem:different} For $t\geq 1$ and $v^{\left( 1 \right)}, \dots,v^{\left( t \right)}\in k^n$ we recall that % \begin{equation*} d^{t}G_{ij}\left( w \right) \left( v^{\left( 1 \right)}, \dots, v^{\left( t \right)} \right)=\sum_{l_1,\dots,l_t=1}^n D_{l_1}\dots D_{l_t} G_{ij}\left( w \right) v^{\left( 1 \right)}_{l_1} \dots v^{\left( t \right)}_{l_t}, \end{equation*} with $v^{(i)}=(v_1^{(i)},\ldots,v_n^{(i)})$ and $D_{l_i}$ are the $l_i$--th partial derivatives of $G$ at $w$ (see Appendix~\ref{ultrametricappendix}). % \end{remark} \indent We can then state the following existence Theorem: \begin{theorem}[Non--analytic case] \label{nonanex} Let $n,l$ be positive integers. Let $u\in k^l$, $w \in k^n$ and for $1 \leq i \leq n$ let $G_i=\left( G_{i1}, \dots ,G_{il}\right)$, with $G_{i}\in k^n \left[ \left[ X_1, \dots, X_n \right] \right]$. Let $\Lambda:k^n\rightarrow k^n$ be a $k^n$--additive, $k^{\prime}$--linear and non--expanding operator. Assume that for $\left( r_1, \dots ,r_n \right)\in \R_+^n$ the $G_i$'s are convergent in $B\left( 0,r_i \right)$, set\footnote{See Appendix\ref{ultrametricappendix} for the definition $||\cdot||_{r}$.} $M_i=||G_i||_{r_i}>0$, $r=\min_i r_i$ and $M=\max_i M_i$. Then equation~\eqref{multilagrange} has the unique solution $$\label{nonansol} H\left( u,w \right)=\Lambda w+ \sum_{N\geq 1}\sum_{\vartheta \in \mathcal{T}_{N}} \vall{\vartheta}{u,w}{\Lambda}\, ,$$ % where ${\it Val}_{\Lambda}: \mathcal{T}_N\times B_{0}(0,r/M)\times B_{0}(0,r) \rightarrow B_{0}(0,r)$, has been defined in (\ref{defvallambda}). Moreover for any fixed $\vartheta$ the function ${\it Val}_{\Lambda}$ is continuous, series~\eqref{nonansol} converges on $B_{0}(0,r/M)\times B_{0}(0,r)$ and the map $(u,w)\mapsto H(u,w)\in B_{0}(0,r)$ is continuous. \end{theorem} \begin{remark} Since $\Lambda$ is not $k^n$--linear, ${\it Val}_{\Lambda}$ and $H$ cannot be analytic. However the non--expanding condition implies that $\Lambda$ is Lipschitz continuous from which the regularity properties of ${\it Val}_{\Lambda}$ and $H$ follow. \indent If $\Lambda$ is $k^n$--linear then we have the following Corollary (particular case of the previous Theorem) with $w^\prime$ instead of $\Lambda w$ and $G^\prime\cdot u^\prime$ instead of $\Lambda (G\cdot u)$, which extends the analytic Lagrange inversion formula~\eqref{inversion}. \end{remark} \begin{corollary}[Analytic case] \label{existence} Let $n,l$ be positive integers. Let $u^\prime\in k^l$, $w^\prime \in k^n$ and for $1 \leq i \leq n$ let $G^\prime_i=\left( G^\prime_{i1}, \dots ,G^\prime_{il}\right)$, with $G^\prime_{i}\in k^n \left[ \left[ X_1, \dots, X_n \right] \right]$. Assume that for $\left( r_1, \dots ,r_n \right)\in \R_+^n$ the $G^\prime_i$'s are convergent in $B\left( 0,r_i \right)$, let $M_i=||G^\prime_i||_{r_i}>0$, $r=\min_i r_i$ and $M=\max_i M_i$. Then % $$\label{solution} H\left( u^\prime,w^\prime \right)=w^\prime+ \sum_{N\geq 1}\sum_{\vartheta \in \mathcal{T}_{N}} \val{\vartheta}{u^\prime,w^\prime}\, ,$$ % is the unique solution of~\eqref{multilagrange} with $\Lambda (G\cdot u)=G^{\prime}\cdot u^\prime$ and $\Lambda w=w^{\prime}$. The function $\val{\vartheta}{u^\prime,w^\prime}$ is nothing else that the function $\vall{\vartheta}{u,w}{\Lambda}$ with $\Lambda (G \cdot u)= G^\prime \cdot u^\prime$ and $\Lambda w=w^\prime$. Moreover $H$ is analytic in $B_0\left( 0,\frac{r}{M} \right) \times B_0\left( 0,r \right)$. \end{corollary} \section{Proofs.} \label{sec:proofs} This section is devoted to the proof of Theorem~\ref{nonanex} and Corollary~\ref{existence}. \noindent{\em Proof of Theorem~\ref{nonanex}.} Using the fact that $\Lambda$ is non--expanding the uniqueness of the solution can be proved easily. Let $H_1$ and $H_2$ be two solutions of~\eqref{multilagrange}, then % \begin{equation*} ||H_1-H_2||=||\Lambda \left[ \left( G\left( H_1 \right) -G\left( H_2 \right) \right) \cdot u \right]|| \leq ||\left( G\left( H_1 \right) -G\left( H_2 \right) \right) \cdot u||, \end{equation*} but for all $i=1,\ldots,n$: $$||\left( G\left( H_1 \right) -G\left( H_2 \right) \right) \cdot u||\leq ||G_i\left( H_1 \right) -G_i\left( H_2 \right)|| \, ||u||, \label{eq:firststep}$$ and by Proposition~\ref{powerseries} $||G_i\left( H_1 \right) -G_i\left( H_2 \right)||\leq \frac{||G_i||_{r_i}}{r_i}||H_1-H_2||$. Then setting $\mu=||u|| \frac{\max M_i}{\min r_i}$, from~\eqref{eq:firststep} we conclude that \begin{equation*} ||H_1-H_2||\leq \mu ||H_1-H_2||. \end{equation*} By hypothesis $||u|| <\frac r M$, then $\mu <1$, from which we conclude that $H_1=H_2$. \indent We now prove existence. Since $G_i$ are convergent, $D^{\alpha}G_i$ also are convergent and Proposition~\ref{Taylorformulas} gives the following estimate % $$||D^{\alpha}G_i||_{r_i}\leq \frac{M_i}{r_i^{\lvert \alpha \rvert}}, \quad \forall \alpha \in \mathbb{N}^n \label{eq:estimcauchy} \, ,$$ % which together with the non--expanding property of $\Lambda$ allows us to prove that for all $N \geq 1$ and all $\vartheta \in \mathcal{T}_{N}$: % \begin{equation*} \Big\lvert \Big\lvert \vall{\vartheta}{u,w}{\Lambda} \Big\rvert \Big\rvert \leq ||u||^{N}\frac{M^N}{r^{N-1}} \, . \end{equation*} % Let us define $H^{(0)}\left( u,w \right)=\Lambda w$ and $H^{(j)}\left( u,w \right)=\Lambda w+\sum_{N=1}^j \sum_{\vartheta \in \mathcal{T}_{N}}\vall{\vartheta}{u,w}{\Lambda}$, clearly $H^{(j)}\rightarrow H$ as $j \rightarrow \infty$ and it is easy to check that: % \begin{equation*} \Big\lvert \Big\lvert H^{(j)}-\Lambda \left[ w-G\left( H^{(j)} \right) \cdot u \right]\Big\rvert\Big\rvert \leq r \left(||u||\frac{M}{r}\right)^{j+1} \end{equation*} % which tends to $0$ as $j \rightarrow \infty$. \endproof We give now the proof of Corollary~\ref{existence}. This one follows closely the one of Theorem~\ref{nonanex} in particular the uniqueness statement, so we will outline only the main differences w.r.t to the previous proof. \proof The hypothesis on $G^{\prime}_i$ gives an estimate similar to~\eqref{eq:estimcauchy}, then by induction on $N$ it is easy to prove that for all $N\geq 1$ and $\vartheta \in \mathcal{T}_{N}$ one has % \begin{equation*} \Big\lvert\Big\lvert\val{\vartheta}{u^{\prime},w^{\prime}}\Big\rvert\Big\rvert\leq ||u^{\prime}||^{N}\frac{M^N}{r^{N-1}}. \end{equation*} % Then if $||u^{\prime}||<\frac{r}{M}$ series~\eqref{solution} converges and if $||w^{\prime}||\leq r$, $H\left(u^{\prime},w^{\prime} \right) \in B_0 \left( 0, r \right)$, in fact % \begin{equation*} ||H||=\Big\lvert\Big\lvert w^{\prime}+\sum_N \sum_{\vartheta \in \mathcal{T}_{N}} \val{\vartheta}{u^{\prime},w^{\prime}}\Big\rvert\Big\rvert \leq \sup \Big \{ ||w^{\prime}||,\sup_{N,\vartheta \in \mathcal{T}_{N}} \Big\lvert\Big\lvert \val{\vartheta}{u^{\prime},w^{\prime}}\Big\rvert\Big\rvert \Big \} < r. \end{equation*} % Now introducing $H^{(0)}\left( u^{\prime},w^{\prime} \right)=w$ and $H^{(j)}\left( u^{\prime},w^{\prime} \right)=w^{\prime}+\sum_{N=1}^j \sum_{\vartheta \in \mathcal{T}_{N}}\val{\vartheta}{u^{\prime},w^{\prime}}$, one can easily prove that $H^{(j)}\rightarrow H$ as $j \rightarrow \infty$. \endproof \begin{remark} In the simplest case $n=l=1$, namely $u,w\in k$ and $G\in k\left[\left[ X \right]\right]$, the solution given by~\eqref{solution} coincides with the classical one of Lagrange~\eqref{eq:lagrangeorig}. One can prove this fact either using the uniqueness of the Taylor development or by direct calculation showing that for all positive integer $N\geq 1$ we have \begin{equation*} \sum_{\vartheta \in \mathcal{T}_{N}}\val{\vartheta}{u,w} =\frac{u^N}{N!}\frac{d^{N-1}}{dw^{N-1}}\left[ G(w)\right]^N . \end{equation*} In the other cases formula~\eqref{solution} is the natural generalization of~\eqref{eq:lagrangeorig}. %When $n\geq 1$ and $l\geq 1$ the combinatorial aspects of the %solution become more complicate because of the number of %dimensions (take care of the scalar products) and the number of %variables (take care of the various possible derivatives). The %tree structure takes care of all this amount of new %difficulties'' allowing us to write the general solution in the %same way of the classical one (when rewritten in terms of trees). \end{remark} \begin{remark} \label{rem:analytic} Series~\eqref{solution} is an analytic function of $u,w$, but it is not explicitly written as $u$--power series. We claim that introducing {\em labeled rooted trees} we can rewrite~\eqref{solution} explicitly as a $u$--power series. %Consider the simplest non--trivial case (to avoid unnecessary %complications due to extra--indices): $h,w,u \in k^n$, $G\in %k\left[\left[X_1,\ldots,X_n\right]\right]$ and $G$ convergent on %some $B(0,r)$. Let $\vartheta$ be a rooted tree of order $N$, and %let $(m_1,\ldots,m_N)$ be its standard decomposition; to the i--th %node, $v_i$, we associate a label %$p^{(i)}=(p^{(i)}_1,\ldots,p^{(i)}_n)\in \mathbb{N}^n$ such that %$|p^{(i)}|=p^{(i)}_1+\ldots+p^{(i)}_n=m_i$. Let us introduce the %{\em total vector label} $P=p^{(1)}+\ldots+p^{(N)}\in %\mathbb{N}^n$, then we claim that this labeled rooted tree %contributes to the solution $H(u,w)$ with a term proportional to %\begin{equation*} %u u^{P} \frac{D^{p^{(1)}}G(w)}{p^{(1)}!}\dots %\frac{D^{p^{(N)}}G(w)}{p^{(N)}!}, %\end{equation*} %where, setting $u=(u_1,\ldots,u_n)$ and $P=(P_1,\ldots,P_n)$, we %used the notation: $u^{P}=u_1^{P_1}\dots u_n^{P_n}$, %$p^{(i)}!=p^{(i)}_1! \dots p^{(i)}_N!$ and %$D^{p^{(i)}}G(w)=D_1^{p_1^{(i)}}\ldots D_n^{p_n^{(i)}}G(w)$. %Because $|P|=m_1+\ldots +m_N=N-1$ this term is of order $||u||^N$. %To be more precise we introduce the set of labeled rooted trees %of order $N$ and total weight $K$: $\mathcal{T}_{N,K}$, namely the %set of couple $(\vartheta,\mathbf{p})$, where $\vartheta \in %\mathcal{T}_N$ and $\mathbf{p}=(p^{(1)},\ldots,p^{(N)})$, %$p^{(i)}\in\mathbb{N}^n$ such that $|p^{(i)}|=m_i$ and %$|p^{(1)}|+\ldots +|p^{(N)}|=K$. Then introducing the function on %labeled rooted trees: %$$%\label{eq:defvallab} %\vval{\vartheta}{\mathbf{p}}=\frac{D^{p^{(1)}}G(w)}{p^{(1)}!} %\dots \frac{D^{p^{(N)}}G(w)}{p^{(N)}!}\, , %$$ %we can prove that the solution $H(u,w)$ of the Lagrange inversion %problem~\eqref{solution} can be rewritten as follows: % %$$%\label{eq:relagrange} H(u,w)=w+\sum_{N\geq 1} %\sum_{\substack{(\vartheta,\mathbf{p}) %\in\mathcal{T}_{N,N-1}\\\mathbf{p}=(p^{(1)},\ldots,p^{(N)})}}u %u^{p^{(1)} + \ldots +p^{(N)}} \vval{\vartheta}{\mathbf{p}}\, . %$$ % \end{remark} \section{The Non--analytic Siegel center problem.} \label{sec:someappl} In this part we show that the problem of the conjugation a (formal) germ of a given function with its linear part near a fixed point (the so called {\em Siegel Center Problem}) can be solved applying Theorem~\ref{nonanex} 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). \subsection{Notations and Statement of the Problem} \label{subsec:siegeln} Let $\alpha =(\alpha_1,\dots,\alpha_n)\in \N^n$, $\lambda=(\lambda_1,\dots,\lambda_n)\in \C^n$ with $\lambda_i \neq \lambda_j$ if $i\neq j$, we will use the compact notation $\lambda^{\alpha}=\lambda_1^{\alpha_1}\dots \lambda_n^{\alpha_n}$ and $|\alpha|=\alpha_1+\dots+\alpha_n$; we will denote the diagonal $n\times n$ matrix with $\lambda_i$ at the $(i,i)$--th place with $\diag{\lambda}$. Let $V=\C^n\left[\left[z_1,\dots,z_n\right]\right]$ be the vector space of the formal power series in the $n$ variables $z_1,\dots,z_n$ with coefficients in $\mathbb{C}^n$: ${\bf f}\in V$ if % \begin{equation*} {\bf f}=\sum_{\alpha \in \mathbb{N}^n} {\bf f}_{\alpha}z^{\alpha} ,\quad {\bf f}_{\alpha}\in \mathbb{C}^n \text{ and } z^{\alpha}=z_1^{\alpha_1}\dots z_n^{\alpha_n} \; \forall \; \alpha =(\alpha_1,\dots,\alpha_n)\in \mathbb{N}^n \, . \end{equation*} % We consider $V$ endowed with the ultrametric absolute norm induced by the ${\bf z}$--adic valuation (${\bf z}=(z_1,\dots,z_n)$): $||{\bf f}||=2^{-v({\bf f})}$, where $v({\bf f})=\inf \{|\alpha|, \alpha \in \mathbb{N}^n: {\bf f}_{\alpha}\neq 0\}$, and for any positive integer $j$ we denote by $V_j = \{{\bf f}\in V :v({\bf f})> j\}$. Let $\mathcal{C}$ be a Class (that we will define later, see paragraph~\ref{classiMn}) 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 diffeomorphism of $(\mathbb{C}^n,0)$ or Gevrey--$s$ Classes, in fact we will see that our classes will contain these special cases. Let $A\in GL(n,\mathbb{C})$ and assume $A$ to be diagonal~\footnote{The case $A$ non--diagonal need some special attentions, see~\cite{Herman} Proposition 3 page 143 and~\cite{Yoccoz} Appendix 1.} with all the eigenvalues distinct. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two classes as stated before, then the {\em Siegel center problem} can be formulated as follow~\cite{Herman,CarlettiMarmi}: % \begin{quotation} Let ${\bf F}({\bf z})=A{\bf z}+{\bf f}({\bf z})\in \mathcal{C}_1$, ${\bf f}\in \mathcal{C}_1\cap V_1$, find necessary and sufficient conditions on $A$ to {\em linearize} in $\mathcal{C}_2$ ${\bf F}$, namely find ${\bf H}\in \mathcal{C}_2\cap V_0$ (the {\em linearization}) solving: % $$\label{eq:siegelformal} {\bf F}\circ {\bf H}({\bf z})={\bf H}(A{\bf z}) \, .$$ % \end{quotation} % Let $A=\diag{\lambda}$ we introduce the operator $D_{\lambda}:V\rightarrow V$: % $$\label{eq:defoperator} D_{\lambda}{\bf g}({\bf z})={\bf g}(A{\bf z})-A{\bf g}({\bf z}) \, ,$$ for any ${\bf g}({\bf z})\in V$. We remark that the action of $D_{\lambda}$ on the monomial ${\bf v}z^{\alpha}$, for any ${\bf v}\in\C^n$ and any $\alpha \in\N^n$, is given by: % $$\label{eq:actionmonomial} D_{\lambda}{\bf v}(z^{\alpha})=(\Omega_{\alpha}{\bf v})z^{\alpha} \, ,$$ % where the matrix $\Omega_{\alpha}$ is defined by $\Omega_{\alpha}= {\it diag}(\lambda^{\alpha}-\lambda_1,\dots,\lambda^{\alpha}-\lambda_n)$ and $(\Omega_{\alpha}{\bf v})=\sum_{i=1}^{n}(\lambda^{\alpha}-\lambda_i)v_i$ is the matrix--vector product. Let $A=\diag{\lambda}\in GL(n,\mathbb{C})$, we say that $A$ is {\em resonant} if there exist $\alpha \in\N^n$, $|\alpha|\geq 2$ and $j\in \{ 1,\dots,n\}$ such that: $$\label{eq:resonant} \lambda^{\alpha}-\lambda_j = 0 \, .$$ If~\eqref{eq:resonant} doesn't hold, we say that $A$ is {\em non--resonant}. If $A$ is non--resonant then $D_{\lambda}$ is invertible on $V_1$ (namely $|\alpha| \geq 2$), it is non--expanding ($||D_{\lambda}{\bf g}||\leq ||{\bf g}||$), and clearly $V$--additive ($D_{\lambda}({\bf f+g})=D_{\lambda}{\bf f}+D_{\lambda}{\bf g}$) and $\C$--linear. Then we claim that the Siegel center problem~\eqref{eq:siegelformal} is equivalent to solve the functional equation: % $$\label{eq:siegelsecvers} D_{\lambda}{\bf h}={\bf f}\circ ( {\bf z}+{\bf h})$$ % where ${\bf z}$ is the identity formal power series, ${\bf f}\in \mathcal{C}_1\cap V_1$ and ${\bf h}\in \mathcal{C}_2\cap V_1$. In fact from~\eqref{eq:siegelformal} we see that the linear part of ${\bf H}$ doesn't play any role, so we can choose ${\bf H}$ tangent to the identity (this normalization assures the uniqueness of the linearization): ${\bf H}({\bf z})={\bf z}+{\bf h}({\bf z})$, with ${\bf h}\in \mathcal{C}_2\cap V_1$. But then~\eqref{eq:siegelformal} can be rewritten as: % $$\label{eq:siegelstep1} {\bf h}(A{\bf z})-A{\bf h}({\bf z})={\bf f}\circ ({\bf z} + {\bf h}({\bf z})) \, ,$$ % and replacing the left hand side with the operator $D_{\lambda}{\bf h}$ we obtain~\eqref{eq:siegelsecvers}. \indent Given ${\bf f}\in V_1$ we consider the function $G_{\bf f}({\bf g})={\bf f}\circ ({\bf z} +{\bf g})$, for all ${\bf g}\in V_1$, assume that we can invert the operator $D_{\lambda}$ (because $v({\bf f}({\bf z} +{\bf g}))\geq v({\bf f})$ we can invert $D_{\lambda}$ whenever ${\bf f}\in V_1$), we can rewrite~\eqref{eq:siegelsecvers} as: % $$\label{eq:siegeln} {\bf h} = D_{\lambda}^{-1} \left(G_{\bf f}({\bf h})\right) \, ,$$ which is a particular case of the non--analytic multidimensional Lagrange inversion formula~\eqref{multilagrange} with $u=1$, $w=0$ and $\Lambda=D^{-1}_{\lambda}$. The following Lemma assures that $G_{\bf f}$ verifies the hypotheses of Theorem~\ref{nonanex}. \begin{lemma} \label{lem:gfv}Given ${\bf f}\in V_1$ the composition ${\bf f}\circ ({\bf z} +{\bf v})$ defines a power series $G_{\bf f}\in V_1\left[\left[ v_1,\dots , v_n\right]\right]$ convergent on $B(0,1/2)=V_0$ and % $$\label{eq:lemmahypot} G_{\bf f}({\bf v})=\sum_{\beta \in\mathbb{N}^n}{\bf g}_{\beta}({\bf f})v^{\beta}$$ % where (if we define ${\bf f}_{\beta}=0$ for $|\beta|=0$ and $|\beta|=1$) series ${\bf g}_{\beta}({\bf f})\in V$ are given by % $$\label{eq:defofg} {\bf g}_{\beta}({\bf f})({\bf z})=\sum_{\alpha \in\mathbb{N}^n}{\bf f}_{\alpha +\beta} \binom{\alpha +\beta}{\beta}z^{\alpha}$$ % Here we used the compact notations $\alpha !=\alpha_1 !\dots \alpha_n!$ and $\binom{\alpha +\beta}{\beta}=\frac{(\alpha + \beta)!}{\alpha !\beta !}$, for $\alpha=(\alpha _1,\dots,\alpha_n)\in \mathbb{N}^n$ and $\beta \in \mathbb{N}^n$. Moreover one has % $$\label{eq:otherforg} ||{\bf g}_{0}||=||{\bf f}||, \, ||{\bf g}_{\beta}||=2||{\bf f}|| \text{ for any |\beta|=1 and } \, ||{\bf g}_{\beta}||\leq 4||{\bf f}|| \text{ for any |\beta|\geq 2};$$ % thus % $$\label{eq:lastforG} ||G_{\bf f}||_{1/2}\leq ||{\bf f}||.$$ % \end{lemma} The proof is straightforward and we omit it. We can thus apply Theorem~\ref{nonanex} to solve~\eqref{eq:siegeln} with $u=1$, $w=0$, $G=G_{\bf f}$, $\Lambda =D^{-1}_{\lambda}$, $r=1/2$ and $M=1/4$, and the unique solution of~\eqref{eq:siegeln} is given by % $$\label{eq:solution} {\bf h}=\sum_{N\geq 1}\sum_{\vartheta \in \forest{N}}\vall{\vartheta}{1,0}{D^{-1}_{\lambda}}.$$ % An explicit expression for the power series coefficients of ${\bf h}$ can be obtained introducing {\em labeled rooted trees} (see Remark~\ref{rem:analytic}). Let us now explain how to do this. Let $N\geq 1$ and let $\vartheta$~\label{pg:label} be a rooted tree of order $N$; to any $v\in \vartheta$ we associate a {\em (node)--label} $\alpha_v \in \N^n$ s.t. $|\alpha_v|\geq 2$ and to the line $\ell_v$ (exiting w.r.t. the partial order from $v$) we associate a {\em (line)--label} $\beta^{\ell_v}\in \N^n$ s.t. $|\beta^{\ell_v}|=1$. We define $\beta_v =\sum_{\ell \in L_v}\beta^{\ell}$ (so $\beta_v \in \N^n$ and $|\beta_{v}|=m_v$) and the {\em momentum} flowing through the line $\ell_v$: $\nu_{\ell_v}=\sum_{w\in \vartheta :w\leq v}(\alpha_w -\beta_w)$. When $v$ will be root of the tree, we will also use the symbol $\nu_{\vartheta}$ (the {\em total momentum} of the tree) instead of $\nu_{\ell_{v}}$. 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 predecessor of $v$, $v > v_i$, we have: $|\nu_v| > |\nu_{v_i}|$). Recalling that the order of $\vartheta$ is $N$ we have: $|\nu_{\vartheta}|\geq N+1$. Let $N\geq 1$, $j\in \{1,\dots,n \}$ and $\alpha \in \N^n$, we finally define $\mathcal{T}_{N,\alpha ,j}$ the {\em forest of rooted labeled trees} of order $N$ with total momentum $\nu_{\vartheta}=\alpha$ and $\beta^{\ell_{v_1}}=e_j$ (being $e_j$ the vector with all zero entries but the $j$--th which is set equal to $1$) for the root line $\ell_{v_1}$. We are now able to prove the following \begin{proposition} \label{thecoefficients} Equation~\eqref{eq:siegeln} admits a unique solution ${\bf h}\in V_1$, ${\bf h}=\sum_{\alpha \in \N ^n}{\bf h}_{\alpha}z^{\alpha}$. For $|\alpha |\ge 2$ the $j$--th component of the coefficient ${\bf h}_{\alpha}$ is given by~\footnote{Compare this expression with equation (3.7) of Proposition 3.1 in~\cite{ChierchiaFalcolini}.}: % $$\label{coefficients} h_{\alpha ,j }=\sum_{N=1}^{|\alpha |-1}\sum_{\vartheta\in \mathcal{T}_{N,\alpha ,j}} ((\Omega^{-1}_{\nu_{\ell_{v_1}}} {\bf f}_{\alpha_{v_1}})\cdot\beta^{\ell_{v_1}}) \prod_{v\in \vartheta } \binom{\alpha_v}{\beta_v} \prod_{\ell_w \in L_v}((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot\beta^{\ell_w})\, ,$$ % where the last product has to be set equal to $1$ whenever $v$ is an end node ($L_v=\emptyset$). \end{proposition} % \begin{remark} By definition $\beta^{\ell_w}$, for any $w \in\vartheta$, has length $1$, so it coincides with an element of the canonical base. Then for $w\in\vartheta$ and any choice of the labels, such that $\beta^{\ell_w}=e_i$, the term $((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot\beta^{\ell_w})$ is nothing else that % \begin{equation*} ((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot \beta^{\ell_w})=\frac{1}{\lambda^{\nu_{\ell_w}}-\lambda_i}f_{\alpha_w,i} \, . \end{equation*} % \end{remark} % \begin{remark}\label{rem:negativecomp}Even if all the nodes labels $\alpha_v$ have non--negative components, the momenta can have (several) negative components. More precisely, let $\vartheta \in \forest{N}$, if the order of the tree if big enough w.r.t. the dimension $n$ ($N\geq n$) then $\nu_{\vartheta}$ can have $n-1$ negative components and their sum can be equal to $1-N$, but we always have $|\nu_{\vartheta}|\geq N+1$. This fact reminds the definitions of the sets ${\bf N}_i$, ${\bf N}^{(m)}$ and ${\bf N}^{(m)}_+$ of~\cite{Bruno}.\end{remark} \proof Let $\vartheta \in \mathcal{T}_{N}$, for $N\ge 1$ and $\alpha \in \N^n$ such that $|\alpha|\geq N+1$. Let us define ${\it Val}(\vartheta)= ((\Omega^{-1}_{\nu_{\ell_{v_1}}} {\bf f}_{\alpha_{v_1}})\cdot \beta^{\ell_{v_1}}) \prod_{v\in \vartheta } \binom{\alpha_v}{\beta_v} \prod_{\ell_w \in L_v}((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot \beta^{\ell_w})$ and % $$\label{eq:firsteqa} h^{\vartheta}_{|\alpha|,N,j}=\sum_{|\nu_{\vartheta}|=|\alpha|}{\it Val}(\vartheta) \, ,$$ % namely for a fixed tree, sum over all possible labels $\alpha_{v_i}$ and $\beta^{\ell_{v_i}}$, with $v_i$ in the tree, in such a way that the total momentum is fixed to $\alpha$ and the root line has label $\beta^{\ell_{v_1}}=e_j$. It is clear that % $$\label{eq:seceqa} {\bf h}=\sum_{|\alpha |\geq 2}z^{\alpha}\sum_{N=1}^{|\alpha|-1}\sum_{\vartheta \in \forest{N}}{\bf h}^{\vartheta}_{|\alpha|,N} \, ,$$ % being ${\bf h}^{\vartheta}_{|\alpha|,N}$ the vector whose $j$--th component is $h^{\vartheta}_{|\alpha|,N,j}$. Let ${\bf h}^{\vartheta}_{N+1}=\sum_{|\alpha |\geq N+1}z^{\alpha}{\bf h}^{\vartheta}_{|\alpha|,N}$, clearly $||{\bf h}^{\vartheta}_{N+1}||\leq 2^{-(N+1)}$ and % $$\label{eq:tereqa} {\bf h}=\sum_{N\geq 1}\sum_{\vartheta \in \forest{N}}{\bf h}^{\vartheta}_{N+1} \, .$$ % Actually in~\eqref{eq:seceqa} ${\bf h}$ is ordered with increasing powers of ${\bf z}$ whereas in~\eqref{eq:tereqa} with increasing order of trees. Convergence in $V_1$ for~\eqref{eq:tereqa} is assured from the estimate $||{\bf h}^{\vartheta}_{N+1}||\leq 2^{-(N+1)}$ and uniform convergence assures that~\eqref{eq:seceqa} and~\eqref{eq:tereqa} coincide. We claim that by induction on the order of the tree, we can prove that for all $N\geq 1$ and all $\vartheta \in \forest{N}$ : % $$\label{eq:induct} {\bf h}^{\vartheta}_{N+1}=\vall{\vartheta}{1,0}{D^{-1}_{\lambda}}$$ % where $\vall{\vartheta}{1,0}{D^{-1}_{\lambda}}$ has been defined in~\eqref{defvallambda}, taking $\Lambda =D^{-1}_{\lambda}$ and $G=G_{\bf f}$, thus establishing the equivalence of~\eqref{coefficients} and~\eqref{eq:solution}. \endproof \subsection{Some known results} \label{subsec:soemknownres} If both Classes $\mathcal{C}_1$ and $\mathcal{C}_2$ are $V_1$ (which verify the hypotheses of stability w.r.t. the derivation, closeness w.r.t. the composition, and the formal Taylor series makes sense), then the {\em Formal} Siegel Center Problem has a solution if the linear part of ${\bf F}$ is non--resonant. A matrix $A=\diag{\lambda}\in GL(n,\mathbb{C})$, is in the {\em Poincar\'e domain} if % $$\label{eq:poincaredomain} \sup_{1\leq j\leq n}|\lambda_j|<1 \quad \text{or} \quad \sup_{1\leq j\leq n}|\lambda_j^{-1}|>1 \, ,$$ % if $A$ doesn't belong to the Poincar\'e domain it will be in the {\em Siegel domain}. In the {\em Analytic} case (both $\mathcal{C}_1$ and $\mathcal{C}_2$ are the ring of the germs of analytic diffeomorphism of $(\mathbb{C}^n,0)$) , let $A$ be the derivative of ${\bf F}$ at the origin, then if $A$ is non--resonant and it is in the {\em Poincar\'e domain}, the Analytic Siegel Center Problem has a solution~\cite{Poincare1,Koenigs} (see also~\cite{Herman} and references therein). Moreover if $A$ is resonant and in the Poincar\'e domain, but ${\bf F}$ is formally linearizable, then ${\bf F}$ 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 $p\in \N$, $p\geq 2$, and let us define % $$\label{eq:piccolidivisori} \Tilde\Omega (p)=\min_{1\leq j \leq n}\inf_{\alpha \in \Z^n: |\alpha|< p} |\lambda^{\alpha}-\lambda_j| \, ,$$ % we remark that even if $A$ is non--resonant, but in the Siegel domain, one has $\lim_{p\rightarrow \infty}\Tilde\Omega (p)=0$; this is the so called {\em small divisors problem}, the main obstruction to the solution of equation~\eqref{eq:siegelformal}. A non--resonant matrix $A$ verifies a {\em Bruno condition}~\footnote{See~\cite{Bruno,Russmann} for various equivalent formulations of this condition.} if there exists an increasing sequence of natural numbers $(p_k)_k$ such that % $$\label{eq:brunocondition} \sum_{k=0}^{+\infty} \frac{\log \Tilde{\Omega}^{-1}(p_{k+1})}{p_{k}}<+\infty \, .$$ % Then if $A$ satisfies a Bruno Condition, the germ is analytically linearizable~\cite{Bruno,Russmann}. For the $1$--dimensional Analytic Siegel Center Problem, Yoccoz~\cite{Yoccoz} proved that the Bruno condition is necessary and sufficient to linearize analytically, in this case the Bruno condition reduces to the convergence of the series % $$\label{eq:1dimbruno} \sum_{k=0}^{+\infty} \frac{\log q_{k+1}}{q_{k}}<+\infty \, ,$$ % where $(q_k)_k$ is the sequence of the convergent to $\omega \in \R \setminus \Q$ such that $\lambda=e^{2\pi i\omega }$. \subsection{A new result: ultradifferentiable Classes} \label{classiMn} Let $(M_{k})_{k\ge 1}$ be a sequence of positive real numbers such that: \label{pg:ultradiff} \item{0)} $\inf_{k\ge 1} M_{k}^{1/k}>0$; \item{1)} There exists $C_{1}>0$ such that $M_{k+1}\le C_{1}^{k+1}M_{k}$ for all $k\ge 1$; \item{2)} The sequence $(M_{k})_{k\ge 1}$ is logarithmically convex; \item{3)} $M_{k}M_{l}\le M_{k+l-1}$ for all $k,l\ge 1$. We define the class $\class{M_k}\subset \C^n\left[\left[z_1,\dots,z_n\right]\right]$ as the set of formal power series ${\bf f}=\sum {\bf f}_{\alpha}z^{\alpha}$ such that there exist $A,B$ positive constant, such that: % $$|{\bf f}_{\alpha}|\leq A B^{|\alpha|}M_{|\alpha|} \quad \forall \alpha \in\mathbb{N}^n \, . \label{eq:classMk}$$ % The hypotheses on the sequence $(M_k)_k$ assure that $\class{M_k}$ 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\footnote{This assures that any term of the Taylor series is well defined.} gives again an element of the class. For example if ${\bf f},{\bf g}\in \class{M_k}$ then also $d{\bf f}({\bf z})({\bf g}({\bf z}))$ belongs to the same class. % \begin{remark} 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. \end{remark} % In~\cite{CarlettiMarmi} a similar problem was studied in the $1$--dimensional case. Here we will extend the results contained there to the case of dimension $n \geq 1$. The main result will be the following Theorem % \begin{theorem} \label{maintheorem} Let $(\lambda_1,\dots,\lambda_n)\in \C^n$ and let $A=\diag{\lambda}$ be non--resonant, $(M_k)_k$ and $(N_k)_k$ be sequences verifying hypotheses 0)--3). Let ${\bf F}\in V_0$, s.t. ${\bf F}({\bf z})=A{\bf z}+{\bf f}({\bf z})$ where ${\bf f}\in V_1$. Then % \begin{enumerate} \item If moreover ${\bf F}\in \class{M_k}\cap V_0$ and $A$ verifies a Bruno condition~\eqref{eq:brunocondition}, then also the linearization ${\bf H}$ belongs to $\class{M_k}\cap V_0$. \item If ${\bf F}$ is a germ of analytic diffeomorphism of $(\C^n,0)$ and there exists an increasing sequence of integer numbers $(p_k)_k$ such that $A$ verifies: % $$\label{eq:firstnewcondition} \limsup_{|\alpha |\rightarrow +\infty} \left( 2\sum_{m=0}^{\kappa (\alpha)} \frac{\log \Omega^{-1}(p_{m+1})}{p_m}-\frac{1}{|\alpha|}\log N_{|\alpha|} \right) \, ,$$ % where $\kappa(\alpha)$ is the integer defined by: $p_{\kappa(\alpha)}\leq |\alpha| 0$). Let us then assume that ${\bf F}\in V_0$, is of the form ${\bf F}({\bf z})=A{\bf z}+{\bf f}({\bf z})$ where $\lambda \in \C^n$, $A=\diag{\lambda}$ and ${\bf f}\in \class{M_k}\cap V_1$. For a fixed rooted labeled tree of order $N \geq 1$ with total momentum equals to $\alpha \in\N^n$, $|\alpha |\geq 2$, we consider the following term of equation~\eqref{coefficients}: % $$\label{eq:coefffix} ((\Omega^{-1}_{\nu_{\ell_{v_1}}} {\bf f}_{\alpha_{v_1}})\cdot \beta^{\ell_{v_1}}) \prod_{v\in \vartheta } \prod_{\ell_w \in L_v}((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot \beta^{\ell_w})$$ % Recalling the definition of scale and the definition on number of lines on scale $k$ we can bound~\eqref{eq:coefffix} with \begin{equation*} \Big\lvert ((\Omega^{-1}_{\nu_{\ell_{v_1}}} {\bf f}_{\alpha_{v_1}})\cdot \beta^{\ell_{v_1}}) \prod_{v\in \vartheta } \prod_{\ell_w \in L_v}((\Omega^{-1}_{\nu_{\ell_w}} {\bf f}_{\alpha_w})\cdot \beta^{\ell_w}) \Big\rvert \leq \prod_{m=0}^{\kappa (\alpha)}\left[2 \Omega^{-1}(p_{m+1})\right]^{N_m(\vartheta)}\prod_{v\in \vartheta}|{\bf f}_{\alpha_v}| \end{equation*} where $\kappa (\alpha)$ is the integer defined by: $p_{\kappa (\alpha)}\leq |\alpha|0$ and the thesis follows dividing by $|\alpha|$ and passing to the limit superior. \subsection{A result for some analytic vector fields of $\C^2$.} \label{ssec:anavfc2} 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. In~\cite{YoccozPerezMarco} authors showed, using a construction of~\cite{MatteiMoussu}, that the problem of the linearization of analytic vector field of $\C^2$ is completely equivalent to the problem of the linearization of analytic germs of $(\C,0)$. More precisely they proved that the analytic vector field $$\label{eq:anavf1} \begin{cases} \dot z_1 &= -z_1 (1+\dots ) \\ \dot z_2 &= \omega z_2(1+\dots) \, , \end{cases}$$ where $\omega >0$ and the suspension points mean terms of order bigger than $1$, has the same analytical classification that the germs of $(\C,0)$: $f(z)=e^{2\pi i \omega}z+\bigo{z^2}$. Using the results of~\cite{Yoccoz} they obtain as corollaries that: if $\omega$ is a Bruno number then the vector field~\eqref{eq:anavf1} is analytically linearizable, whereas if $\omega$ is not a Bruno number then there exist analytic vector fields of the form~\eqref{eq:anavf1} which 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 $\rho$ which satisfies the same lower bound (in term of the Bruno function) as the radius of convergence of the linearizing function of the germ~\cite{CarlettiMarmi,Yoccoz} does. To do this we must introduce some {\em normalization condition} for the vector field; let $\omega >0$, we consider the family $\mathcal{F}_{\omega}$ of analytic vector fields $\mathbf{F}:\mathbb{D}\times\mathbb{D} \rightarrow \mathbb{C}^2$ of the form $$\begin{cases} F_1(z_1,z_2) &=-z_1+\sum_{|\alpha|\geq 2}f_{\alpha,1}z^{\alpha} \\ F_1(z_1,z_2) &=\omega z_2+\sum_{|\alpha|\geq 2}f_{\alpha,2}z^{\alpha} \, , \end{cases} \label{eq:vectfieldform}$$ with $|f_{\alpha,j}|\leq 1$ for all $|\alpha|\geq 2$ and $j=1,2$. For power series in several complex variables the analogue of the disk of convergence is the {\em complete Reinhardt domain of center $0$}, $\mathcal{R}_0$, 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: $\mathbf{f}=\sum_{|\alpha|\geq 2}\mathbf{f}_{\alpha}z^{\alpha}$, this distance is given by $d_{\mathbf{F}}=\inf_{(z_1,z_2)\in \mathcal{R}_0}(|z_1|^2+|z_2|^2)^{1/2}$. The family $\mathcal{F}_{\omega}$ is compact w.r.t. the uniform convergence on compact subsets of $\mathbb{D}\times\mathbb{D}$ (use Weierstrass Theorem and Cauchy's estimates in $\C^2$, see for example~\cite{Shabat}) so we can define $d_{\omega}=\inf_{\mathbf{F}\in\mathcal{F}_{\omega}}d_{\mathbf{F}}$. Let $\rho_{\mathbf{F}}>0$ and let us introduce $P(0,\rho_{\mathbf{F}})=\{ (z_1,z_2)\in \C^2 : |z_i|<\rho_{\mathbf{F}},i=1,2 \}$, the biggest polydisk of center $0$ contained in $\mathcal{R}_0$, whose radius depends on the vector field $\mathbf{F}$. Trivially $\rho_{\mathbf{F}}$ and $d_{\mathbf{F}}$ are related by a coefficient depending only on the dimension: $\sqrt{2}\rho_{\mathbf{F}}=d_{\mathbf{F}}$. We can then prove \begin{theorem}[Lower bound on $d_{\omega}$] \label{the:mainvfn2} Let $\omega >0$ be a Bruno number, then there exists an universal constant $C$ such that: $$\label{eq:lowboundd} \log d_{\omega} \geq -B(\omega)-C \, ,$$ where $B(\omega)$ is the value of the Bruno function~\cite{MMY} on $\omega$. \end{theorem} We don't prove this Theorem being its proof very close to the one of Theorem~\ref{the:mainvf} case 1), we only stress that the use of the continued fraction allows us to give an optimal'' counting Lemma as done in~\cite{CarlettiMarmi,Davie}, which essentially bounds the number of lines on scale $k$ in a rooted labeled tree of order $N$ and total momentum $\nu_{\vartheta}$, with $\Big\lfloor \frac{\bar{\nu}_{\vartheta}}{q_k}\Big\rfloor$, being $(q_k)_k$ the denominators of the convergent to $\omega$. In the case of analytic germs of $(\C,0)$ Yoccoz~\cite{Yoccoz} proved that the same bound holds from above for the radius of convergence of the linearization, then using the results of~\cite{YoccozPerezMarco} we conjecture that the following bound holds \begin{equation*} \log d_{\omega} \leq -B(\omega)+C^{\prime} \, , \end{equation*} for some universal constant $C^{\prime}$. 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 $\omega$ is not a Bruno number (a similar result has been proved in~\cite{Yoccoz} Appendix $2$ and in~\cite{CarlettiMarmi} paragraph 2.4 for germs). \begin{remark}[Ultradifferentiable vector fields of $\C^2$] In the more general case of ultradifferentiable vector fields of $\C^2$ we can improve Theorem~\ref{the:mainvf} showing that we can linearize the vector field under weaker conditions. \begin{theorem} Let $\omega >0$ and let $(p_k/q_k)_k$ be its convergents. Let $\mathbf{F}$ be a vector field of the form~\eqref{eq:vectfieldform} (without additional hypotheses on the coefficients $\mathbf{f}_{\alpha}$), let $(M_n)_n$ and $(N_n)_n$ be two sequences verifying conditions 0)--3) of section~\ref{classiMn}. Then % \begin{enumerate} \item If moreover $\mathbf{F}$ belongs to $\class{M_n}$ and $\omega$ is a Bruno number then also the linearization ${\bf h}$ belongs to $\class{M_k}\cap V_1$. \item If ${\bf F}$ is a germ of analytic diffeomorphism of $(\C^2,0)$ and $\omega$ verifies: % $$\limsup_{|\alpha |\rightarrow +\infty} \left( \sum_{m=0}^{\kappa (\alpha)} \frac{\log q_{m+1}}{q_m}-\frac{1}{|\alpha|}\log N_{|\alpha|} \right) \, ,$$ % where $\kappa(\alpha)$ is the integer defined by: $q_{\kappa(\alpha)}\leq |\alpha| \alpha \}$, for $\alpha \geq 0$, are ideals of $A_v$. $I_0^{\prime}$ is the maximal ideal of $A_v$ and it is an open set in the topology induced by the ultrametric absolute value defined on $k$: % \begin{equation*} I_0^{\prime} = \left\{ x \in A_v \mid v \left( x \right) > 0 \right\} = \left\{ x \in A_v \mid \lvert x \rvert < 1 \right\} = B_{0} \left( 0,1 \right), \end{equation*} % where $B_{0} \left( x,r \right)=\{y\in k\, , \, |x-y|0$, if: $\sum_{\alpha\in\mathbb{N}^n}||F_{\alpha}||r^{|\alpha|} <+\infty$, where $|\alpha|=\sum_{i=1}^n \alpha_i$. $F$ will be said convergent in $B_{0}(0,r)$ if it is convergent in $B(0,r^\prime )$ for all $00$ such that % $$\label{Cauchy} ||F_{\alpha}||\le Mr^{-|\alpha|} \quad \forall \alpha\in\mathbb{N}^n \, .$$ % \item{2.} If there exists $M>0$ such that (\ref{Cauchy}) holds for all $\alpha\in\mathbb{N}^n$, $F$ converges in $B_{0}(0,r)$ and uniformly in $B(0,r^\prime )$ for all $00$ such that: % \begin{enumerate} \item $B_{0} \left( x,r \right) \subset U$, \item $G$ converges in $B_{0} \left( 0,r \right)$, and for all $y \in B_{0} \left( 0,r \right)$, $\Tilde{G} \left( x + y \right)= G \left( y \right)$. \end{enumerate} With a slight abuse of notation we will omit the superscript $\tilde{}$ to distinguish analytic functions from convergent power series. If $F$ is a convergent power series on $B(0,r)$ we denote $$\label{norm} ||F||_r=\sup_{\alpha\in\mathbb{N}^n}||F_{\alpha}||r^{|\alpha|}\, ,$$ and we define $\mathcal{A}_r(k^n)=\left\{F\in S_n: ||F||_r <+\infty \right\}$. \indent Let $U$ be an open set of $k^n$, $x$ a point of $U$, let us consider $G :U \rightarrow V\subset k^m$. A linear function $L: k^n \rightarrow k^m$ is called a {\em derivative} of $G$ at $x$ if: \begin{equation*} \lim_{\substack{||y || \rightarrow 0 \\ y \neq 0}} \frac {|| G \left( x + y \right) - G \left( x \right) - Ly ||} {|| y ||}=0. \end{equation*} % Clearly if the limit exists then the derivative is unique and it will be denoted by $dG \left( x \right)$. Let $\delta_i=\left( 0, \dots,1,\dots,0 \right)\in k^n$ the vector with $1$ at the i--th place, we call % $$D_iG\left( x \right) =dG\left( x \right) \left( \delta_i \right)\in k^m \label{partialder}$$ % the i-th partial derivative of $G$ at $x$. Higher order derivatives are defined analogously. \indent Let $G = \sum G_{\alpha}X^{\alpha}$ be an element of $S_n$, then $G_{\alpha} = \frac{D^{\alpha}G(x)}{\alpha !}$, where for $\alpha=(\alpha_1,\ldots, \alpha_n)\in\mathbb{N}^n$, $D^{\alpha}=D^{\alpha_1}_1\ldots D^{\alpha_n}_n$ and $\alpha!=\alpha_1!\ldots \alpha_n!$. Thus $G$ is just the Taylor series of $G$ at the point $x$. It is not difficult to prove that any power series $G\in S_{n}$ 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 $x\in U$ such that $B_{0}(x,r)\subset U$ does not necessarily converge on all of $B_{0}(0,r)$. This is true if one assumes $k$ to be algebraically closed. \indent Let $F \in S_{n}$, $F=\sum_{\alpha \in \N^n}F_{\alpha}X^{\alpha}$, let $\beta \in \N^n$ we define the formal derivative of $F$ by % $$\Delta^{\beta} F= \sum_{\alpha \in \N^n,\alpha \geq \beta} F_{\alpha} \binom{\alpha}{\beta}X^{\alpha-\beta}\, , \label{formalder}$$ % where we used the notations: $\alpha \geq \beta$ if $\alpha_i \geq \beta_i$ for $1 \leq i \leq n$, $\binom{\alpha}{\beta}=\frac{\alpha!}{\beta!\left( \alpha - \beta \right)!}$, for $\alpha,\beta \in \N^n$ and one can then prove~\cite{Serre}: $\alpha ! \Delta^{\alpha}=D^{\alpha}, \quad \binom{\alpha + \beta}{\alpha} \Delta^{\alpha + \beta}= \Delta^{\alpha} \Delta^{\beta}$. Finally we note that the composition of two analytic functions is analytic (\cite{Serre}, p. 70) and that the following Cauchy estimates and Taylor formula hold (\cite{HermanYoccoz}, pp. 421-422): \begin{proposition} \label{Taylorformulas} Let $r>0$, $s>0$, let $F\in\mathcal{A}_r(k^n)$, and let $G\, , \, H$ be two elements of $\mathcal{A}_s(k^n)$, with $||G||_{s}\le r$ and $||H||_{s}\le r$. 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