# Bruno ideal and the variety of centers for singular germs of vector fields

Maria Martín-Vega, Daniel Panazzolo

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Maria Martín-Vega, Daniel Panazzolo. Bruno ideal and the variety of centers for singular germs of vector fields. 2026. hal-05510537

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# BRUNO IDEAL AND THE VARIETY OF CENTERS FOR SINGULAR GERMS OF VECTOR FIELDS

MARÍA MARTÍN-VEGA AND DANIEL PANAZZOLO

ABSTRACT. Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial)$ defined by the collinearity locus of the semi-simple and nilpotent components of $\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial)$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial)$ is an analytic variety, and the foliation defined by $\partial|_V$ is analytically linearizable.

## 1. INTRODUCTION

Based on ideas of A.D. Bruno, the aim of this paper is to study the existence of analytic invariant sets related to the convergence of the Poincaré-Dulac normalization for germs of singular analytic vector fields.

Let us start by briefly recalling some basic concepts and results of normal forms. The detailed definitions will be given in the next subsection. A germ of singular vector field $\partial$ is called a *Poincaré-Dulac normal form* (or simply *normal form*) if we can decompose it in the form

$$ \partial = S + R $$

where:

(1) $S = \sum_{i=1}^{n} \lambda_{i}x_{i}\frac{\partial}{\partial x_{i}}$ is linear diagonal vector field

(2) $R$ is a non-linear vector field with a nilpotent linear part.

(3) $S$ and $R$ commute, i.e. the Lie Bracket $[S, R]$ vanishes.

As remarked by Martinet in [15], these properties are equivalent to say that $S$ and $R$ are respectively the semi-simple and nilpotent components in the *Jordan decomposition* of $\partial$ (see subsection 2.3). This definition extends to the case where $\partial$ is a singular *formal* vector field, i.e. to a vector field of the form $\partial = \sum f_i \frac{\partial}{\partial x_i}$ where each component $f_i$ is a formal series without constant term.

Let us express the condition $[S, R] = 0$ more explicitly. Using the logarithmic basis $\{x_i \frac{\partial}{\partial x_i}\}_{i=1}^n$ for vector fields, we can expand $R$ in the form

$$ \sum_{i=1}^{n} \sum_{\underline{m} \in M_i} a_{\underline{m}} \mathbf{x}^{\underline{m}} x_i \frac{\partial}{\partial x_i} $$

where $M_i$ is a subset of $\mathbb{Z}_{\ge 0}^{i-1} \times \mathbb{Z}_{\ge -1} \times \mathbb{Z}_{\ge 0}^{n-i-1}$. The condition $[S, R] = 0$ requires that the non-zero terms in the above expansion of $R$ are such that $\tilde{S}(\mathbf{x}^{\underline{m}}) = 0$. In other words, that $\langle \lambda, \underline{m} \rangle = \sum \lambda_i m_i$ vanishes. These terms are called *resonant* and the relations $\langle \lambda, \underline{m} \rangle = 0$ are called *resonances*. We say a resonance is *negative* if one of the components $m_i$ of $\underline{m}$ is equal to $-1$, and *positive* otherwise.

From the classical results by H. Poincaré [19] and H. Dulac [7], it is known that every singular analytic vector field $\partial$ is formally conjugate to a normal form. In other words, there exists a formal change of coordinates, which we identify to an automorphism $\Phi$ of the ring of formal series, such that

$$ \Phi \partial \Phi^{-1} = S + R $$

where $S, R$ are *formal* vector fields satisfying the above conditions (1), (2) and (3). In this case, we say that $\Phi \partial \Phi^{-1}$ is a *formal normal form* of $\partial$ and that the conjugating automorphism defines a formal *normalization*.

The analogous problem of obtaining an *analytic* normalization presents two main difficulties. Either the normal form itself is not analytic—reflecting the fact that the formal series expansion of $R$ given above has zero radius of convergence—or the conjugating automorphism $\Phi$ fails to be analytic.
Let us now discuss one of the main results on analytic normalisation proved by Bruno in [2]. We start by the introducing the following notation. To a vector of complex numbers $\lambda = (\lambda_1, \dots, \lambda_n) \in \mathbb{C}^n$, we associate the sequence $\omega_k = \omega_k(\lambda)$, indexed by $k \in \mathbb{N}$, defined by

$$ \omega_k = \min\{|\langle\lambda, \underline{m}\rangle| : \underline{m} \in M_1 \cup \dots \cup M_n, \underline{m} \text{ non-resonant, } \|\underline{m}\| \le 2^k\}. $$

where we note $\|\underline{m}\| = \sum_{i=1}^n |m_i|$ and the sets $M_i$ are defined as above.

We now state the following conditions on a germ of singular vector field $\partial$:

**$\omega$-condition:** We say that $\partial$ satisfies the *Bruno's arithmetic $\omega$-condition* (or simply the $\omega$-condition) if it has a normal form $S+R$ with semi-simple part $S = \sum_{i=1}^n \lambda_i x_i \frac{\partial}{\partial x_i}$ such that the sequence $\omega_k = \omega_k(\lambda)$ satisfies

$$ \sum_{k=0}^{\infty} -\frac{\log(\omega_k)}{2^k} < \infty. $$

**Geometric $A$-condition:** We say that $\partial$ satisfies the *geometric $A$-condition* if it has a normal form $S+R$ such that

$$ R = fS $$

for some formal series $f \in C[[\mathbf{x}]]$. In other words, the nilpotent component is *collinear* to the semi-simple component.

The $\omega$-condition is easily seen to be independent of the choice of normal form, since the spectrum $\lambda = (\lambda_1, \dots, \lambda_n)$ is clearly invariant under coordinate changes. As we shall see later, as a simple consequence of the uniqueness of the Jordan decomposition, the geometric $A$-condition is also independent of the choice of normal form. Consequently, both the $\omega$-condition and the geometric $A$-condition are intrinsically determined by $\partial$.

*Remark 1.1.* We remark that the geometric $A$-condition formulated here is strictly stronger than the original $A$-condition formulated by Bruno, but it is better adapted to the formulation of our main result. We refer the reader to [2] or to [24, sec. 10.3] for the precise statement of the $A$-condition.

**Theorem 1.2** (Bruno, 1971). Let $\partial$ be a germ of singular analytic vector field fulfilling the $\omega$-condition and the geometric $A$-condition. Then, $\partial$ is analytically conjugated to a normal form.

This result of Bruno has been generalized in several different directions. For instance, L. Stolovitch proved a normalization result for foliations generated by perturbations of commuting semi-simple vector fields under conditions of complete integrability. C. Chavaudret [5] proved an analog of Bruno's result for the normalization of a vector field in the vicinity of a $n$-torus.

We observe that, in the absence of resonances, the geometric $A$-condition is automatically satisfied and the $\omega$-condition implies that $\partial$ is analytically conjugated to a linear diagonal vector field. In this case, we simply say that $\partial$ is *analytically linearizable*.

In this context, the Theorem 1.2 represents a significant weakening the arithmetic conditions for linearizability previously imposed by other authors, such as C.-L. Siegel [22] and V.-A. Pliss [18]. We also note that such condition is satisfied by almost all tuples of complex number, and that Perez-Marco and Yoccoz [17] proved it to be a necessary and sufficient condition for analytic linearizability in dimension $n \le 2$.

On the other hand, in the presence of resonances, the geometric $A$-condition is rarely satisfied and imposes a strong restriction in order to guarantee analytic normalization.

In this setting, Bruno's insight, which we want to investigate in this paper, can be summarized as follows: rather than imposing the geometric $A$-condition from the outset, one should consider the problem of analytic linearizability restricted to the *formal locus* where this condition holds.

To enunciate our main result, we need to introduce the notion of *Bruno ideal*. In what follows, we will always assume that $\partial$ is a *logarithmic vector field*, i.e. a vector field preserving the coordinate hyperplanes $x_i = 0$, for $i = 1, \dots, n$.

Suppose initially that $\partial$ is a (possibly formal) logarithmic vector field in normal form with nonvanishing semisimple part $S$. Then $\partial$ can be written as

$$ \partial = (1 + f) S + \sum_{j=1}^{n-1} g_j T_j, $$
where

$$T_j = \sum_{i=1}^{n} \mu_{ji} x_i \frac{\partial}{\partial x_i}$$

is an arbitrarily chosen family of diagonal vector fields such that the set {$S, T_1, ..., T_{n-1}$} is $\mathbb{C}$-linearly independent, and where $f, g_j \in \mathbb{C}[[x]]$ are formal power series without constant term. The normal form condition (3) stated above is equivalent to requiring

$$S(f) = S(g_j) = 0, \quad j = 1, ..., n-1.$$

where $S$ acts on $\mathbb{C}[[x]]$ as a derivation. In this setting, we define the *Bruno ideal* of $\partial$ by

$$B(\partial) = \langle g_1, ..., g_{n-1} \rangle \subset \mathbb{C}[[x]].$$

More generally, for an arbitrary singular vector field $\partial$, we define $B(\partial)$ as follows: Choose a (formal) conjugated normal form $\delta = \Phi\partial\Phi^{-1}$, and define $B(\partial)$ as the pullback of $B(\delta)$ under the coordinate change induced by $\Phi$. As we shall prove, it turns out that the resulting formal ideal $B(\partial)$ is independent of the choice of the conjugated normal form and is therefore intrinsically attached to $\partial$. Additionally, such formal ideal is invariant by $\partial$, i.e.

$$\partial(B(\partial)) \subset B(\partial)$$

and we can consider the problem of reduction to normal form modulo $B(\partial)$. Here, we refer the reader to subsection 2.5 for the precise definition of a normal form modulo an invariant ideal.

Our main result can now be stated as follows:

**Main Theorem.** Let $\partial$ be an analytic logarithmic vector field satisfying the $\omega$-condition. Then, the ideal $B(\partial)$ is analytic and there exists an analytic automorphism $\Phi$ such that the conjugated derivation $\delta = \Phi\partial\Phi^{-1}$ is in normal form modulo $B(\delta)$.

We recall that a formal ideal $I \subset \mathbb{C}[[x]]$ is called *analytic* if it is generated by analytic germs, i.e. if

$$I = (I \cap \mathbb{C}\{x\}) \mathbb{C}[[x]]$$

Therefore, it follows from the Theorem that the vanishing locus of $B(\partial)$ defines a germ of analytic subvariety $V(B(\partial)) \subset (\mathbb{C}^n, 0)$, which is moreover invariant by $\partial$. We say such variety is the *Bruno variety* of $\partial$.

The following result is an important *dynamical* consequence of the theorem.

**Corollary 1.3.** Let $\partial$ be an analytic vector field satisfying the $\omega$-condition. Then, $\partial$ defines a linear foliation in restriction to its Bruno variety.

1.1. **Previous works.** In [3] and, more recently, in [4], Bruno introduced two formal ideals $B(\partial) \supseteq A(\partial)$ that are closely related to the ideal $B(\partial)$ defined above.

In these references, he asserts that, if the eigenvalues of $S$ (the semi-simple part of $\partial$) are pairwise commensurable then $A(\partial)$ is analytic. More generally, if these eigenvalues are purely imaginary and satisfy the $\omega$-condition, then ideal $B(\partial)$ is analytic. Unfortunately, these assertions are not proved in the cited works, and we have not been able to locate any complete proofs in the references quoted by Bruno.

Note that, under the additional assumption that $\partial$ is logarithmic, the ideal $B(\partial)$ originally defined by Bruno coincides with the ideal $B(\partial)$ as defined above.

The Main Theorem is also closely related to a result of L. Stolovitch [23] on the existence of analytic invariant varieties, which itself generalizes to higher dimensions a classical result of Dulac [8].

More precisely, let us state Stolovitch's result in the present logarithmic setting. Suppose that all resonant monomials associated with the diagonal vector field $S$ belong to the semigroup generated by the monomials $x^{m_1}, ..., x^{m_s}$, for some vectors $\underline{m}_1, ..., \underline{m}_s \in \mathbb{N}^n \setminus \{0\}$. Assume moreover that $S$ satisfies the $\omega$-condition. Then, Stolovitch proves that, up to analytic conjugacy, any analytic logarithmic vector field with semisimple part $S$ can be expanded in the form

$$\partial = S + \sum_{i=1}^{n} \left( \sum_{j=1}^{s} g_{ij}(x) x^{\underline{m}_j} \right) x_i \frac{\partial}{\partial x_i},$$

where the $g_{ij}$ are analytic germs.

Note that Stolovitch's result remains valid without the assumption that $\partial$ is logarithmic, provided instead that all resonances are positive. We refer the reader to the original paper for the detailed definitions. We also
emphasize that the above expansion is particularly useful for controlling the growth of solution curves of the vector field, since it allows one to view them as *perturbations* of the resonant monomials. Indeed, for each *j*, one has $\partial(x^{m_j})$ lying in the ideal generated by $\langle x^{m_1}, \dots, x^{m_s} \rangle$. We observe however that the above expansion is not a *normal form* in the sense (1), (2), (3) since the non-linear part does not necessarily commute with *S*.

In the logarithmic setting, our main result implies Theorem 1.2. In fact, the proof will follow closely the inductive scheme originally introduced by Bruno in [2], but conveniently adapted and reformulated using the elegant approach developed by Martinet in his Bourbaki survey [15]. In that survey, Martinet provides a detailed sketch of the proof of Bruno's theorem in the non-resonant case. His method relies on the Jordan decomposition and on the natural *S*-graduation of the space of vector fields, which significantly simplifies the analytic estimates required to establish the analyticity of the conjugating map.

During the preparation of this work, we became aware of a recent preprint by Romanovski and Walcher [20], in which the authors propose a simplified proof of Theorem 1.2. Their approach makes systematic use of the natural homogeneous *S*-graduation, in the same spirit as Martinet's method, but without using the formalism of Jordan decompositions.

In [9], Ecalle uses the powerful formalism of mould-comould expansions and arborifications to give alternative proof of Theorem 1.2 in the non-resonant case (see also [11]). It would be interesting to study if Ecalle's formalism can be adapted to prove our main theorem.

**1.2. Plan of the paper.** In Section 2, we introduce the basic concepts and definitions, as well as several classical results that will be used in the proof of the Main Theorem. We also prove that the first part of the Main Theorem (analyticity of $B(\partial)$) is indeed a consequence of the second part (existence of an analytic normalization modulo $B(\partial)$).

The proof of the theorem is divided into two clearly distinguished parts, presented in Sections 3 and 4. In Section 3, we establish a purely formal normalization result by developing the inductive procedure required to reduce a formal vector field to a normal form modulo its Bruno ideal.

In Section 4, we show that, under the $\omega$-condition, it is possible to control the domains of analyticity of the successive coordinate changes constructed in the previous section when they are applied to an analytic vector field.

Finally, in Section 5, we discuss several examples of Bruno ideals and present some applications of the main result.

**2. DERIVATIONS AND NORMAL FORMS MODULO IDEALS**

In this section, we present some basic definitions and results that will be used in the paper. Since the normal form results will be explained in the formalism of a Jordan decomposition, we will hereby consider vector fields as *derivations*.

**2.1. Some facts about free modules and derivations.** Let $R$ be a ring $R$ such that $Q \subset R$ and let $M$ be a free $R$-module. We list some useful facts.

(M.1) For each ideal $I \subset R$, the quotient module $M/IM$ is a free module over the ring $R/I$.

(M.2) The second exterior power $M \wedge M$ is a free module over $R$.

The *coefficient ideal* of an element $m \in M$ is the ideal $\Gamma(m) \subset R$ generated by the coefficients of the expansion of $m$ with respect to any arbitrary free basis $\{e_\alpha\}_{\alpha \in A}$ of $M$.

The *ideal of collinearity* of two elements $m, n \in M$ is the ideal $\Gamma(m \wedge n)$, i.e. the coefficient ideal of $m \wedge n$, seen as an element of the second exterior power $M \wedge M$.

Consider now the $R$-module $\text{Der}(R)$ of derivations on $R$. Given an ideal $I \subset R$, we denote by $\text{Der}(-\log I)$ the submodule of derivations $\partial \in \text{Der}(R)$ such that $\partial(I) \subset I$. In this case, we will also say that $I$ is $\partial$-invariant or that $I$ is *preserved by* $\partial$. Note that $\text{Der}(-\log I)$ contains $\text{Der}(R)$ as a submodule.

A derivation $\partial \in \text{Der}(-\log I)$ induces a derivation in the quotient ring $A = R/I$, which we will denote by $\partial_A$. Given an element $f \in R$, we can write $(f\partial)_A = f_A\partial_A$, where $f_A$ denotes the class of $f$ in $A$. The following fact will also be useful:

(D.1) Suppose that $\partial_1, \partial_2 \in \text{Der}(-\log I)$ are such that $(\partial_1)_{A} = (\partial_2)_{A}$ in $A = R/I$. Then $\partial_1 - \partial_2$ belongs to the submodule $\text{IDer}(R)$.
(D.2) If $\partial \in \text{Der}(-\log I)$ then $\partial(\text{rad}(I)) \subset \text{rad}(I)$, where $\text{rad}(I)$ denotes the radical of $I$. Moreover, if $I$ is
radical and $I = P_1 \cap \cdots \cap P_n$ is its minimal prime decomposition, then $\partial(P_i) \subset P_i$ for each $i$. See for
instance [21].

2.2. Formal and analytic logarithmic derivations. Graduations and Collinearity. We denote by $\mathcal{O} = \mathbb{R}\{x\}$ or $\mathbb{C}\{x\}$ the ring of germs of real (resp. complex) analytic functions in $n$ variables $x_1, \dots, x_n$ with maximal ideal $\mathfrak{m} = (x_1, \dots, x_n)$.

Let $\tilde{\mathcal{O}} = \mathbb{R}[[x]]$ or $\mathbb{C}[[x]]$ be the completion of $\mathcal{O}$ with respect to the Krull topology defined by $\{\mathfrak{m}^k\}_{k \in \mathbb{N}}$.
Given an ideal $I \subset \mathcal{O}$, we denote by $I\tilde{\mathcal{O}}$ the ideal generated by $I$ in $\tilde{\mathcal{O}}$. Since $\mathcal{O} \subset \tilde{\mathcal{O}}$ is a faithfully flat
extension, we have

$I\tilde{\mathcal{O}} \cap \mathcal{O} = I.$

We say that an ideal $\tilde{J} \subset \tilde{\mathcal{O}}$ is analytic if it has the form $\tilde{J} = I\tilde{\mathcal{O}}$ for some $I \subset \mathcal{O}$. Each element $f \in \tilde{\mathcal{O}}$ is defined by a formal series

$$f = \sum_{\mathfrak{m} \in \mathbb{N}^n} a_{\mathfrak{m}} x^{\mathfrak{m}}$$

where $a_{\mathfrak{m}} \in \mathcal{C}$ and $x^{\mathfrak{m}} = x_1^{m_1} \cdots x_n^{m_n}$. We define the support of $f$ by

$$\text{supp}(f) = \{\mathfrak{m} : a_{\mathfrak{m}} \neq 0\}$$

and define the order and degree of $f$ by

$$\text{ord}(f) = \inf\{\|\underline{m}\| : \underline{m} \in \text{supp}(\partial)\} \quad \text{and} \quad \text{deg}(f) = \sup\{\|\underline{m}\| : \underline{m} \in \text{supp}(\partial)\} \quad (1)$$

where $\|\underline{m}\| = \sum_i |m_i|$. As usual we convention that $\text{deg}(0) = -\infty$ and $\text{ord}(0) = \infty$. Note that $f$ is a polynomial iff $\text{deg}(f) < \infty$, or equivalently, if $\text{supp}(f)$ is finite.

We denote by $\text{Der}(\mathcal{O})$ the module of derivations in $\mathcal{O}$. An element of $\text{Der}(\mathcal{O})$ is fully determined by its
action on the variables $x_1, \dots, x_n$. Hence, we can identify $\text{Der}(\mathcal{O})$ with the free module of vector fields

$$\sum_{i=1}^{n} a_i \frac{\partial}{\partial x_i}$$

with coefficients $a_i \in \mathcal{O}$. A derivation $\partial \in \text{Der}(\mathcal{O})$ is called local if $\partial(\mathfrak{m}) \subset \mathfrak{m}$. We also define $\text{Der}_{\text{log}}(\mathcal{O}) \subset \text{Der}(\mathcal{O})$ as the submodule of derivations preserving the ideal $I_{\text{log}} = (x_1 \cdots x_n)$. It is a free module with basis $\{x_i \frac{\partial}{\partial x_i}\}$.

We define similarly the free modules $\text{Der}_{\text{log}}(\tilde{\mathcal{O}}) \subset \text{Der}(\tilde{\mathcal{O}})$ of derivations and logarithmic derivations
in the ring $\tilde{\mathcal{O}}$. Each derivation $\partial \in \text{Der}(\tilde{\mathcal{O}})$ extends uniquely to a derivation in $\text{Der}(\tilde{\mathcal{O}})$. We say that a
derivation $\partial$ in $\text{Der}(\tilde{\mathcal{O}})$ is analytic if $\partial$ is the extension of a derivation in $\text{Der}(\tilde{\mathcal{O}})$.

In this paper, we consider only logarithmic derivations. To simplify the notation, we will henceforth denote
the modules $\text{Der}_{\text{log}}(\mathcal{O})$ and $\text{Der}_{\text{log}}(\tilde{\mathcal{O}})$ simply by $\mathcal{D}$ and $\tilde{\mathcal{D}}$, respectively.

We will frequently consider logarithmic derivations of the form

$x^{\underline{m}} L(\lambda)$

where $\underline{m} \in \mathbb{N}^n$, and $L(\lambda) := \sum \lambda_j x_j \frac{\partial}{\partial x_j}$ for some $\lambda \in \mathbb{C}^n$. We will call them monomial derivations. Note
that a monomial derivation acts on monomials by the formula

$$x^{\underline{m}} L(\lambda) (x^{\underline{n}}) = \langle \lambda, \underline{n} \rangle x^{\underline{m}+\underline{n}} \quad (2)$$

and Lie bracket of two monomial derivations is given by

$$[x^{\underline{m}} L(\lambda), x^{\underline{n}} L(\mu)] = \langle \lambda, \underline{n} \rangle x^{\underline{m}+\underline{n}} L(\mu) - \langle \mu, \underline{m} \rangle x^{\underline{m}+\underline{n}} L(\lambda) \quad (3)$$

and in particular, $[L(\lambda), L(\mu)] = 0$. We further observe that each derivation $\partial \in \tilde{\mathcal{D}}$ has a unique expansion
as a sum of monomial derivations,

$$\partial = \sum_{\mathfrak{m} \in \mathbb{N}^n} \mathbf{x}^{\mathfrak{m}} L(\lambda_{\mathfrak{m}}) \quad (4)$$

with each $\lambda_{\mathfrak{m}} \in \mathbb{C}^n$. We call this the monomial expansion of $\partial$.
Based on this expansion, define the support of $\partial$ by

$$ \mathrm{supp}(\partial) = \{\underline{m} : \lambda_{\underline{m}} \neq 0\} $$

and define the order and degree of $\partial$ by

$$ \mathrm{ord}(\partial) = \inf\{\|\underline{m}\| : \underline{m} \in \mathrm{supp}(\partial)\} \quad \text{and} \quad \mathrm{deg}(\partial) = \sup\{\|\underline{m}\| : \underline{m} \in \mathrm{supp}(\partial)\} \qquad (5) $$

We say that $\partial$ is $k$-flat if $\mathrm{ord}(\partial) \ge k$. Note that this is equivalent to say that

$$ \partial(\mathbf{m}) \subset \mathbf{m}^{k+1}. $$

In analogy to the case of germs, we say that $\partial$ polynomial if $\mathrm{deg}(\partial) < \infty$, or equivalently, if $\mathrm{supp}(\partial)$ is finite.

It will be also convenient to consider expansions of derivations with respect to a fixed set of diagonal derivations. A collection of diagonal derivations $L(\mu_0), \dots, L(\mu_{n-1})$ will be called a logarithmic basis if the vectors $\mu_0, \dots, \mu_{n-1} \in \mathbb{C}^n$ are linearly independent. In this case, each derivation $\partial \in \mathcal{D}$ has a unique expansion

$$ \partial = \sum_{i=0}^{n-1} g_i L(\mu_i) \qquad (6) $$

with coefficients $g_0, \dots, g_n \in \tilde{\mathcal{O}}$. We call it the logarithmic expansion of $\partial$ with respect to the basis $\{L(\mu_i)\}$.

**Remark 2.1.** The transition between the monomial expansion (4) and the above expansion can be easily obtained by expressing each diagonal derivation $L(\lambda_{\underline{m}})$ in the basis $\{L(\mu_0), \dots, L(\mu_{n-1})\}$ and then regrouping the terms. In other words, it suffices to consider the $n \times n$ matrix $\mathbf{T}$ of change of basis between the canonical basis $\{e_1, \dots, e_n\}$ and the basis $\{\mu_0, \dots, \mu_{n-1}\}$ of $\mathbb{C}^n$.

We shall frequently use the following fact: A diagonal derivation $S = L(\lambda)$ induces a graduation

$$ \tilde{\mathcal{O}} = \bigoplus_{\alpha \in \mathbb{C}} \mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S), \qquad (7) $$

where $\mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S)$ is the linear subspace of series $f \in \tilde{\mathcal{O}}$ such that $S(f) = \alpha f$. We also observe that $S$ induces a graduation on the formal derivations,

$$ \tilde{\mathcal{D}} = \bigoplus_{\alpha \in \mathbb{C}} \mathrm{Gr}_{\alpha}(\tilde{\mathcal{D}}, S), \qquad (8) $$

where $\mathrm{Gr}_{\alpha}(\tilde{\mathcal{D}}, S) = \{\delta \in \tilde{\mathcal{D}} : [S, \partial] = \alpha \partial\}$, where $[\cdot, \cdot]$ denotes the Lie bracket.

More concretely, $\mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S)$ and $\mathrm{Gr}_{\alpha}(\tilde{\mathcal{D}}, S)$ are respectively the spaces of series and derivations with expansion of the form

$$ f = \sum_{\underline{m}} a_{\underline{m}} x^{\underline{m}}, \qquad \partial = \sum_{\underline{m}} \mathbf{x}^{\underline{m}} L(\lambda_{\underline{m}}) $$

where the sums are taken over all $\underline{m} \in \mathbb{N}^n$ satisfying $\langle \lambda, \underline{m} \rangle = \alpha$. We also recall the basic graduated algebra relations

$$ \mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S) \cdot \mathrm{Gr}_{\beta}(\tilde{\mathcal{O}}, S) \subset \mathrm{Gr}_{\alpha+\beta}(\tilde{\mathcal{O}}, S), \quad [\mathrm{Gr}_{\alpha}(\tilde{\mathcal{D}}, S), \mathrm{Gr}_{\beta}(\tilde{\mathcal{D}}, S)] \subset \mathrm{Gr}_{\alpha+\beta}(\tilde{\mathcal{D}}, S) \qquad (9) $$

Also,

$$ \mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S) \cdot \mathrm{Gr}_{\beta}(\tilde{\mathcal{D}}, S) \subset \mathrm{Gr}_{\alpha+\beta}(\tilde{\mathcal{D}}, S) $$

for all $\alpha, \beta \in \mathbb{C}$. Similar graduations are defined over the analytic ring $\mathcal{O}$ and analytic derivations $\mathcal{D}$.

We conclude by observing that if $\partial$ is a local derivation of $\mathcal{O}$ then its exponential

$$ \Phi = \exp(\partial) = \sum_{k=0}^{\infty} \frac{\partial^k}{k!} = \mathbf{Id} + \partial + \frac{\partial^2}{2} + \dots \qquad (10) $$

is an automorphism of $\mathcal{O}$. Moreover, since we will always assume that $\partial$ is logarithmic, we can write

$$ \Phi(x_i) = x_i u_i, \quad i = 1, \dots, n $$

for some units $u_i \in \mathcal{O}$. In particular, we can consider conjugation morphism acting on $\mathcal{D}$, defined by either one of the following equalities

$$ \delta \mapsto \Phi \delta \Phi^{-1} = \exp(\mathrm{ad}_{\partial}) \delta = \sum_{n=0}^{\infty} \frac{1}{n!} \mathrm{ad}_{\partial}^{n}(\delta) \qquad (11) $$
where $ad_\partial(\delta) = [\delta, \partial]$ and $ad_\partial^n = ad_\partial \circ \cdots \circ ad_\partial$.

All these properties also hold for formal derivations.

2.3. **Jordan decomposition.** Let $\mathcal{J}^k = \mathcal{O}/\mathfrak{m}^{k+1}$ denote the ring of $k$-jets, which we identify, as a vector space, with the polynomials of degree at most $k$.

A local derivation is called *semi-simple/nilpotent* if the quotient derivation $\partial_{\mathcal{J}^k} := \partial \mod \mathfrak{m}^{k+1}$ is a semi-simple/nilpotent endomorphism of $\mathcal{J}^k$, for each $k \in \mathbb{N}$. Each local derivation $\partial \in \tilde{\mathcal{D}}$ has a unique *Jordan* decomposition as a sum $\partial = \partial_{ss} + \partial_{nilp}$, where $\partial_{ss}, \partial_{nilp} \in \tilde{\mathcal{D}}$ are respectively semi-simple and nilpotent derivations and $[\partial_{ss}, \partial_{nilp}] = 0$.

**Remark 2.2.** (1) The assumption that $\partial \in \tilde{\mathcal{D}}$ implies that $\partial_{ss}$ and $\partial_{nilp}$ also belong to $\tilde{\mathcal{D}}$.

(2) An ideal $J \subset \tilde{\mathcal{O}}$ is $\partial$-invariant if and only if it is both $\partial_{ss}$ and $\partial_{nilp}$ invariant

(3) If $J$ is a $\partial$-invariant ideal, one can define the relative Jordan decomposition of the induced derivation $\partial_A$ on the quotient algebra $A = \tilde{\mathcal{O}}/J$. The construction is exactly the same as above, but carried out on the sequence of relative jet spaces $\mathcal{J}_A^k = A/(\mathfrak{m}^{k+1}/J)$. It follows from item (2) and from the uniqueness of the Jordan decomposition that this relative decomposition is given by

$$ \partial_A = (\partial_{ss})_A + (\partial_{nilp})_A $$

In fact, one can prove that the semisimple–nilpotent decomposition is *functorial*, meaning that it is preserved under arbitrary (formal) ring morphisms.

We observe that, from the definitions, it follows that any semi-simple derivations is formally diagonalizable over $\mathbb{C}$. More precisely, eventually extending the base field from $\mathbb{R}$ to $\mathbb{C}$, there exists a formal automorphism $\Phi$ of $\tilde{\mathcal{O}}$ such that we can write

$$ S := \Phi \partial_{ss} \Phi^{-1} = \sum_{i=1}^{n} \lambda_i x_i \frac{\partial}{\partial x_i} $$

In particular, any derivation $\partial \in \tilde{\mathcal{D}}$ is formally conjugated to a Poincaré-Dulac normal form, according to the definition given in the Introduction.

Moreover, it follows from (7) that $\partial_{ss}$ defines a graduation

$$ \tilde{\mathcal{O}} = \bigoplus_{\alpha \in \mathbb{C}} \mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, \partial_{ss}) \qquad (12) $$

where the degree $\alpha$-component, $\mathrm{Gr}_{\alpha}(\mathcal{O}, \partial_{ss})$ is the set of series $f \in \tilde{\mathcal{O}}$ such that $\partial_{ss}(f) = \alpha f$. Note that we can also write

$$ \mathrm{Gr}_{\alpha}(\mathcal{O}, \partial_{ss}) = \Phi(\mathrm{Gr}_{\alpha}(\mathcal{O}, S)) $$

i.e. $\mathrm{Gr}_{\alpha}(\mathcal{O}, \partial_{ss})$ is the image of the subspace $\mathrm{Gr}_{\alpha}(\mathcal{O}, S)$ under the diagonalizing automorphism $\Phi$ defined above.

It is important to remark that, even if $\partial$ is analytic, its semi-simple and nilpotent components $\partial_{ss}$ and $\partial_{nilp}$ are not necessarily analytic, as the following example illustrates.

**Example 2.3.** Consider a germ of analytic saddle-node singularity $\partial = y \frac{\partial}{\partial y} + f(x, y)x \frac{\partial}{\partial x}$ with $f \in \mathfrak{m}$ and suppose that $\partial_{ss}$ and $\partial_{nilp}$ are both analytic. This implies that $\partial_{ss}$ is analytically conjugated to $y \frac{\partial}{\partial y}$ and therefore, up to analytic conjugacy, we can assume that $[\partial_{nilp}, y \frac{\partial}{\partial y}] = 0$, which implies that we can write $\partial_{nilp} = F(x)x \frac{\partial}{\partial x}$. Then, by a further analytic coordinate change involving only the $x$-variable, we can assume that $\partial$ has the form

$$ \partial = y \frac{\partial}{\partial y} + \frac{x^{k+1}}{1 + \lambda x^k} \frac{\partial}{\partial x}. $$

However, this special form implies the holonomy of $\partial$ along the strong separatrix $x = 0$ can be embedded into a flow. This is a very restrictive condition. Indeed, for a *generic* choice of $f$ one can show that the holonomy of $\partial$ along the strong separatrix cannot be embedded into a flow (see e.g. [16]). We refer the reader to [10] for a systematic discussion of a large classes of examples presenting similar phenomena.

**Assumption:** *From now on, we shall tacitly make the assumption that the derivation $\partial$ satisfies the non-degeneracy condition*

$$ \partial_{ss} \neq 0. $$
**2.4. Collinearity and Bruno ideal.** We recall that the ideal of collinearity of two derivations $\partial_1, \partial_2 \in \tilde{D}$ is the ideal

$$ \Gamma(\partial_1 \wedge \partial_2) \subset \tilde{\mathcal{O}} $$

as defined in subsection 2. We observe that if $\partial_1, \partial_2$ are analytic then $\Gamma(\partial_1 \wedge \partial_2)$ is analytic.

More generally, suppose that $\partial_1, \partial_2$ preserve some ideal $I \subset \tilde{\mathcal{O}}$ (i.e. that $\partial_i(I) \subset I$ for $i = 1, 2$). Then, we define the *mod-$I$ ideal of collinearity* by the ideal sum

$$ \Gamma(\partial_1 \wedge \partial_2) + I $$

**Remark 2.4.** Let us suppose that $I$ is an analytic ideal and that $\partial$ is an analytic derivation. Then the condition of invariance $\partial(I) \subset I$ is equivalent to saying that the germ of analytic variety $V(I)$ has the following property:

$$ \forall p \in V(I) : \partial_p \in T_p V(I) $$

where $\partial_p$ denote the vector defined by $\partial$ in the tangent space $T_p\mathbb{C}^n$ at $p$ and $T_pV(I)$ is the Zariski tangent space of $V(I)$ at $p$.

Let us now assume that $\partial_1, \partial_2$ are analytic derivations satisfying $\partial_1(I), \partial_2(I) \subset I$ and consider the *mod-$I$ ideal of collinearity* $J = \Gamma(\partial_1 \wedge \partial_2) + I$. Then the subvariety $V(J) \subset V(I)$ is defined by

$$ V(J) = \{p \in V(I) : \partial_{1,p} \wedge \partial_{2,p} = 0\} $$

i.e. the set of points in $V(I)$ where the 2-vector $\partial_1 \wedge \partial_2$ vanishes.

The following ideal plays a prominent role in the present work.

**Definition 2.5.** The logarithmic Bruno ideal, or simply, the Bruno ideal of a derivation $\partial \in \tilde{D}$ is the formal ideal defined by

$$ B(\partial) = \Gamma(\partial_{ss} \wedge \partial_{nilp}). $$

In other words, $B(\partial)$ is the ideal of collinearity of the semi-simple and nilpotent components of $\partial$.

**Lemma 2.6.** $B(\partial)$ is $\partial$-invariant and, in the quotient algebra $A = \tilde{\mathcal{O}}/B(\partial)$, we can write

$$ \partial_A = (1 + h_A)(\partial_{ss})_A $$

where $h$ is a element of $\text{Gr}_0(\tilde{\mathcal{O}}, \partial_{ss}) \cap \mathfrak{m}$.

We recall that $\text{Gr}_0(\tilde{\mathcal{O}}, \partial_{ss})$ denotes the degree-0 component of the graduation given in (12).

*Proof.* Using the non-degeneracy assumption, we complete $\partial_{ss}$ to a commuting logarithmic basis $\{\partial_{ss}, L_1, \dots, L_{n-1}\}$ of $\tilde{D}$. Then we can write the expansion

$$ \partial = \partial_{ss} + \partial_{nilp} = \partial_{ss} + h\partial_{ss} + \sum_{k=1}^{n-1} b_k L_k $$

where $h, b_1, \dots, b_{n-1} \in \mathfrak{m}$. The commutativity relation $[\partial_{ss}, \partial_{nilp}] = 0$ implies that $h_0, b_1, \dots, b_{n-1} \in \text{Gr}_0(\tilde{\mathcal{O}}, \partial_{ss})$. Moreover, the definition of $B(\partial)$ implies that it is generated by $\langle b_1, \dots, b_{n-1} \rangle$. An easy computation shows that $B(\partial)$ is both $\partial_{ss}$ and $\partial_{nilp}$ invariant. Hence, it is $\partial$-invariant. The last statement is obvious. $\square$

**Remark 2.7.** As remarked above, even if we assume that $\partial$ is an analytic derivation, we cannot in general guarantee that $B(\partial)$ is an analytic ideal since the semi-simple and nilpotent components of $\partial$ are not necessarily analytic.

For an ideal $J \subset \tilde{\mathcal{O}}$, we denote by $\Phi^*J$ the ideal generated by $\{f \circ \phi | f \in J\}$, with $\phi$ being the coordinate change associated with $\Phi$. The following result shows the intrinsic nature of the Bruno ideal.

**Lemma 2.8.** For any automorphism $\Phi$ of $\tilde{\mathcal{O}}$, one has

$$ B(\Phi \partial \Phi^{-1}) = \Phi^* B(\partial). $$
Proof. Let $\delta = \Phi\partial\Phi^{-1}$. Since the Jordan decomposition is preserved under conjugation, we have $\delta_{ss} = \Phi\partial_{ss}\Phi^{-1}$ and $\delta_{nilp} = \Phi\partial_{nilp}\Phi^{-1}$. Therefore, computing in $\tilde{D} \wedge \tilde{D}$, we obtain the identity

$$\delta_{ss} \wedge \delta_{nilp} = \Phi(\partial_{ss} \wedge \partial_{nilp})\Phi^{-1}.$$

If we choose an arbitrary free basis $(B_i \wedge B_j)_{i<j}$ of $\tilde{D} \wedge \tilde{D}$, and write

$$\partial_{ss} \wedge \partial_{nilp} = \sum_{i<j} a_{ij} B_i \wedge B_j.$$

Then the above identity gives

$$\delta_{ss} \wedge \delta_{nilp} = \Phi(\partial_{ss} \wedge \partial_{nilp})\Phi^{-1} = \sum_{i<j} (\Phi^* a_{ij}) (\Phi B_i \Phi^{-1}) \wedge (\Phi B_j \Phi^{-1}).$$

Hence, as $B(\partial)$ is the ideal generated by the coefficients $\{a_{ij}\}_{i<j}$, the ideal $B(\delta)$ is generated by the transformed coefficients $\{\Phi^* a_{ij}\}_{i<j}$. $\square$

Note that if we consider the associated quotient algebras

$$A = \tilde{O}/B(\partial) \quad \text{and} \quad A' = \tilde{O}/B(\Phi\partial\Phi^{-1}).$$

Then the above Lemma implies that the automorphism $\Phi$ induces a natural algebra morphism (which we denote by the same letter) $\Phi: A \to A'$, and the induced derivations satisfy

$$(\Phi\partial\Phi^{-1})_{A'} = \Phi\partial_A\Phi^{-1}.$$

The following easy Lemma will play an important role in the proof of the Proposition 3.3.

**Lemma 2.9.** Suppose that $I$ is a $\partial$-invariant ideal and that there exists two derivations $S, R \in \tilde{D}$ satisfying the following conditions:

a) $S$ is semi-simple, $R$ is nilpotent and $[S, R] = 0$

b) The ideal $I$ is $(S+R)$-invariant and, in the quotient algebra $A = \tilde{O}/I$ we have the equality

$$\partial_A = (S + R)_A$$

Then the ideal $I$ is both $S$-invariant and $R$-invariant. Moreover

$$(\partial_{ss})_A = S_A, \quad (\partial_{nilp})_A = R_A \qquad (13)$$

Proof. Consider the derivation $\delta := S + R$, which has Jordan decomposition $\delta_{ss} = S$ and $\delta_{nilp} = R$. The item (b) of the enuciate implies that $\partial - \delta \in I\tilde{D}$, by property (D.1) in Section 2.1. Note also that $I$ is $\delta$-invariant. Hence, by the Remark 2.2, $I$ is both $S$ and $R$ invariant. The final statement is obvious. $\square$

**Remark 2.10.** Using the same argument as in the proof above, we can establish the following relative version of the Lemma: suppose that $J \subset I$ are ideals in $\tilde{O}$ such that $J$ is invariant under $\partial$, $S$, and $R$, and that conditions (a) and (b) hold upon restriction to the quotient algebra $C = \tilde{O}/J$. Then the ideal $K = I/J$ is invariant under $S$ and $R$, and property (13) holds in the quotient algebra $A = C/K$.

**2.5. Normal form and normal form modulo an ideal.** From now on, we fix a non-zero diagonal derivation $S = L(\lambda)$.

**Definition 2.11.** An $S$-perturbation is a formal derivation $\partial \in \tilde{D}$ with a monomial expansion of the form

$$\partial = S + \sum_{\|m\| \ge 1} x^m L(\lambda_m) \qquad (14)$$

As in the introduction, we say that $\partial$ is in normal form if $\partial_{ss} = S$, or, equivalently, if

$$\mathrm{supp}(\partial) \subset \{\underline{m} : \langle \lambda, \underline{m} \rangle = 0\}.$$

We now introduce a relative version of such concept, by considering the restriction of $\partial$ to a quotient algebra by an invariant ideal.

**Definition 2.12.** Suppose that $I \subset \tilde{O}$ is a $\partial$-invariant ideal. We say that an $S$-perturbation $\partial$ is in normal form modulo $I$ if the following two conditions hold:

<footnote n="1">¹This follows from the natural action of $\Phi$ on $\tilde{D} \wedge \tilde{D}$.</footnote>
* The ideal I is S-invariant and

* $(\partial_{ss})_A = S_A$ in the quotient algebra $A = \tilde{O}/I$.

Note that, when $I = 0$, we recover the usual notion of normal form. In the next enunciate, we suppose that $\partial$ is given by (14) and consider the nilpotent derivation

$$R = \partial - S = \sum_{\|m\| \ge 1} x^m L(\lambda_m)$$

**Proposition 2.13.** If $\partial$ is in normal form modulo $I$ then:

(1) $I$ is $R$-invariant and

(2) $(\partial_{nilp})_A = R_A$

As a consequence, $[S_A, R_A] = 0$ (i.e. $S$ and $R$ commute modulo $I$).

*Proof.* The first statement is clear. Furthermore, in the quotient algebra we have $R_A = (\partial - S)_A = \partial_A - S_A = \partial_A - (\partial_{ss})_A = (\partial_{nilp})_A$. $\square$

Let us observe that the relation $[S_A, R_A] = 0$ is equivalent to state that

$$\left[ S, \sum_{\|m\| \ge 1} x^m L(\lambda_m) \right] \in I \tilde{D}$$

Applying (3), this is also equivalent to say that $S(x^m) \in I$ for all $m \in \text{supp}(\partial)$.

We shall mainly be interested in the case where the invariant ideal $I$ in the definition 2.12 is precisely the Bruno ideal $B(\partial)$ of $\partial$. Let us show that in this case the induced derivation $\partial_A$ in the quotient algebra

$$A = \tilde{O}/B(\partial)$$

assumes a very special form.

Let us fix an arbitrary collection of diagonal vector fields vectors $T_1 = L(\mu_1), \dots, T_{n-1} = L(\mu_{n-1})$ such that $\{S, T_1, \dots, T_{n-1}\}$ forms a logarithmic basis. Then, as in subsection 2.2, we can consider the logarithmic expansion

$$\partial = S + R = S + \left( f S + \sum_{j=0}^{n-1} g_j T_j \right) \qquad (15)$$

with $f, g_j \in \mathfrak{m}$.

**Corollary 2.14.** Suppose that $\partial$ is in normal form modulo $B(\partial)$, where $B(\partial)$ is the Bruno ideal of $\partial$. Then, we can write

$$\partial_A = (1 + f_0)S_A$$

where $f_0 \in \text{Gr}_0(\tilde{O}, S)$ is the degree 0 component of $f$ with respect to the graduation (7). In particular, the elements

$$f - f_0, g_1, \dots, g_{n-1}$$

belong to the Bruno ideal.

*Proof.* By the Lemma 2.6, we have

$$\partial_A = (1 + h)(\partial_{ss})_A = (1 + h)S_A$$

which shows that

$$\sum g_j T_j \in B(\partial) \tilde{D}.$$

Since $\{S, T_1, \dots, T_{n-1}\}$ forms a basis of $\tilde{D}$, we conclude that $g_j \in B(\partial)$. Moreover, $1+h$ is congruent to $1+f$ modulo $B(\partial)$. As we remarked above $S(f) = S(f - f_0) \in B(\partial)$, which implies that $f - f_0 \in B(\partial)$ ($2$). Therefore, $(1+f)S_A = (1+f_0)S_A$. $\square$

<footnote>$^2$Since $B(\partial)$ is $S$-invariant, it is a $S$-graded ideal. In particular, if $h = \sum_{\alpha} h_{\alpha}$ is the $S$-graded expansion of a $h \in \mathcal{O}$ then $h \in B(\partial)$ if and only if $h_{\alpha} \in B(\partial)$ for each degree $\alpha$. $S(h) = \sum_{\alpha} \alpha h_{\alpha} \in B(\partial) \implies h_{\alpha} \in B(\partial) (\forall \alpha \neq 0) \implies h - h_0 \in B(\partial)$</footnote>
**2.6. Analyticity of the Bruno ideal and Corollary 1.3.** Using the results of the previous subsection, we will show that the first statement of the Main Theorem is in fact consequence of the second one.

More precisely, for an analytic derivation $\partial$, the analyticity of the Bruno ideal $B(\partial)$ holds if we assume that $\partial$ is in normal form modulo $B(\partial)$ (see Definition 2.12).

Therefore, in this subsection we will assume that $\partial = S + R$ is an analytic derivation which is an $S$-perturbation (i.e. $R$ is an analytic derivation satisfying $\mathrm{ord}(R) \ge 1$).

We also fix an arbitrary logarithmic expansion as in (15) where now $f, g_j$ are analytic germs. Based on this expansion, we define the following analytic ideals, using the notation introduced in subsection 2.1:

(1) The ideal of collinearity $I_1 := \Gamma(S \wedge R)$, which is generated by $g_1, \dots, g_{n-1}$.

(2) The ideal of commutativity $I_2 := \Gamma([S, R])$, which is generated by $S(f), S(g_1), \dots, S(g_{n-1})$.

(3) The ideal sum $W = I_1 + I_2$

(4) The differential closure $I = S[W]$ of $W$ with respect to the derivation $S$ (in the ring $\mathcal{O}$).

where we recall that, in a ring $R$, the differential closure of an ideal $J \subset R$ with respect to a derivation $\delta$ is the smallest ideal $\delta[I] \subset R$ containing $I$ which is stable by $\delta$.

**Proposition 2.15.** Suppose that $\partial$ is in normal form modulo $B(\partial)$. Then $B(\partial) = I\tilde{\mathcal{O}}$. In particular $B(\partial)$ is an analytic ideal.

*Remark 2.16.* We note that this result imposes no arithmetic condition on the eigenvalues of $S$.

*Proof.* It follows from Corollary 2.14 that $g_1, \dots, g_{n-1} \in B(\partial)$. From the remark before that Corollary, we also have $S(f), S(g_1), \dots, S(g_{n-1}) \in B(\partial)$. Therefore, $W \subset B(\partial)$. Since $B(\partial)$ is invariant by $S$, the differential closure $I = S[W]$ of $W$ is still contained in $B(\partial)$. Therefore, $I\tilde{\mathcal{O}} \subset B(\partial)$.

We now prove the inverse inclusion

$$B(\partial) \subset I\tilde{\mathcal{O}}.$$

Note that $I$ is both $S$-invariant and $R$-invariant and therefore also $\partial, \partial_{ss}$ and $\partial_{nilp}$-invariant. By the definition of $I$, the derivations $S_C$ and $R_C$ commute in the quotient algebra $C = \tilde{\mathcal{O}}/I\tilde{\mathcal{O}}$. Since they are respectively semi-simple and nilpotent, it follows from Remark 2.10 that

$$(\partial_{ss})_C = S_C, \quad \text{and} \quad (\partial_{nilp})_C = R_C$$

As a consequence, $\partial_{ss} - S$ and $\partial_{nilp} - R$ lie in the submodule $I\tilde{\mathcal{D}}$ (according to the general fact (D.1) of Section 2.1). Therefore,

$$I\tilde{\mathcal{O}} = (\Gamma(S \wedge R) + I)\tilde{\mathcal{O}} = \Gamma(\partial_{ss} \wedge \partial_{nilp}) + I\tilde{\mathcal{O}} = B(\partial) + I\tilde{\mathcal{O}},$$

which shows that $B(\partial)$ is contained in $I\tilde{\mathcal{O}}$. $\square$

We conclude this section proving Corollary 1.3, which follows immediately from the above Proposition.

*Proof of Corollary 1.3.* Suppose that $\partial$ is in normal form modulo $B(\partial)$ and recall that $V(B(\partial))$ is an analytic invariant variety for $\partial$. Moreover, if we set $A = \mathcal{O}/B(\partial)$ then

$$\partial_A = (1 + f_0)S_A$$

for some analytic germ $f_0 \in Gr_0(\mathcal{O}, S)$. Hence, in restriction $V(B(\partial))$, the foliation defined by $\partial$ is linear. $\square$

### 3. FORMAL REDUCTION TO NORMAL FORM MODULO THE BRUNO IDEAL

The Proposition 2.15 from the last subsection implies that the Main Theorem will hold if we prove that, under the $\omega$-condition, any analytic derivation is analytically conjugated to a derivation in normal form modulo its Bruno ideal.

To prove this result, we adapt the Newton-type inductive scheme introduced by Bruno. The main idea is to construct a sequence of coordinate changes that successively transform $\partial$ into a (relative) normal form, modulo truncations to jet spaces of increasing order. As in Newton's method, the order of truncation is doubled at each step.

As mentioned in the introduction, we begin by describing how the normalization procedure works for formal derivations, for which (formal) convergence is straightforward to establish and requires no arithmetic assumptions.
**3.1. The adjoint operator.** The inductive reduction to normal form relies on the invertibility of the following adjoint operator, acting on $\tilde{D}$,

$$ \mathrm{ad}_{(1+f_0)S}(\cdot) = [(1+f_0)S, \cdot] $$

where $f_0$ is a series belonging to $\mathfrak{m} \cap \mathrm{Gr}_0(\tilde{\mathcal{O}}, S)$. As a first remark, we observe that such operator restricts to a linear map

$$ \mathrm{Gr}_{\alpha}(\tilde{D}, S) \to \mathrm{Gr}_{\alpha}(\tilde{D}, S) $$

for each $\alpha \in \mathbb{C}$, as it easily follows from (9). In other words, $\mathrm{ad}_{(1+f_0)S}$ is a graduated operator of degree 0. The following result will play a crucial role in the formal and analytic reduction to a normal form.

**Lemma 3.1.** Let $f_0$ be as above and suppose that $\alpha \in \mathbb{C} \setminus \{0\}$. Then, $\mathrm{ad}_{(1+f_0)S}$ restricts to an isomorphism of $\mathrm{Gr}_{\alpha}(\tilde{D}, S)$. More precisely, its inverse is given explicitly by

$$ \mathrm{ad}_{(1+f_0)S}^{-1} x^m L(\mu) = \frac{x^m}{\alpha(1+f_0)} L(\mu) + \frac{x^m L(\mu)(f_0)}{\alpha^2(1+f_0)^2} S $$

where $L(\mu)$ is any diagonal derivation and $x^m$ denotes a monomial in $\mathrm{Gr}_{\alpha}(\tilde{\mathcal{O}}, S)$.

*Proof.* This is a straightforward computation using the following decomposition

$$ \mathrm{ad}_{(1+f_0)S}(U) = [(1+f_0)S, U] = \alpha(1+f_0) \cdot \left( U - \frac{U(f_0)}{\alpha(1+f_0)}S \right) $$

where $U \in \mathrm{Gr}_{\alpha}(\tilde{D}, S)$. Note that the linear operator inside the parenthesis has the form $\mathbf{Id} - \mathbf{N}$ with $\mathbf{N}$ being the linear operator

$$ U \mapsto \mathbf{N}(U) = \frac{U(f_0)}{\alpha(1+f_0)}S $$

and, since $S(f_0) = 0$, we have $\mathbf{N}^2 = 0$ (i.e. $\mathbf{N}$ is nilpotent of degree 2). Furthermore, it commutes with the scalar operator of multiplication by $\alpha(1+f_0)$. Therefore, the inverse of $\mathrm{ad}_{(1+f_0)S}$ can be written as $\frac{1}{\alpha(1+f_0)}(\mathbf{Id} - \mathbf{N})$. This is precisely the expression given in the enunciate. $\square$

Let us state a simple Corollary which will be used later. Given a derivation $\delta \in \tilde{D}$, we denote by

$$ \delta = \sum_{\alpha} \delta_{\alpha} $$

its decomposition with respect to the graduation given in (8), and let $\delta_* = \delta - \delta_0$ denote the sum of those graduated components of $\delta$ of non-zero degree.

**Corollary 3.2.** (1) Given a derivation $\delta \in \tilde{D}$, the equation

$$ [(1 + f_0)S, U] = \delta_* $$

has a solution $U \in \tilde{D}$. Moreover, such solution is unique if we assume that $U = U_*$.

(2) If we further assume that $\mathrm{ord}(\delta) \ge K$, i.e. that $\delta \in \mathfrak{m}^K \tilde{D}$ for some $K \in \mathbb{N}$, then the same holds for the unique solution defined above.

The second part of the enunciate is an immediate consequence of the explicit form $\mathrm{ad}_{(1+f_0)S}^{-1}$ given by the Lemma.

**3.2. The inductive formal reduction.** From now on, we will adopt the following notation. We denote by $\mathfrak{m}^{(k)} = \mathfrak{m}^{2^k}$ the $2^k$-st power of the maximal ideal, for $k \in \mathbb{N}$, and let

$$ \mathcal{J}^{(k)} = \mathcal{O}/\mathfrak{m}^{(k)}. $$

denote the corresponding jet space. Given a derivation $\partial$, we denote by

$$ \partial^{(k)} \in \mathrm{Der}(\tilde{\mathcal{O}}) $$

its $\mathfrak{m}^{(k)}$-truncation, i.e. the derivation on the quotient algebra $\mathcal{J}^{(k)}$. Also, we define the $\mathfrak{m}^{(k)}$-truncated Bruno ideal of $\partial$ by

$$ B(\partial^{(k)}) = \Gamma(\partial_{ss}^{(k)} \wedge \partial_{nilp}^{(k)}) $$
Note that, from Remark 2.2.(3) we conclude that

$$B(\partial^{(k)}) = B(\partial)/\mathfrak{m}^{(k)}$$

In other words, the definition of the Bruno ideal commutes with the operation of truncation modulo $\mathfrak{m}^{(k)}$.

The reduction to the formal normal form its modulo the Bruno ideal will be carried out inductively with respect to $k$. Consequently, we assume that the following induction hypothesis holds for a given $k \in \mathbb{N}$:

**Hypothesis** ($\mathbf{H}_k$) $\partial^{(k)}$ is in normal form modulo $B(\partial^{(k)})$.

We recall that this hypothesis corresponds to the following two requirements:

($\mathbf{H}_k^{(1)}$): $B(\partial^{(k)})$ is $S$-invariant, and

($\mathbf{H}_k^{(2)}$): We have the equality

$$(\partial_{ss}^{(k)})_{A^{(k)}} = S_{A^{(k)}}$$

in restriction to the quotient algebra $A^{(k)} = \mathcal{J}^{(k)}/B(\partial^{(k)})$.

It follows from Corollary 2.14 that we can write

$$(\partial^{(k)})_{A^{(k)}} = (1 + f_0^{(k)}) S_{A^{(k)}} \qquad (16)$$

for some $f_0 \in \text{Gr}_0(\mathcal{O}, S)$. Here $f_0^{(k)}$ can be seen as the polynomial part of degree at most $2^k$ in the power series expansion of $f_0$. Using this notation, let us consider the next quotient algebra

$$A^{(k+1)} = \mathcal{J}^{(k+1)}/B(\partial^{(k+1)})$$

associated to the Bruno ideal of $\partial^{(k+1)}$, seen as a derivation in the jet space $\mathcal{J}^{(k+1)}$.

It follows from (16) that the restriction of $\partial^{(k+1)}$ to $A^{(k+1)}$ can be written as (3)

$$(\partial^{(k+1)})_{A^{(k+1)}} = ((1 + f_0^{(k)})S + W)_{A^{(k+1)}} \qquad (17)$$

where $W$ is a polynomial derivation such that

$$\text{ord}(W) \geq 2^k, \quad \text{and} \quad \text{deg}(W) \leq 2^{k+1}$$

according to the definition in subsection 2.2. As previously, we write

$$W = W_0 + W_*,$$

where $W_0$ is the degree 0 component of $W$ with respect to the $S$-graduation. Let $U$ be the derivation satisfying the equations

$$[(1 + f_0)S, U] = -W_*, \quad U = U_* \qquad (18)$$

whose existence and unicity has been established in Corollary 3.2. The following result constitutes the key step in the inductive procedure.

**Proposition 3.3.** Consider the automorphism $\Phi = \exp(U)$. Then, the conjugated derivation

$$\delta = \Phi \partial \Phi^{-1}$$

is such that its truncation $\delta^{(k+1)}$ satisfies hypothesis ($\mathbf{H}_{k+1}$).

*Proof.* It follows from Lemma 2.8 that $B(\delta^{(k+1)}) = \Phi^*B(\partial^{(k+1)})$. Moreover, if we consider the quotient algebra $\Lambda^{(k+1)} = \mathcal{J}^{(k+1)}/B(\delta^{(k+1)})$ with respect to Bruno ideal of $\delta$, then we can write

$$\begin{aligned} (\delta^{(k+1)})_{\Lambda^{(k+1)}} &= \Phi((1 + f_0^{(k)})S + W)_{\Lambda^{(k+1)}} \Phi^{-1} \\ &= (\Phi(1 + f_0^{(k)})S \Phi^{-1} + \Phi W \Phi^{-1})_{\Lambda^{(k+1)}} \end{aligned}$$

Since $\Phi = \exp(U)$ and $\text{ord}(U) \geq 2^k$ (as it follows from Corollary 3.2.(2)), the following two equalities hold modulo $\mathfrak{m}^{(k+1)}$,

$$\Phi(1 + f_0)S \Phi^{-1} = (1 + f_0^{(k)})S + [U, (1 + f_0^{(k)})S] \quad \text{mod} \quad \mathfrak{m}^{(k+1)}$$

$$\frac{\mathcal{J}^{(k+1)}/\mathfrak{m}^{(k)}}{B(\partial^{(k+1)})/\mathfrak{m}^{(k)}} = \mathcal{J}^{(k)}/B(\partial^{(k)})$$

<footnote n="3">³We have $B(\partial^{(k+1)})/\mathfrak{m}^{(k)} = B(\partial^{(k)})$, and therefore</footnote>
and

$$ \Phi W \Phi^{-1} = W \quad \mod \mathbf{m}^{(k+1)} $$

Therefore, by the choice of $U$,

$$ (\delta^{(k+1)})_{\Lambda^{(k+1)}} = ((1 + f_0^{(k)})S + W_0)_{\Lambda^{(k+1)}} $$

We now observe that $S$ commutes with $f_0^{(k)}S + W_0$. Moreover $S$ and $f_0^{(k)}S + W_0$ are respectively semi-simple and nilpotent. Hence, by Lemma 2.9, the ideal $B(\delta^{(k+1)})$ is $S$-invariant and we have

$$ S_{\Lambda^{(k+1)}} = (\delta_{ss}^{(k+1)})_{\Lambda^{(k+1)}} $$

Therefore, the hypothesis $(H_{k+1}^{(1)})$ and $(H_{k+1}^{(2)})$ hold for $\delta^{(k+1)}$. $\square$

<figure><img src="image_2.png" /></figure>

**Remark 3.4.** (1) For later use, we note that, since the above proof only involves truncations modulo $\mathbf{m}^{(k+1)}$, the same argument applies if $U$ is replaced by any solution of the following truncated bracket equation:

$$ [(1 + f_0)S, U] = -W_* \quad \mod \mathbf{m}^{(k+1)}. $$

In particular, $U$ may be chosen to be a polynomial derivation of degree strictly less than $2^{k+1}$.

(2) For the same reason, if $U$ is assumed to be a polynomial solution of the truncated equation above, we may replace the coordinate change defined by $\exp(U)$ with the polynomial coordinate change

$$ \varphi(x) = x + U(x), $$

which corresponds to retaining only the first two terms of the exponential series (10). This is precisely the choice adopted in the work of Bruno in [2].

Applying recursively the previous Proposition, we obtain the following formal normal form result:

**Theorem 3.5.** Let $\partial$ be a formal derivation in $\tilde{D}$. Then, there exist a formal automorphism $\Phi$ such that the conjugated derivation

$$ \delta = \Phi \partial \Phi^{-1} $$

is in normal form modulo its Bruno ideal $B(\delta)$.

*Proof.* For each $k \in \mathbb{N}$, consider the automorphism $\Phi_k = \exp(U_k)$, where $U = U_k$ is defined by the Proposition 3.3. Since $\mathrm{ord}(U_k) \ge 2^k$, it follows from the exponential series (10) that we can write

$$ \Phi_k = \mathrm{Id} + \phi_k $$

where $\phi_k$ is an endomorphism mapping $\tilde{O}$ to $\mathbf{m}^{(k)}$. Hence, the sequence of automorphisms

$$ \Phi_k \circ \dots \circ \Phi_1, \quad k \in \mathbb{N} $$

converges (with respect to the Krull topology) to a formal automorphism $\Phi$.

Moreover, the resulting derivation $\delta = \Phi \partial \Phi^{-1}$ is such that, for all $k \in \mathbb{N}$, its truncation $\delta^{(k)}$ satisfies the hypothesis $(H_k)$. Therefore, by considering the (inductive) limit, we conclude that $\delta$ is in normal form modulo $B(\delta)$. $\square$

<figure><img src="image_3.png" /></figure>

## 4. ANALYTIC REDUCTION TO NORMAL FORM MODULO THE BRUNO IDEAL

In this section, we prove that under the $\omega$-condition, the formal reduction to normal form constructed in the previous section is analytic, provided that $\partial$ is analytic.

As we mentioned in the introduction, the key analytic estimates rely on the original ideas of Bruno, detailed in [2]. However, we will reformulate these estimates in terms of the natural $S$-graduation of the space of vector fields given by (8), in the spirit of Martinet's survey [15].

For the sake of clarity, the next subsection is devoted to establishing some basic general facts about majorant norms for analytic germs and analytic derivations.
4.1. Analytic r-norms. We denote by $D_r = \{\mathbf{x} \in \mathbb{C}^n : |x_i| < r, i = 1, \dots, n\}$ polydisk of radius $r > 0$, and by $\mathcal{O}(D_r)$ the ring of series which are absolutely convergent on the closure $D_r$. In other words, a series

$$f = \sum_{\underline{m} \in \mathbb{N}^n} a_{\underline{m}} \mathbf{x}^{\underline{m}}$$

belongs to $\mathcal{O}(D_r)$ if its r-majorant norm (or simply r-norm)

$$\|f\|_r := \sum_{\underline{m} \in \mathbb{N}^n} |a_{\underline{m}}| r^{\|\underline{m}\|}$$

is finite, where we note $\|\underline{m}\| = \sum_{i=1}^n |m_i|$. Observe that each $f \in \mathcal{O}(D_r)$ is an analytic function on $D_r$, and hence defines an analytic germ. Reciprocally, every analytic germ admits a representative in $\mathcal{O}(D_r)$ for some $r > 0$. We also have

$$\|f \cdot g\|_r \le \|f\|_r \cdot \|g\|_r$$

which implies that $\mathcal{O}(D_r)$ is a Banach algebra with respect to the above norm.

Similarly, we define the r-norm for a logarithmic derivation $\partial$ as follows. We consider its monomial expansion (see 4)

$$\partial = \sum_{\underline{m} \in \mathbb{N}^n} \mathbf{x}^{\underline{m}} L(\lambda_{\underline{m}})$$

and set

$$\|\partial\|_r := \sum_{\underline{m} \in \mathbb{N}^n} r^{\|\underline{m}\|} \|\lambda_{\underline{m}}\|$$

where, as previously, we define $\|\lambda\| = \sum_{i=1}^n |\lambda_i|$. We denote by $\mathcal{D}(D_r)$ the space of logarithmic derivations with finite r-norm. As above, we observe that each $\partial \in \mathcal{D}(D_r)$ defines a germ of analytic derivation, i.e. an element of $\mathcal{D}$. We also have

$$\|f \cdot \partial\|_r \le \|f\|_r \cdot \|\partial\|_r$$

for all $f \in \mathcal{O}(D_r)$ and $\partial \in \mathcal{D}(D_r)$.

Remark 4.1. Consider the expansion of $\partial$ with respect to an arbitrary logarithmic basis $\{L(\mu_0), \dots, L(\mu_{n-1})\}$

$$\partial = \sum_{i=0}^{n-1} g_i L(\mu_i),$$

as in subsection 2.2. Then it follows from Remark 2.1 that there exists two constants $0 < c < d$ (depending only on $\mu_0, \dots, \mu_{n-1}$) such that

$$c \|\partial\|_r \le \sum_{i=0}^{n-1} \|g_i\|_r |\mu_i| \le d \|\partial\|_r$$

In particular, $\partial \in \mathcal{D}(D_r)$ if and only if $g_0, \dots, g_{n-1} \in \mathcal{O}(D_r)$.

We now list some useful properties of the r-norm.

(1) Let $\varphi = (\varphi_1, \dots, \varphi_n) \in \mathcal{O}(D_\rho)^n$ be such that $\max\{\|\varphi_1\|_\rho, \dots, \|\varphi_n\|_\rho\} \le r$ for some $\rho, r > 0$. Then, $\varphi(D_\rho) \subseteq D_r$ and the composed function $f \circ \varphi = f(\varphi_1, \dots, \varphi_n)$ satisfies

$$\|f \circ \varphi\|_{\rho} \le \|f\|_{r}$$

for all functions $f \in \mathcal{O}(D_r)$.

(2) Suppose that $\mathrm{ord}(f) \ge k$ (i.e. that $f \in \mathfrak{m}^k$), and let $0 < \rho < r$. Then, we have the following relation between the $\rho$-norm and the $r$-norm.

$$\|f\|_{\rho} = \sum_{\underline{m} \in \mathbb{N}^n} |a_{\underline{m}}| \rho^{\|\underline{m}\|} = \sum_{\underline{m} \in \mathbb{N}^n} |a_{\underline{m}}| \left(\frac{\rho}{r}\right)^{\|\underline{m}\|} r^{\|\underline{m}\|} \le \left(\frac{\rho}{r}\right)^k \|f\|_r,$$

where the last inequality follows from the fact that $a_{\underline{m}}$ vanishes for $\|\underline{m}\| < k$.

(3) Similarly for $\partial \in \mathcal{D}(D_r)$ a derivation such that $\mathrm{ord}(\partial) \ge k$ and $0 < \rho < r$, we have

$$\|\partial\|_{\rho} \le \left(\frac{\rho}{r}\right)^k \|\partial\|_{r}.$$
(4) Given a monomial derivation $x^mL(\lambda)$ and a monomial $x^n$, the identity (2) gives the estimate

$$ \|x^m L(\lambda) x^n\|_r \le |\langle\lambda, \underline{n}\rangle| \|x^{m+n}\|_r \le \|\underline{n}\| \|x^m L(\lambda)\|_r \|x^n\|_r \quad (19) $$

where we have used the trivial inequality $|\langle u, v\rangle| \le \|u\| \|v\|$. Similarly, given two monomial derivations $x^mL(\lambda)$, $x^nL(\mu)$, the identity (3) yields the following key estimate for their Lie bracket:

$$ \begin{aligned} \| [x^m L(\lambda), x^n L(\mu)] \|_r &\le \|\lambda\| \|\underline{n}\| \|x^{m+n} L(\mu)\|_r \\ &\quad + \|\mu\| \|\underline{m}\| \|x^{m+n} L(\lambda)\|_r \\ &\le (\|\underline{n}\| + \|\underline{m}\|) \|x^m L(\lambda)\|_r \|x^n L(\mu)\|_r \end{aligned} $$

(5) More generally, if we consider two derivations $H, K \in \mathcal{D}(D_r)$ with monomial expansions

$$ H = \sum_{\underline{m} \in \mathbb{N}^n} \mathbf{x}^m L(\lambda_{\underline{m}}), \quad K = \sum_{\underline{n} \in \mathbb{N}^n} \mathbf{x}^n L(\mu_{\underline{n}}) $$

Then, the following estimate holds for the Lie Bracket $[H, K]$,

$$ \| [H, K] \|_r \le \sum_{\underline{m}, \underline{n}} (\|\underline{m}\| + \|\underline{n}\|) \|\mathbf{x}^m L(\lambda_{\underline{m}})\|_r \|\mathbf{x}^n L(\mu_{\underline{n}})\|_r $$

In particular, if we assume that $H$ and $K$ are both polynomial derivations (see 5), then

$$ \|[H, K]\|_r \le (\deg(H) + \deg(L)) \|H\|_r \|K\|_r $$

(6) Similarly, if $f = \sum_{\underline{m}} a_{\underline{m}} x^m$ is a polynomial function and $\partial \in \mathcal{D}(D_r)$ is an arbitrary derivation then
$$ \|\partial(p)\|_r \le \deg(f) \|\partial\|_r \|f\|_r $$

We now derive a simple estimate for the flow maps $(t, x) \mapsto \Phi_{\partial}(t, x)$ associated with analytic derivations $\partial$. This estimate will be used to control the majorant norm under coordinate changes defined by such flows.

Assuming that $\partial \in \mathcal{D}(D_r)$, we say the flow $\Phi_{\partial}$ is defined up to time 1 at a point $x \in \mathbb{C}^n$ if

$$ \Phi_{\partial}(t, x) \in D_r $$

for all complex times $t \in \mathbb{D} = \{z \in \mathbb{C} : |z| \le 1\}$.

**Lemma 4.2.** Suppose that there exists a constant $0 \le c < r$ such that

$$ \|\partial\|_r \le c/r $$

Then, the flow of $\partial$ is defined up to time 1 for all points lying in the polydisk $D_{r-c}$ of radius $r-c$.

*Proof.* Let us note by $\Phi_1(t, x), \dots, \Phi_n(t, x)$ the $n$-components of the flow $\Phi = \Phi_{\partial}$ of $\partial$. We observe that, for all $x \in D_r$ and all sufficiently small $t \in \mathbb{C}$ (depending on $x$), each component $\Phi_i$ of $\Phi$ satisfies the integral equation

$$ \Phi_i(t, x) = x_i + \int_0^t f_i(\Phi_i(s, x)) ds \quad (20) $$

where the integral is computed over segment $[0, t]$ and $f_i = \partial(x_i)$ is the function obtained by applying $\partial$, as a derivation, to the coordinate $x_i$. We now observe, using the monomial expansion of $\partial$, that

$$ f_i = \sum_{\underline{m} \in \mathbb{N}^n} \mathbf{x}^m L(\lambda_{\underline{m}}) x_i = \left( \sum_{\underline{m} \in \mathbb{N}^n} \mathbf{x}^m \cdot \lambda_{\underline{m}, i} \right) x_i $$

where $\lambda_{\underline{m}, i}$ denotes the $i$th component of $\lambda_{\underline{m}}$. As a consequence,

$$ \|f_i\|_r \le r \|\partial\|_r \le c $$

Suppose now that the initial point $x$ belongs to the poly-disk $D_{r-c}$. Then, assuming that $\Phi(t, x)$ is defined for $|t| \le T$, the equation (20) gives

$$ |\Phi_i(t, x)| \le |x_i| + Tc $$

Therefore, suppose by contradiction that there exists a time $t_0$ with $|t_0| \le 1$ such that the solution curve $t \mapsto \Phi(t, x)$ exits the polydisk $D_r$ at some first exit time $t_1 \in [0, t_0]$. This immediately contradicts the inequality above. $\square$
The following Corollary is an immediate consequence of the Lemma and property (1). It will allow to estimate the norm of a function under the coordinate change given by a time-1 flow.

**Corollary 4.3.** Let $\partial$ be as above and let $x \mapsto \varphi(x) = \Phi_{\partial}(1, x)$ denote the time-1 flow of $\partial$. Then

$$ \varphi(D_{r-c}) \subset D_r $$

As a consequence, for any function $f \in \mathcal{O}(D_r)$, the property (1) gives the estimate

$$ \| \exp(\partial)(f) \|_{r-c} \le \|f\|_r \qquad (21) $$

since the action of the exponential automorphism $f \mapsto \Phi(f) = \exp(\partial)(f)$ (see (10)) is precisely given by $f \mapsto f \circ \varphi$.

We conclude this subsection by studying how the $r$-norm of a derivation behaves under the coordinate change defined by $\Phi = \exp(\partial)$. We recall from subsection 2.2 that, for a derivation $\delta$, we can write

$$ \Phi \delta \Phi^{-1} = \exp(\mathrm{ad}_{\partial}) \delta = \sum_{n=0}^{\infty} \frac{1}{n!} \mathrm{ad}_{\partial}^{n}(\delta) $$

where $\mathrm{ad}_{\partial}(\delta) = [\partial, \delta]$.

We are particularly interested in the case where $\partial$ is a polynomial derivation and $\delta = L(\mu)$ is a diagonal derivation. Therefore, we start by considering the action of the iterated adjoint map

$$ \mathrm{ad}_{\partial}^{n} = \mathrm{ad}_{\partial} \circ \cdots \circ \mathrm{ad}_{\partial}, \quad n \in \mathbb{N} $$

in such case. The next enunciates refer to the order and degree of $\partial$, as defined in (5).

**Lemma 4.4.** Let $\partial$ be a polynomial derivation. Then,

$$ \deg (\mathrm{ad}_{\partial}^{n}(L(\mu))) \le n \cdot \deg(\partial) \quad \text{and} \quad \frac{1}{n!} \| \mathrm{ad}_{\partial}^{n}(L(\mu)) \|_{r} \le \left[ \deg(\partial) \left( \|\partial\|_{r} \right) \right]^{n} \|\mu\| $$

for all $\mu \in \mathbb{C}^n$.

*Proof.* Let $d = \deg(D)$. Using the properties (4) and (5) listed above, we obtain

$$ \deg (\mathrm{ad}_{\partial}^{k-1}(L(\mu))) \le d, \quad \text{and} \quad \| \mathrm{ad}_{\partial}^{k-1}(L(\mu)) \|_{r} \le d \|\partial\|_{r} \|\mu\| $$

We now proceed by induction. Assuming that

$$ \deg (\mathrm{ad}_{\partial}^{k-1}(L(\mu))) \le (k-1)d, \quad \text{and} \quad \| \mathrm{ad}_{\partial}^{k-1}(L(\mu)) \|_{r} \le (k-1)! d^{k-1} \|\partial\|_{r}^{k-1} \|\mu\| $$

The induction step follows immediately by applying again properties (4) and (5). This concludes the proof. $\square$

<figure><img src="image_4.png" /></figure>

The following Corollary is an immediate consequence of the Lemma and property (3):

**Corollary 4.5.** Let $\partial$ be a polynomial derivation. Then, for all $0 < \rho < r$ and $n \in \mathbb{N}$,

$$ \frac{1}{n!} \| \mathrm{ad}_{\partial}^{n}(L(\mu)) \|_{\rho} \le \left[ \deg(\partial) \left( \frac{\rho}{r} \right)^{\mathrm{ord}(\partial)} \|\partial\|_{r} \right]^{n} \|\mu\| $$

**4.2. Bounds on $r$-norms under conjugation.** The following technical lemma provides the first key ingredient in the induction procedure for analytic normalization. Its purpose is to control the norm of a derivation under a coordinate change defined by the flow of a polynomial vector field. In general, such a norm cannot be controlled on a polydisk of fixed radius, since it tends to blow up as one approaches the boundary. The central idea is therefore to work with three carefully chosen nested polydisks, under the constraint that their radii do not shrink too much.

We emphasize that this result is a direct consequence of several technical lemmas established in [2]. For the sake of completeness, we state it separately and provide an independent proof.

Let $k \in \mathbb{N}$, and consider three radii $0 < \rho_1 < r < \rho$ (depending on $k$) satisfying the following conditions

$$ \frac{\rho_1}{r} = \left( \frac{1}{C2^k} \right)^{C/2^k}, \quad \frac{r}{\rho} = \left( \frac{1}{C2^k} \right)^{C/2^k} \Omega_k \qquad (22) $$
for some constant $C \ge 1$ to be specified later and $(\Omega_k)_k$ some sequence of real numbers satisfying $\lim_{k \to \infty} \Omega_k = 1$, also to be specified later. Further, let $U \in \mathcal{D}(D_r)$ be a polynomial derivation satisfying the following conditions

$$ \mathrm{ord}(U) \ge 2^k, \quad \mathrm{deg}(U) \le 2^{k+1}, \quad \|U\|_r \le \frac{1}{2^k} $$

**Lemma 4.6.** There exists a $k_0 \in \mathbb{N}$ (depending only on $C$ and the sequence $(\Omega_k)$) such that, for all integers $k \ge k_0$, the following conditions hold:

1. $\varphi(D_{\rho_1}) \subset D_r$, where $\varphi$ is the time-1 flow of $U$.

2. For an arbitrary function $f \in \mathcal{O}(D_\rho)$,

$$ \| \exp(U) f \|_{\rho_1} \le \| f \|_{\rho} $$

3. If $R \in \mathcal{D}(D_\rho)$ is a derivation with $\mathrm{ord}(R) \ge 1$ then

$$ \| \exp(\mathrm{ad}_U) R \|_{\rho_1} \le \left( 1 - \frac{Ck}{2^k} \right) \| R \|_{\rho} $$

*Proof.* The item (1) will follow directly from Corollary 4.3 if we show that $\rho_1 \le r - r \|U\|_r$. Under the above hypothesis on $\|U\|_r$, this is equivalent to prove the inequality

$$ 1 - \left( \frac{1}{C2^k} \right)^{C/2^k} \ge \frac{1}{2^k} $$

But notice that the left-hand side of this inequality is equivalent to $\frac{kC \log 2}{2^k}$ as $k \to \infty$. Therefore, the inequality holds for all sufficiently large $k$.

The item (2) is an immediate consequence item (1) and equation (21).

Let us prove item (3). We initially consider the case of a monomial derivation $R = x^m L(\mu)$ and observe that we can write

$$ \exp(\mathrm{ad}_U) (x^m L(\mu)) = \exp(U) x^m \cdot \exp(\mathrm{ad}_U) L(\mu) $$

Therefore, applying separately the item (2) of the present Lemma to the monomial $x^m$ and Corollary 4.5 to the derivation $L(\mu)$, we obtain

$$ \begin{aligned} \|\exp(\mathrm{ad}_U) (x^m L(\mu))\|_{\rho_1} &\le \|\exp(U) x^m\|_{\rho_1} \cdot \|\exp(\mathrm{ad}_U) L(\mu)\|_{\rho_1} \\ &\le \|x^m\|_r \sum_{n \ge 0} \left[ \mathrm{deg}(U) \left(\frac{\rho_1}{r}\right)^{\mathrm{ord}(U)} \|U\|_r \right]^n \|\mu\| \\ &\le \|x^m\|_r \sum_{n \ge 0} \left[ 2 \left(\frac{1}{C2^k}\right)^C \right]^n \|\mu\| \end{aligned} $$

We can assume that the constant $A = 2 \left(\frac{1}{C2^k}\right)^C$ is $< 1$ by choosing $k$ sufficiently large. Therefore, applying property (1) to $x^m$ and summing the geometric series, we obtain the estimate

$$ \|\exp(\mathrm{ad}_U) (x^m L(\mu))\|_{\rho_1} \le \frac{(r/\rho)^{\|m\|}}{1-A} \|x^m\|_{\rho} \|\mu\| = \frac{(r/\rho)^{\|m\|}}{1-A} \|x^m L(\mu)\|_{\rho} $$

In the general case, under the hypothesis that $\mathrm{ord}(R) \ge 1$, we can write the monomial expansion of $R$ as

$$ R = \sum_{\|m\| \ge 1} x^m L(\lambda_m) $$

Applying the above estimate to each monomial derivation $x^m L(\lambda_m)$, we conclude that

$$ \| \exp(\mathrm{ad}_U) R \|_{\rho_1} \le \frac{r/\rho}{1-A} \| R \|_{\rho} $$

Using that $\frac{1}{1-A} \le 1 + 2A$ for $A \le 1/2$ we obtain the equivalence

$$ \frac{r/\rho}{1-A} \le \Omega_k \frac{1}{(C2^k)^{C/2^k}} \left( 1 + 4 \frac{1}{C^2 2^{kC}} \right) = 1 - \frac{Ck \log 2}{2^k} (1 + o(1)) $$
where $o(1)$ indicates a term which goes to zero as $k \to \infty$. As a consequence, there exists a $k_0 \in \mathbb{N}$ such that

$$ \| \exp(\mathrm{ad}_U) R \|_{\rho_1} \le \left(1 - \frac{Ck}{2^k}\right) \|R\|_{\rho} $$

for all $k \ge k_0$.

<figure><img src="image_5.png" /></figure>

We now suppose that $\partial \in \mathcal{D}(D_\rho)$ has the form of an $S$-perturbation (see Definition 2.11). In other words, that $\partial$ can be written as

$$ \partial = S + R $$

where $R$ is a derivation with $\mathrm{ord}(R) \ge 1$. Note that the conjugated derivation $\partial_1 = \exp(\mathrm{ad}_U)\partial$ also has the form of a $S$-perturbation. Indeed, we can write $\partial_1 = S + R_1$ with

$$ R_1 = (\exp(\mathrm{ad}_U) - \mathrm{Id})S + \exp(\mathrm{ad}_U)R $$

which implies that $\mathrm{ord}(R_1) \ge \min\{\mathrm{ord}(U), \mathrm{ord}(R)\} \ge 1$. Using the previous Lemma, we obtain the following:

**Corollary 4.7.** Suppose given a constant $\Delta > 0$. Then, under the above hypothesis, there exists a constant $k_0 \in \mathbb{N}$ (depending only on $\lambda, \Delta, C$ and the sequence $(\Omega_k)$) such that

$$ \|R\|_{\rho} \le \Delta \Rightarrow \|R_1\|_{\rho_1} \le \Delta $$

for all $k \ge k_0$.

*Proof.* Using the estimates of Corollary 4.5 and the fact that $S = L(\lambda)$ we have

$$ \|(\exp(\mathrm{ad}_U) - \mathrm{Id})S\| \le \frac{A}{1-A} \|\lambda\| $$

where $A = 2 \left(\frac{1}{C2^k}\right)^C$ as in the previous Lemma. In particular,

$$ \frac{A}{1-A} \|\lambda\| \sim 2 \left(\frac{1}{C2^k}\right)^C \|\lambda\| $$

as $k \to \infty$. As a consequence, there exists a value of $k_0$ (depending on $\Delta$) such that

$$ \frac{A}{1-A} \|\lambda\| \le \frac{Ck}{2^k} \Delta $$

for all $k \ge k_0$. Therefore, based on the item (3) of the previous Lemma, one obtains

$$ \|R_1\|_{\rho_1} \le \frac{A}{1-A} \|\lambda\| + \left(1 - \frac{Ck}{2^k}\right) \|R\|_{\rho} \le \frac{Ck}{2^k} \Delta + \left(1 - \frac{Ck}{2^k}\right) \Delta = \Delta $$

for sufficiently large values of $k$.

<figure><img src="image_6.png" /></figure>

**4.3. Analytic estimates for the bracket equation.** We now provide the second main ingredient in the induction procedure for analytic normalization. The goal is to control the norm of the solution $U$ of the bracket equation (18) in terms of the norm of $W$. The $\omega$-condition on the eigenvalues of $S$ will be one of the essential ingredient here (even if it is not used in its full-force).

Let $S = L(\lambda)$ be a diagonal derivation satisfying the $\omega$-condition. Given $k \in \mathbb{N}$, we consider again three radii $0 < \rho_1 < r < \rho$ (depending on $k$) defined by (22), where we now additionally suppose that

$$ C \ge 3, \quad \Omega_k := (\omega_k)^{C/2^k} $$

where $(\omega_k)_k = (\omega_k(\lambda))_k$ is the sequence defined in the Introduction in terms of the eigenvalues of $S$. Note that the $\omega$-condition implies that

$$ \lim_{k \to \infty} \Omega_k = 1 $$

Suppose given also an analytic germ $f_0 \in \mathcal{O}$ belonging to $\mathrm{Gr}_0(\mathcal{O}, S)$, i.e. such that $S(f_0) = 0$.

**Lemma 4.8.** Suppose that $W$ is a polynomial derivation satisfying the following conditions:

$$ \mathrm{ord}(W) \ge 2^k, \quad \mathrm{deg}(W) \le 2^{k+1} $$

Then, there exists a unique polynomial derivation $U$ which solves the truncated bracket equation

$$ [(1 + f_0)S, U] = -W_* \mod \mathbf{m}^{(k+1)} \quad (23) $$
and additionally satisfies the following three conditions

$$ \mathrm{ord}(U) \ge 2^k, \quad \mathrm{deg}(U) < 2^{k+1} \quad \text{and} \quad U = U_* $$

Moreover, there exists a $k_0 \in \mathbb{N}$ (depending only on $\|\lambda\|$ and the sequence $(\omega_k)$) such that, for all $k \ge k_0$, the conditions

$$ \|f_0\|_{\rho} \le \frac{1}{2} \quad \text{and} \quad \|W\|_{\rho} \le 1 $$

imply that $\|U\|_r \le \frac{1}{2^k}$.

**Remark 4.9.** Notice that we only require $U$ to solve the bracket equation up to flat terms of order $2^{k+1}$. This is the reason why we can obtain a polynomial solution.

*Proof.* We have seen in Corollary 3.2 that the equation

$$ [(1 + f_0)S, V] = -W_* $$

has a unique solution $V \in \mathcal{D}$ such that $\mathrm{ord}(V) \ge 2^k$ and $V = V_*$. Let us write the monomial expansion of such solution $V$ as

$$ V = \sum_{\|m\| \ge 2^k} x^m L(\lambda_m) $$

Then, the derivation defined by its truncation at degree $2^{k+1}$,

$$ U = \sum_{2^k \le \|m\| < 2^{k+1}} x^m L(\lambda_m) $$

is the unique polynomial solution satisfying both the bracket equation modulo $m^{(k+1)}$, and the required conditions on the order and the degree stated in the enunciate.

It remains to show the bound on the $r$-norm $\|U\|_r$. We initially consider the case where $W = x^m L(\mu)$ is a monomial derivation belonging to $\mathrm{Gr}_\alpha(S, \mathcal{D})$, i.e. such that $\langle \lambda, \underline{m} \rangle = \alpha$. Then, from the explicit formula for $\mathrm{ad}_{(1+f_0)S}^{-1}$ given in Lemma 3.1 we obtain the estimate

$$ \begin{aligned} \| \mathrm{ad}_{(1+f_0)S}^{-1} (x^m L(\mu)) \|_{\rho} &\le \left( \frac{\|\mu\|}{\alpha(1-\|f_0\|_{\rho})} + \frac{\|L(\mu)f_0^{(k)}\|_{\rho}}{\alpha^2(1-\|f_0\|_{\rho})^2} \|\lambda\| \right) \|x^m\|_{\rho} \\ &\le \left( \frac{1}{\omega_k(1-\|f_0\|_{\rho})} + \frac{2^k \|f_0\|_{\rho}}{\omega_k^2(1-\|f_0\|_{\rho})^2} \|\lambda\| \right) \|x^m\|_{\rho} \|\mu\| \\ &\le \left( \frac{2}{\omega_k} + \frac{2^{k+1}}{\omega_k^2} \|\lambda\| \right) \|x^m L(\mu)\|_{\rho} \\ &\le \left( \frac{2}{\omega_k} \right) \left( 1 + \frac{2^k}{\omega_k^2} \|\lambda\| \right) \|x^m L(\mu)\|_{\rho} \end{aligned} $$

where we used property (6) of subsection 4.1 to write that

$$ \|L(\mu)f_0^{(k)}\|_{\rho} \le \|\mu\| \deg(f_0^{(k)}) \|f_0^{(k)}\| \le \|\mu\| 2^k \|f_0\|_{\rho} $$

We consider now the case where $W$ is an arbitrary polynomial derivation satisfying the conditions of the enunciate. Note that we can write $S$-weighted decomposition of $W_*$ as

$$ W_* = \sum_{\alpha \ne 0} W_\alpha $$

where each $W_\alpha \in \mathrm{Gr}_\alpha(S, \mathcal{D})$ is a finite sum of monomial derivations. Applying the above estimate, and the triangle inequality, we obtain

$$ \|U\|_{\rho} \le \left( \frac{2}{\omega_k} \right) \left( 1 + \frac{2^{k+2}}{\omega_k} \|\lambda\| \right) \|W\|_{\rho} $$

Since $\mathrm{ord}(U) \ge 2^k$, it follows from property (3) of subsection 4.1 that

$$ \|U\|_r \le \left( \frac{r}{\rho} \right)^{2^k} \|U\|_{\rho} $$
In view of the definition of the ratio $r/\rho$ in (22), we get

$$ \|U\|_r \le \left(\frac{\omega_k}{C2^k}\right)^C \left(\frac{2}{\omega_k}\right) \left(1 + \frac{2^k}{\omega_k}\|\lambda\|\right) \sim \frac{2\omega_k^{C-2}}{C^C} \|\lambda\| \frac{1}{2^{k(C-1)}} $$

Since $C \ge 3$ and $\omega_k \le \|\lambda\|$ for all $k$, we conclude that $\|U\|_r \le \frac{1}{2^k}$ for all sufficiently large values of $k$. $\square$

### 4.4. Proof of the Main theorem: Analytic normalization modulo the Bruno ideal.
We will fix once and for all a logarithmic basis $\{S, T_1, \dots, T_n\}$ so that each analytic derivation $R$ has an expansion

$$ R = fS + \sum_{j=1}^{n-1} g_j T_j $$

with $f, g_j$ analytic germs. Further, it follows from Remark 4.1 that there exists an absolute constant $d > 0$ (depending only on the choice of the basis $\{S, T_1, \dots, T_{n-1}\}$) such that for all derivations $R$ and all radius $\rho > 0$ we have

$$ \max\{\|f\|_{\rho}, \|g_1\|_{\rho}, \dots, \|g_{n-1}\|_{\rho}\} \le d\|R\|_{\rho} \quad (24) $$

In order to start the inductive procedure, we chose the following constants

$$ \Delta = \min \left\{ 1, \frac{1}{2d} \right\}, \quad C = 3 $$

and choose $k_0 \in \mathbb{N}$ as the maximum of the integers $k_0$ given by Corollary 4.7 and Lemma 4.8 (once fixed the above choices of $\Delta$ and $C$).

Consider now an analytic $S$-perturbation $\partial = S+R$. Before applying the analytic normalization procedure, we need to perform the following preparation steps:

(1) Using the Remark 3.4, we can assume, up to a polynomial change of coordinates, that the hypothesis $(H_{k_0})$ of subsection 3.2 holds for $\partial$, i.e. that the $m^{2k_0}$-truncation of $\partial^{(k_0)}$ is in normal form modulo $B(\partial^{(k_0)})$.

(2) Up to a homothecy, we can assume that $R \in \mathcal{D}(D_{\rho})$ with $\rho = 1$.

(3) Up to replacing $R$ by some multiple $(1/\mu)R$ with $\mu > 0$, we assume that

$$ \|R\|_{\rho} \le \Delta $$

Under these conditions, the following properties hold for $k = k_0$:

**$P_k(1)$:** If we denote by $f_0$ the degree-0 component of $f$ with respect to the $S$-graduation then

$$ \|f_0\|_{\rho} \le \|f\|_{\rho} \le \frac{1}{2}. $$

Note that the leftmost inequality follows simply from the fact that the power series of $f_0$ is extracted from the power series of $f$ and the rightmost inequality follows from (24).

**$P_k(2)$:** If we denote by $W_*$ the polynomial derivation defined by (17) then

$$ \|W_*\|_{\rho} \le \|R\|_{\rho} \le 1 $$

As above, the leftmost inequality follows from the fact that the monomial expansion of $W_*$ is extracted from the monomial expansion of $R$.

As a consequence, we are precisely in the conditions of Lemma 4.8, which guarantees that the polynomial solution $U$ of the truncated bracket equation (23) satisfies the estimate

$$ \|U\|_r \le \frac{1}{2^k} $$

Therefore, we can apply Corollary 4.7 which this choice of $U$. It guarantees that under the coordinate change given by the automorphism $\Phi = \exp(U)$ one obtains a conjugated derivation $\partial_1 = S + R_1$ such that

$$ \|R_1\|_{\rho_1} \le \Delta $$

where we recall that $\rho_1$ is given in terms of $\rho$ by (22).

Expanding the new derivation $\partial_1$ with respect to the fixed logarithmic basis $\{S, T_j\}$, one concludes that it satisfies the conditions $P_k(1)$ and $P_k(2)$ with now $k = k_0 + 1$. Therefore, the induction procedure can be continued.
As in the proof of Theorem 3.5, this inductive procedure produces a sequence of automorphisms

$$ \Phi_1, \Phi_1 \circ \Phi_2, \dots, \Phi_1 \circ \dots \circ \Phi_s, \dots $$

which converges (in the Krull topology) to a formal automorphism $\Phi$. However, the above analytic estimates allows also to consider the associated sequence of coordinate changes

$$ \begin{align*} \varphi_1 : D_{\rho_1} &\to D_{\rho} \\ \varphi_1 \circ \varphi_2 : D_{\rho_2} &\to D_{\rho} \\ \varphi_1 \circ \dots \circ \varphi_s : D_{\rho_s} &\to D_{\rho}, \quad s \ge 1 \end{align*} $$

where $\rho_s$ is given by

$$ \rho_s = \rho \prod_{k=k_0}^{k_0+s} \left( \frac{\omega_k}{C^{2k}} \right)^{\frac{C}{2k}} $$

To conclude the proof, it suffices to show that the sequence of radii $(\rho_s)_{s \ge 1}$ converges to a strictly positive limit $\bar{\rho}$. Indeed, this will imply that the formal automorphism $\Phi$ constructed above defines an analytic coordinate change with domain the polydisk $D_{\bar{\rho}}$.

Since the general term of the above product converges to 1, it is enough to establish the convergence of the series

$$ \sum_{k \ge k_0} \frac{\log(\omega_k)}{C 2^k} - \sum_{k \ge k_0} \frac{2Ck \log 2 + C \log C}{2^k} $$

This follows immediately from the $\omega$-condition. The proof of the Main Theorem is therefore complete.

### 5. EXAMPLES AND APPLICATIONS OF THE BRUNO IDEAL

We now present applications of the Main Theorem and its corollary, starting with two-dimensional vector fields.

*Example 5.1.* Let $\partial = S + R$ be a formal logarithmic vector field with linear part

$$ S = \lambda x \frac{\partial}{\partial x} + \mu y \frac{\partial}{\partial y}, $$

where $\lambda \in \mathbb{C}^*$ and $\mu \in \mathbb{C}$. Assuming that $\partial$ is in normal form, the perturbation term $R$ can be written as

$$ R = GS + F y \frac{\partial}{\partial y}, $$

where $F, G \in \mathbb{C}[[x, y]]$ are formal power series without constant term having an expansion of the form

$$ F = \sum_{\substack{(k,l) \in \mathbb{N}^2 \\ \lambda k + \mu l = 0}} a_{k,l} x^k y^l, \quad G = \sum_{\substack{(k,l) \in \mathbb{N}^2 \\ \lambda k + \mu l = 0}} b_{k,l} x^k y^l. $$

The Bruno ideal is therefore given by $B(\partial) = \langle F \rangle$ and we distinguish two cases:

* Resonant case. If $\mu/\lambda \in \mathbb{Q}_{\ge 0}$, then writing $\mu/\lambda = -m/n$ in irreducible terms, we have

$$ F, G \in \mathbb{C}[[x^n y^m]] \cap \mathfrak{m}. $$

* Non-resonant case. If $\mu/\lambda \notin \mathbb{Q}_{\ge 0}$, then $F = 0, G = 0$.

Let us now consider an arbitrary analytic vector field $\delta$ that is formally conjugated to $\partial$, and examine the information provided by the Main Theorem.

Assume first that $\partial$ is in the resonant case. In this situation, the $\omega$-condition is always satisfied, and therefore

$$ V(B(\delta)) \subset (\mathbb{C}^2, 0) $$

is a germ of an analytic variety.

We further distinguish two subcases. If $F = 0$, then $V(B(\delta)) = (\mathbb{C}^2, 0)$, and, up to an analytic change of coordinates, we may write

$$ \delta = u \left( nx \frac{\partial}{\partial x} - my \frac{\partial}{\partial y} \right), $$
for some analytic unit $u \in \mathbb{C}\{x^n y^m\}$. On the other hand, if $F \neq 0$, then $V(B(\delta))$ defines a germ of an analytic variety with either one irreducible smooth component (if $\mu = 0$) or two irreducible smooth components (if $\mu \neq 0$), intersecting transversely.

Finally, assume that $\partial$ is in the non-resonant case and that the $\omega$-condition holds. Then $V(B(\delta)) = (\mathbb{C}^2, 0)$, and the Main Theorem implies that $\delta$ is analytically linearizable.

We observe that the resonant case above with $F=0$ corresponds to a classical theorem of Dulac: if a germ of a two-dimensional vector field with non-vanishing linear part admits a formal first integral, then it also admits an analytic first integral.

More interesting examples arise in higher dimension. The following result provides an alternative proof of Lyapunov's theorem for logarithmic vector fields. Consider a germ of real analytic vector field of the form

$$ \partial = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y} + \lambda z \frac{\partial}{\partial z} + R $$

where $\lambda \in \mathbb{R}^*$ and $R$ is 1-flat. By the Center Manifold Theorem, $\partial$ always has a formal invariant manifold $\tilde{W}^c$ tangent to the $z = 0$ direction, called its center manifold. We say that $\partial$ if formally integrable on its center manifold if the restriction of $\partial$ to $\tilde{W}^c$ has a formal fist integral. On the other hand, we say that $\partial$ is analytic integrable on its center manifold if $\tilde{W}^c$ is a germ of analytic variety and the restriction of $\partial$ to $\tilde{W}^c$ has an analytic first integral.

**Proposition 5.2** (Lyapunov's theorem [14]). $\partial$ is formally integrable on its center manifold if and only if it is analytic integrable on its center manifold.

*Proof.* Suppose that $B(\partial) = 0$, then the result is obvious since the foliation is necessarily linearizable, being $z = 0$ its analytically integrable center manifold. Now suppose $B(\partial) \neq 0$ and that the vector field is formally integrable in $\tilde{W}^c$. Then it follows from the definition of $B(\partial)$ that $\tilde{W}^c$ is an irreducible component of $B(\partial)$, therefore it is analytic. $\square$

Related to Lyapunov's theorem, we get also a relative easier proof for the following result, see [1, 6, 13].

**Proposition 5.3.** Let $\partial$ be a vector field in the hypotheses of the previous Proposition. There is an infinite family $\{C_i\}_{i \in I}$ of cycles accumulating at the origin if and only if $\partial$ is analytically integrable on its center manifold.

*Proof.* We only prove the direct implication since the converse is a direct consequence of the previous result. We note that there are many steps in common with respect to the proof in [6]. Suppose that there is an infinite family $\{C_i\}_{i \in I}$ of cycles accumulating at the origin. Since the eigenvalues of the linear part of $\partial$ are $\lambda, \pm i$, there is an analytic one-dimensional stable (unstable) manifold $\Omega$ associated to $\lambda < 0$ ($\lambda > 0$) and a formal two-dimensional center manifold associated to $i, -i$.

We start making a cylindric blow-up centered at the stable manifold. The fiber $\pi^{-1}(0)$ is the only cycle defined by the strict transform $\tilde{\partial}$ of $\partial$ in $\pi^{-1}\Omega$. Based on this cycle, we can define the Poincaré map on an analytic transverse section $\Delta$. The Poincaré map $P: \Delta' \subset \Delta \to \Delta$ is an analytic map given by the first intersection of the trajectory starting at a point $p \in \Delta'$ with $\Delta$. We will study the cycles of $\tilde{\partial}$ near $\pi^{-1}(0)$ by means of studying the periodic points of the Poincaré map, since they are one-to-one related.

The map $P$ has eigenvalues $1, e^{\lambda}$. Therefore, it has a one-dimensional stable (unstable) manifold associated to the eigenvalue $e^{\lambda} < 1$ ($e^{\lambda} > 1$), and for each $k \in \mathbb{N}$ a $C^k$ one-dimensional center manifold $\Gamma_k$ associated to the eigenvalue 1. By the properties of the center manifolds [12], the curve $\Gamma$ contains the periodic points of $P$. On the other hand, the saturation of such curve provides an invariant surface, which means that the cycles are contained in a surface, and consequently, they make a single turn and they must be one-to-one related to the fixed points of $P$. Then, since $\{C_i\}_{i \in I}$ is an infinite family of cycles accumulating to 0, the Poincaré map has an infinite family of fixed points accumulating to $\pi^{-1}(0) \cap \Delta$. Since $\text{Fix}(P) \subseteq \Gamma_k$ is an analytic set, we have that $\text{Fix}(P)$ must be a curve, and its saturation provides an analytic surface $X$.

The surface $X$ is also invariant for its semi-simple part $\partial_{ss}$ of $\partial$. Notice that $\partial_{ss}$ only has two possible invariant manifolds: one curve tangent to the eigenspace of $\lambda$ and a surface tangent to the eigenspaces of $i, -i$. By the two-dimensional Dulac's theorem, we conclude the analytic integrability of $\partial$ in $X$. $\square$
Suppose that the linear part of $\partial$ is as in Proposition 5.2 but now supposing that $\lambda = 0$. Since there is a zero eigenvalue, more intricate Bruno varieties can arise.

*Example 5.4. Let*

$$ \partial = ix \frac{\partial}{\partial x} - iy \frac{\partial}{\partial y} + (xy - z^2)(x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}) + (xy - z^2)z \frac{\partial}{\partial z} $$

be a vector field written on the logarithmic basis $S = ix \frac{\partial}{\partial x} - iy \frac{\partial}{\partial y}$, $T_1 = x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$, $T_2 = z \frac{\partial}{\partial z}$. As in the previous examples, $\partial$ is already in normal form, being $\partial_{ss} = ix \frac{\partial}{\partial x} - iy \frac{\partial}{\partial y}$ and $\partial_{nilp} = (xy - z^2)(x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}) + (xy - z^2)z \frac{\partial}{\partial z}$. We have $\partial_{ss} \wedge \partial_{nilp} = (xy - z^2)(S \wedge T_1 + S \wedge T_2)$, thus

$$ B(\partial) = \langle xy - z^2 \rangle. $$

It produces an invariant analytic variety given by $V(B(\partial)) = \{xy - z^2 = 0\}$, which is a surface in $\mathbb{C}^3$.

This example has been carefully chosen in order to show the existence of surfaces filled with cycles. In particular, notice that the real vector field

$$ \delta = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y} + (x^2 + y^2 - z^2)(x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}) + (x^2 + y^2 - z^2)z \frac{\partial}{\partial z}, $$

is conjugated to $\partial$ by the automorphism $\Phi(x, y, z) = (x + iy, x - iy, z)$. We observe that the real analytic surface $V(B(\partial)) = V(\Phi^*(B(\partial))) = \{x^2 + y^2 - z^2 = 0\}$ is filled with cycles.

We conclude this section by considering the quasi-periodic behavior of the foliation defined by $\partial$ in restriction to its Bruno ideal.

Let us assume that $\partial = S + R$ is an analytic $S$-perturbation such that $B(\partial)$ is analytic. Considering the quotient algebra $A = \mathcal{O}/B(\partial)$, it follows from Corollary 2.14 that we can write

$$ \partial_A = (1 + f_0)S_A $$

for some analytic germ $f_0 \in \text{Gr}_0(\mathcal{O}, S) \cap \mathfrak{m}$.

Therefore, following [4], one can further consider the period fibration on $V(B(\partial))$ defined by $\{f_0 = \text{cte}\}$. It is easy to prove that each level set

$$ F_a = V(B(\partial)) \cap f_0^{-1}(\{a\}) \qquad (25) $$

is an $\partial$-invariant analytic set. Moreover, in restriction to $F_a$, the local flow of $\partial$ is simply given by

$$ \Phi(t, x_1, \dots, x_n) = (e^{t(1+a)\lambda_1} x_1, \dots, e^{t(1+a)\lambda_n} x_n) $$

Let us now assume $\partial = S + R$ be a real analytic derivation such that

$$ S = \omega_1 \left( -y_1 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial y_1} \right) + \dots + \omega_n \left( -y_n \frac{\partial}{\partial x_n} + x_n \frac{\partial}{\partial y_n} \right) $$

with $\omega_1, \dots, \omega_n \in \mathbb{R}^*$ and $R$ 1-flat. Suppose further that the vector

$$ \lambda = (\pm \omega_1, \dots, \pm \omega_n) \in \mathbb{R}^{2n} $$

satisfies the $\omega$-condition. Then it follows from the Main theorem that $B(\partial)$ is analytic and that $V = V(B(\partial))$ is a (germ of) analytic variety.

**Lemma 5.5.** For each point $p \in V$ there exists a constant $a \in \mathbb{R}$ such that the orbit of $\partial$ through $p$ lies in a quasi-periodic torus $\mathbb{T}_p \subset V$ with a set of frequencies in $\{(1+a)\omega_1, \dots, (1+a)\omega_n\}$.

*Proof.* Notice that the foliation defined by $S$ fills the space with invariant tori, i.e. each trajectory is a quasi-periodic orbit contained in a tori of dimension ranging from 0 to $n$. In fact, $k$-dimensional tori for $k < n$ are contained in some $2k$-plane $\{x_{i_1} = y_{i_1} = \dots = x_{i_{n-k}} = y_{i_{n-k}} = 0\}$ for $1 \le i_1 < \dots < i_{n-k} \le n$. Each point $p \in V$ lies in some level set $F_a$ defined by (25) and we have $\partial = (1+a)S$ in restriction to $F_a$. Therefore, $\mathbb{T}_p \subset L$ is a $n$-dimensional quasi-periodic torus with frequencies $(1+a)\omega_1, \dots, (1+a)\omega_n$. $\square$
## ACKNOWLEDGEMENT

The second author is partially supported by the ANR Project NonSper ANR-23-CE40-0028. The first author is partially supported by the ANR Project DiffeRS ANR-25-ERCC-0003-01. The first author has also been partially supported by the mobility program "Ayudas para estancias breves en el desarrollo de tesis doctorales. Convocatoria 2025" of the University of Valladolid and by the project PID2022-139631NB-I00 funded by the Agencia Estatal de Investigación - Ministerio de Ciencia e Innovación during the realization of this work.

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UNIVERSITÉ PARIS CITÉ, SORBONNE UNIVERSITÉ, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE
*Email address: martinvega@imj-prg.fr*

DÉPARTEMENT DE MATHÉMATIQUES – IRIMAS–UHA, 18 RUE DES FRÈRES LUMIÈRE, 68093 MULHOUSE, FRANCE.
*Email address: daniel.panazzolo@uha.fr*