# On Milnor and Tjurina numbers of foliations

Arturo Fernández-Pérez  Department of Mathematics. Federal University of Minas
Gerais. Av. Antônio Carlos, 6627 CEP 31270-901
Pampulha, Belo Horizonte, Brazil
<EMAIL_ADDRESS>,  Evelia R. García Barroso  Dpto. Matemáticas, Estadística
e Investigación Operativa
Sección de Matemáticas
Universidad de La Laguna. Apartado de Correos 456. 38200 La Laguna, Tenerife,
Spain<EMAIL_ADDRESS>and  Nancy Saravia-Molina  Dpto. Ciencias - Sección
Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801,
San Miguel, Lima 32, Peru<EMAIL_ADDRESS>
###### Abstract.

We study the relationship between the Milnor and Tjurina numbers of a singular
foliation $\mathcal{F}$, in the complex plane, with respect to a balanced
divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$. For that, we
associated with $\mathcal{F}$ a new number called the $\chi$-number and we
prove that it is a $C^{1}$ invariant for holomorphic foliations. We compute
the polar excess number of $\mathcal{F}$ with respect to a balanced divisor of
separatrices $\mathcal{B}$ for $\mathcal{F}$, via the Milnor number of the
foliation, the multiplicity of some hamiltonian foliations along the
separatrices in the support of $\mathcal{B}$ and the $\chi$-number of
$\mathcal{F}$. On the other hand, we generalize, in the plane case and the
formal context, the well-known result of Gómez-Mont given in the holomorphic
context, which establishes the equality between the GSV-index of the foliation
and the difference between the Tjurina number of the foliation and the Tjurina
number of a set of separatrices of $\mathcal{F}$. Finally, we state numerical
relationships between some classic indices, as Baum-Bott, Camacho-Sad, and
variational indices of a singular foliation and its Milnor and Tjurina
numbers; and we obtain a bound for the sum of Milnor numbers of the local
separatrices of a holomorphic foliation on the complex projective plane.

###### Key words and phrases:

Tjurina number, Milnor number, $\chi$-number, Dicritical foliation.

###### 2010 Mathematics Subject Classification:

32S65 (primary), 37F75 (secundary)

The first-named author is partially supported by CNPq-Brazil Grant Number
302790/2019-5, Pronex-FAPERJ and the Research Program 2021 of Universidad de
La Laguna. The second-named and third-named authors were partially supported
by the Pontificia Universidad Católica del Perú project CAP 2019-ID 717. The
second-named author was supported by the grant PID2019-105896GB-I00 funded by
MCIN/AEI/10.13039/501100011033.

##  1\. Introduction

The Milnor and Tjurina numbers are classical invariants in the theory of
complex analytic hypersurfaces with isolated singularity. The Milnor number is
the number of spheres in the Milnor fibre of the hypersurface. The notion of
the Milnor number for hypersurfaces was introduced in [22, Section 7]. The
Tjurina number is the dimension of the base space of a semi-universal
deformation of the hypersurface. Semi-universal deformations were studied in
[29] and as far as our knowledge reaches, the name Tjurina number appears for
the first time in [16]. The Milnor number is a topological invariant and the
Tjurina number is an analytic invariant of the singularity. There is an
abundant and varied bibliography on the Milnor number of singular
hypersurfaces. The Tjurina number has not been as well studied, perhaps
because it is an analytical invariant, but in recent years new studies have
been published. In the context of singular foliations, the notion of Milnor
number appears for the first time with that name in [5], although this notion
is found in previous works such as that of Van den Essen [30], where a new
proof of Seidenberg’s theorem is given. As in the case of hypersurfaces, the
Milnor number of a one-dimensional holomorphic foliation is a topological
invariant and there is a varied bibliography on it. However, the concept of
the Tjurina number of a foliation has been less studied and according to our
knowledge, always related to the Gómez-Mont-Seade-Verjovsky index after [14].
We emphasize that Gómez-Mont did not use the terminology of the Tjurina number
of a foliation, such a name appears for the first time in [7, Page 159]. In
[18, Corollary 2.7] there is a bound of the Tjurina number of a
$\mathcal{F}$-invariant curve $C$ as a function of the Tjurina number of
$\mathcal{F}$ with respect to $C$. Some verifications on the relationship
between the Milnor and Tjurina numbers of non-dicritical foliations and their
total union of separatrices are collected in [11].

In this work we deepen into the study of the Tjurina number of a singular
foliation along a reduced curve of separatrices and its relationship with the
Milnor number and other invariants and indices associated with the foliation
such as the polar excess number, Baum-Bott, Camacho-Sad, and variational
indices. Taking into account that a singular foliation could admit many
infinitely of separatrices – dicritical foliation – the Tjurina number of a
foliation will be associated with a balanced divisor of separatrices. The
notion of balanced divisor of separatrices for a foliation was introduced by
Genzmer in [12]. This is a geometric object formed by a finite set of
separatrices, choosing all isolated separatrices and some separatrices from
the ones associated to dicritical components, with weights, possibly negative
(those that correspond to poles). In the non-dicritical case, this notion
coincides with the total union of separatrices of the foliation. A balanced
divisor of separatrices provides a control of the algebraic multiplicity of
the foliation and of its separatrices (see Proposition 2.4). In Foliation
Theory it is not easy to determine whether an invariant associated with a
foliation is topological (remember that, for example, it is not yet known if
the algebraic multiplicity is). However, it has been shown that some indices
are analytical invariants, such as the Baum-Bott index (see [9, Remark 3.2]),
Camacho-Sad and Variational indices (a good exercise for beginners in the
world of foliations). We hope that this article contributes for understanding
the dicritical foliations and the invariants associated with them.

The paper is organized as follows. In Section 2, we recall some preliminary
notions that are necessary in the paper. The first motivation for this work
was to understand the relationship between the Milnor and Tjurina numbers of a
singular foliation $\mathcal{F}$ and a balanced divisor of separatrices for
$\mathcal{F}$. It is for that in Section 3 we associate a new number
$\chi_{p}(\mathcal{F})$ to any (dicritical or non dicritical) singular
foliation $\mathcal{F}$ at $(\mathbb{C}^{2},p)$. This number is defined in
function of the algebraic multiplicities and excess tangency indices of the
strict transforms of $\mathcal{F}$ at infinitely near points of $p$. Hence
$\chi_{p}(\mathcal{F})$ is a $C^{1}$ invariant for holomorphic foliations. We
study the properties of the $\chi$-number in Proposition 3.1. In particular,
we prove that it is a nonnegative integer number and any foliation of
algebraic multiplicity bigger than 1 is of second type if and only if the
$\chi$-number equals zero.

Section 4 is devoted to the indices associated with a foliation making use of
a generic polar curve of it, as the polar intersection and the polar excess.
Proposition 4.2 provides a formula to compute the polar excess number of a
foliation with respect to the zero divisor of a reduced balanced divisor of
separatrices, and as a consequence, we obtain a characterization of
generalized curve foliations, which generalizes [7, Proposition 2] to
dicritical context. In Proposition 4.7, we establish the relationship between
the Milnor number of a foliation, the multiplicity of the foliation along the
separatrices (of a balanced divisor) and the $\chi$-number of the foliation,
generalizing [7, Corollary 2] to dicritical foliations. As a consequence we
give a new proof, using foliations, of the well-known relationship between the
Milnor number of a reduced plane curve and the Milnor numbers of its
irreducible components (see Proposition 4.8). Theorem A is the main result in
this section and one of the main results in the paper, where we compute the
polar excess number of a singular foliation $\mathcal{F}$ with respect to a
balanced divisor of separatrices $\mathcal{B}$,
$\Delta_{p}(\mathcal{F},\mathcal{B})$, via the Milnor number of the foliation,
$\mu_{p}(\mathcal{F})$, the multiplicity of some hamiltonian foliations along
the separatrices in the support of $\mathcal{B}$, $\mu_{p}(d{F}_{B},B)$, and
the $\chi$-number of $\mathcal{F}$. More precisely

######  Theorem A.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\Delta_{p}(\mathcal{F},\mathcal{B})=\mu_{p}(\mathcal{F})-\sum_{B}a_{B}\mu_{p}(d{F}_{B},B)+\deg(\mathcal{B})-1-\chi_{p}(\mathcal{F}),$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$. Moreover, if $\mathcal{F}$ of a second type foliation, then

$\Delta_{p}(\mathcal{F},\mathcal{B})=\mu_{p}(\mathcal{F})-\sum_{B}a_{B}\mu_{p}(d{F}_{B},B)+\deg(\mathcal{B})-1.$

As a consequence, in Corollary 4.10, we give a new characterization of
generalized curve foliations in the non-dicritical case.

In Section 5, we study the Gómez-Mont-Seade-Verjovsky index (GSV-index). In
particular, in Corollary 5.3, we compute the GSV-index of a foliation
$\mathcal{F}$ with respect the zero divisor of a reduced balanced divisor of
separatrices for $\mathcal{F}$. In Proposition 5.4, we generalize, to the
dicritical case, [7, Proposition 4] which establishes the equality between the
GSV-index of a foliation $\mathcal{F}$ (containing perhaps purely formal
branches) with respect to an $\mathcal{F}$-invariant curve $C:f(x,y)=0$ and
the intersection multiplicity of $C$ with a divisor which zero divisor is a
generic polar curve of $\mathcal{F}$ and its pole divisor is a generic polar
curve of $C$. We finish this section establishing, in Proposition 5.7, a
relationship between the GSV-index and the multiplicity of a foliation along a
fixed separatrix.

In Section 6, we introduce the notion of Tjurina number of a singular
foliation $\mathcal{F}$ along a reduced curve of separatrices $C$, denoted by
$\tau_{p}(\mathcal{F},C)$. Gómez-Mont proved, for a singular foliation with a
set of convergent separatrices $C$, that the difference between the Tjurina
number of the foliation and the Tjurina number of $C$, $\tau_{p}(C)$, equals
to the GSV-index (see [14, Theorem 1]). In Proposition 6.2, we show that this
result also holds, in the formal context, for the Tjurina number of a singular
foliation along a reduced curve of separatrices. As a consequence we get the
next corollary for non-dicritical foliations:

######  Corollary B.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C$ is the total union of separatrices of
$\mathcal{F}$. Then

$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},C)=\mu_{p}(C)-\tau_{p}(C)+\chi_{p}(\mathcal{F}).$

Moreover, if $\mathcal{F}$ is of second type then
$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},C)=\mu_{p}(C)-\tau_{p}(C)$.

The main result in this section, and other of the main results in the paper,
is Theorem C, where given a balanced divisor of separatrices
$\mathcal{B}=\sum_{B}a_{B}B$ of the singular foliation $\mathcal{F}$, we
compute the difference of the Milnor of $\mathcal{F}$ and the sum
$T_{p}(\mathcal{F},\mathcal{B})=\sum_{B}a_{B}\tau_{p}(\mathcal{F},B)$ of
Tjurina numbers of $\mathcal{F}$ along the components of $\mathcal{B}$:

######  Theorem C.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\displaystyle\mu_{p}(\mathcal{F})-T_{p}(\mathcal{F},\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}[\mu_{p}(dF_{B},B)-\tau_{p}(B)]-\deg(\mathcal{B})+1+\chi_{p}(\mathcal{F})$
$\displaystyle-\sum_{B}a_{B}[i_{p}(B,(F_{B})_{0}\setminus
B)-i_{p}(B,(F_{B})_{\infty})],$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$.

As a consequence, we precise this relationship for second type foliations in
Corollary 6.6 and for non-dicritical foliations in:

######  Corollary D.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C=\cup_{j=1}^{\ell}C_{j}$ is the total
union of separatrices of $\mathcal{F}$. Then

$\mu_{p}(\mathcal{F})-T_{p}(\mathcal{F},C)=\mu_{p}(C)-\sum_{j=1}^{\ell}\tau_{p}(C_{j})+\chi_{p}(\mathcal{F})-\sum_{j=1}^{\ell}i_{p}(C_{j},C\setminus
C_{j}),$

where $\tau_{p}(C_{j})$ is the Tjurina number of $C_{j}.$

We complete this section with several examples. In particular, in Example 6.5,
we construct a family of dicritical foliations which are not of second type.
We finish Section 6 stating numerical relationships between some classic
indices, as Baum-Bott, Camacho-Sad, and variational indices, of a singular
foliation and the Milnor and Tjurina numbers. Finally, in Section 7, we obtain
a bound for the sum of Milnor numbers of the local separatrices of a
holomorphic foliation on the complex projective plane.

##  2\. Basic tools

In order to fix the terminology and the notation, we recall some basic
concepts of local Foliation Theory. Unless we specify otherwise, throughout
this text $\mathcal{F}$ denotes a germ of a singular (holomorphic or formal)
foliation at $(\mathbb{C}^{2},p)$. In local coordinates $(x,y)$ centered at
$p$, the foliation is given by a (holomorphic or formal) 1-form

(2.1) $\omega=P(x,y)dx+Q(x,y)dy,$

or by its dual vector field

(2.2)
$v=-Q(x,y)\frac{\partial}{\partial{x}}+P(x,y)\frac{\partial}{\partial{y}},$

where $P(x,y),Q(x,y)\in{\mathbb{C}}[[x,y]]$ are relatively prime, where
${\mathbb{C}}[[x,y]]$ is the ring of complex formal power series in two
variables. The algebraic multiplicity $\nu_{p}(\mathcal{F})$ is the minimum of
the orders $\nu_{p}(P)$, $\nu_{p}(Q)$ at $p$ of the coefficients of a local
generator of $\mathcal{F}$.

Let $f(x,y)\in\mathbb{C}[[x,y]]$. We say that $C:f(x,y)=0$ is invariant by
$\mathcal{F}$ or $\mathcal{F}$-_invariant_ if

$\omega\wedge df=(f.h)dx\wedge dy,$

for some $h\in\mathbb{C}[[x,y]]$. If $C$ is an irreducible
$\mathcal{F}$-invariant curve then we will say that $C$ is a separatrix of
$\mathcal{F}$. The separatrix $C$ is analytical if $f$ is convergent. We
denote by $\mbox{\rm Sep}_{p}(\mathcal{F})$ the set of all separatrices of
$\mathcal{F}$. When $\mbox{\rm Sep}_{p}(\mathcal{F})$ is a finite set we will
say that the foliation $\mathcal{F}$ is non-dicritical and we call total union
of separatrices of $\mathcal{F}$ to the union of all elements of $\mbox{\rm
Sep}_{p}(\mathcal{F})$. Otherwise we will say that $\mathcal{F}$ is a
dicritical foliation.

A point $p\in\mathbb{C}^{2}$ is a reduced or simple singularity for
$\mathcal{F}$ if the linear part ${\rm D}v(p)$ of the vector field $v$ in
(2.2) is non-zero and has eigenvalues $\lambda_{1},\lambda_{2}\in\mathbb{C}$
fitting in one of the two following cases:

  1. (1)

$\lambda_{1}\lambda_{2}\neq 0$ and
$\lambda_{1}/\lambda_{2}\not\in\mathbb{Q}^{+}$ (in which case we say that $p$
is a non-degenerate or complex hyperbolic singularity).

  2. (2)

$\lambda_{1}\neq 0$ and $\lambda_{2}=0$ (in which case we say that $p$ is a
saddle-node singularity).

The reduction process of the singularities of a codimension one singular
foliation over an ambient space of dimension two was achieved by Seidenberg
[25]. Van den Essen gave a new proof in [30].

A singular foliation $\mathcal{F}$ at $(\mathbb{C}^{2},p)$ is a generalized
curve foliation if it has no saddle-nodes in its reduction process of
singularities, that is, the case (1). This concept was defined by Camacho-Lins
Neto-Sad [5, Page 144]. In this case, there is a system of coordinates $(x,y)$
in which $\mathcal{F}$ is induced by the equation

(2.3) $\omega=x(\lambda_{1}+a(x,y))dy-y(\lambda_{2}+b(x,y))dx,$

where $a(x,y),b(x,y)\in{\mathbb{C}}[[x,y]]$ are non-units, so that $\mbox{\rm
Sep}_{p}(\mathcal{F})$ is formed by two transversal analytic branches given by
$\\{x=0\\}$ and $\\{y=0\\}$. In the case (2), up to a formal change of
coordinates, the saddle-node singularity is given by a 1-form of the type

(2.4) $\omega=x^{k+1}dy-y(1+\lambda x^{k})dx,$

where $\lambda\in\mathbb{C}$ and $k\in\mathbb{Z}_{>0}$ are invariants after
formal changes of coordinates (see [20, Proposition 4.3]). The curve
$\\{x=0\\}$ is an analytic separatrix, called strong, whereas $\\{y=0\\}$
corresponds to a possibly formal separatrix, called weak or central.

Given a foliation $\mathcal{F}$ at $(\mathbb{C}^{2},p)$ we follow [10, page
1115] to introduce the set $\mathcal{I}_{p}(\mathcal{F})$ of infinitely near
points of $\mathcal{F}$ at $p$. This is defined in a recursive way along the
reduction process of the singularities of $\mathcal{F}$. We do as follows.
Given a sequence of blow-ups
$\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ — an intermediate step in
the reduction process, where $\mathcal{D}=\pi^{-1}(p)$ and $\tilde{X}$ is the
ambient space containing $\mathcal{D}$ — and a point $q\in\mathcal{D}$ we set:

  * •

if $\tilde{\mathcal{F}}$ is $\mathcal{D}$-reduced at $q$, then
$\mathcal{I}_{q}(\tilde{\mathcal{F}})=\\{q\\}$;

  * •

if $\tilde{\mathcal{F}}$ is $\mathcal{D}$-singular but not
$\mathcal{D}$-reduced at $q$, we perform a blow-up
$\sigma:(\hat{X},\hat{\mathcal{D}})\to(\tilde{X},\mathcal{D})$ at $q$, where
$\hat{\mathcal{D}}=\sigma^{-1}(\mathcal{D})=(\sigma^{*}\mathcal{D})\cup D$ and
$D=\sigma^{-1}(q)$ (here $\sigma^{*}\mathcal{D}$ denotes the strict transform
of $\mathcal{D}$). If $q_{1},\ldots,q_{\ell}$ are all
$\hat{\mathcal{D}}$-singular points of
$\hat{\mathcal{F}}=\sigma^{*}\tilde{\mathcal{F}}$ on $D$, then

$\mathcal{I}_{q}(\tilde{\mathcal{F}})=\\{q\\}\cup\mathcal{I}_{q_{1}}(\hat{\mathcal{F}})\cup\ldots\cup\mathcal{I}_{q_{\ell}}(\hat{\mathcal{F}}).$

In order to simplify notation, we settle that a numerical invariant for a
foliation $\mathcal{F}$ at $q\in\mathcal{I}_{p}(\mathcal{F})$ actually means
the same invariant computed for the transform of $\mathcal{F}$ at $q$.

For a fixed reduction process of singularities
$\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ for $\mathcal{F}$, a
component $D\subset\mathcal{D}$ can be:

  * •

non-dicritical, if $D$ is $\tilde{\mathcal{F}}$-invariant. In this case, $D$
contains a finite number of simple singularities. Each non-corner singularity
of $D$ carries a separatrix transversal to $D$, whose projection by $\pi$ is a
curve in $\mbox{\rm Sep}_{p}(\mathcal{F})$. Remember that a corner singularity
of $D$ is an intersection point of $D$ with other irreducible component of
$\mathcal{D}$.

  * •

dicritical, if $D$ is not $\tilde{\mathcal{F}}$-invariant. The reduction
process of singularities gives that $D$ may intersect only non-dicritical
components of $\mathcal{D}$ and $\tilde{\mathcal{F}}$ is everywhere transverse
to $D$. The $\pi$-image of a local leaf of $\tilde{\mathcal{F}}$ at each non-
corner point of $D$ belongs to $\mbox{\rm Sep}_{p}(\mathcal{F})$.

Let $\sigma$ be the blow-up of the reduction process of singularities $\pi$
that generated the component $D\subset\mathcal{D}$. We will say that $\sigma$
is _non-dicritical_ (respectively _dicritical_) if $D$ is non-dicritical
(respectively dicritical).

Denote by $\mbox{\rm Sep}_{p}(D)\subset\mbox{\rm Sep}_{p}(\mathcal{F})$ the
set of separatrices whose strict transforms by $\pi$ intersect the component
$D\subset\mathcal{D}$. If $B\in\mbox{\rm Sep}_{p}(D)$ with $D$ non-dicritical,
$B$ is said to be isolated. Otherwise, it is said to be a dicritical
separatrix. This determines the decomposition $\mbox{\rm
Sep}_{p}(\mathcal{F})=\mbox{Iso}_{p}(\mathcal{F})\cup\mbox{Dic}_{p}(\mathcal{F})$,
where notations are self-evident. The set $\mbox{Iso}_{p}(\mathcal{F})$ is
finite and contains all purely formal separatrices. It subdivides further in
two classes: weak separatrices — those arising from the weak separatrices of
saddle-nodes — and strong separatrices — corresponding to strong separatrices
of saddle-nodes and separatrices of non-degenerate singularities. On the other
hand, if $\mbox{Dic}_{p}(\mathcal{F})$ is non-empty then it is an infinite set
of analytic separatrices. Observe that a foliation $\mathcal{F}$ is dicritical
when $\mbox{\rm Sep}_{p}(\mathcal{F})$ is infinite, which is equivalent to
saying that $\mbox{Dic}_{p}(\mathcal{F})$ is non-empty. Otherwise,
$\mathcal{F}$ is non-dicritical.

Throughout the text, we would rather adopt the language of divisors of formal
curves. More specifically, a divisor of separatrices for a foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$ is a formal sum

(2.5) $\mathcal{B}=\sum_{B\in\text{Sep}_{p}(\mathcal{F})}a_{B}\cdot B,$

where the coefficients $a_{B}\in\mathbb{Z}$ are zero except for finitely many
$B\in\mbox{\rm Sep}_{p}(\mathcal{F})$. The set of separatrices
$\\{B\;:\;a_{B}\neq 0\\}$ appearing in (2.5) is called the _support_ of the
divisor $\mathcal{B}$ and it is denoted by $\hbox{\rm supp}(\mathcal{B})$. The
_degree_ of the divisor $\mathcal{B}$ is by definition
$\deg\mathcal{B}=\sum_{B\in\hbox{\rm supp}(\mathcal{B})}a_{B}$. We denote by
$\mbox{\rm Div}_{p}(\mathcal{F})$ the set of all these divisors of
separatrices, which turns into a group with the canonical additive structure.
We follow the usual terminology and notation:

  * •

$\mathcal{B}\geq 0$ denotes an effective divisor, one whose coefficients are
all non-negative;

  * •

there is a unique decomposition
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$, where
$\mathcal{B}_{0},\mathcal{B}_{\infty}\geq 0$ are respectively the zero and
pole divisors of $\mathcal{B}$;

  * •

the algebraic multiplicity of $\mathcal{B}$ is
$\nu_{p}(\mathcal{B})=\displaystyle\sum_{B\in\hbox{\rm
supp}(\mathcal{B})}\nu_{p}(B).$

Given a foliation $\mathcal{F}$ and a formal meromorphic equation
${F}(x,y)=\prod_{i=1}^{s}f_{i}(x,y)^{a_{i}}$, whose irreducible components
define separatrices $B_{i}:f_{i}(x,y)=0$ of $\mathcal{F}$ , we associate the
divisor $({F})=\sum_{i}a_{i}\cdot B_{i}$. A curve of separatrices ${C}$,
associated with a reduced equation ${F}(x,y)$, is identified to the divisor
$({F})$ and we write ${C}=({F})$. Such an effective divisor is named reduced,
that is, all coefficients are either $0$ or $1$. In general,
$\mathcal{B}\in\mbox{\rm Div}_{p}(\mathcal{F})$ is reduced if both
$\mathcal{B}_{0}$ and $\mathcal{B}_{\infty}$ are reduced divisors. A divisor
$\mathcal{B}$ is said to be adapted to a curve of separatrices ${C}$ if
$\mathcal{B}_{0}-{C}\geq 0$.

Following [12, page 5] and [13, Definition 3.1], we remember the following
notion:

######  Definition 2.1.

A _balanced divisor of separatrices_ for $\mathcal{F}$ is a divisor of the
form

$\mathcal{B}\ =\ \sum_{B\in{\rm Iso}_{p}(\mathcal{F})}B+\sum_{B\in{\rm
Dic}_{p}(\mathcal{F})}\ a_{B}\cdot B,$

where the coefficients $a_{B}\in\mathbb{Z}$ are non-zero except for finitely
many $B\in\mbox{Dic}_{p}(\mathcal{F})$, and, for each dicritical component
$D\subset\mathcal{D}$, the following equality is respected:

$\sum_{B\in{\text{Sep}_{p}(D)}}a_{B}=2-\text{Val}(D).$

The integer $\text{Val}(D)$ stands for the valence of a component
$D\subset\mathcal{D}$ in the reduction process of singularities, that is, it
is the number of components of $\mathcal{D}$ intersecting $D$ other from $D$
itself.

Observe that the notion of balanced divisor of separatrices generalizes, to
dicritical foliations, the notion of total union of separatrices for non-
dicritical foliations.

A balanced divisor $\mathcal{B}=\sum_{B}a_{B}B$ of separatrices of
$\mathcal{F}$ is called primitive if, $a_{B}\in\\{-1,1\\}$ for any
$B\in\hbox{\rm supp}(\mathcal{B})$. A balanced equation of separatrices is a
formal meromorphic function ${F}(x,y)$ whose associated divisor
${C}={C}_{0}-{C}_{\infty}$ is a balanced divisor. A balanced equation is
reduced, primitive or adapted to a curve $C$ if the same is true for the
underlying divisor.

Remember that the intersection number of two formal curves $C$ and $D$ at
$(\mathbb{C}^{2},p)$ is by definition

$i_{p}(C,D)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(g,h),$

where $C:g(x,y)=0$, $D:h(x,y)=0$, and $(g,h)$ denotes the ideal generated by
$g$ and $h$ in $\mathbb{C}[[x,y]]$. The intersection number for formal curves
at $(\mathbb{C}^{2},p)$ is canonically extended in a bilinear way to divisors
of curves.

Let $\mathcal{F}$ be a foliation at $(\mathbb{C}^{2},p)$, given by a $1$-form
as in (2.1), with reduction process
$\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ and let
$\tilde{\mathcal{F}}=\pi^{*}\mathcal{F}$ be the strict transform of
$\mathcal{F}$. Denote by $\textsf{Sing}(\cdot)$ the set of singularities of a
foliation. A saddle-node singularity $q\in\textsf{Sing}(\tilde{\mathcal{F}})$
is said to be a tangent saddle-node if its weak separatrix is contained in the
exceptional divisor $\mathcal{D}$, that is, the weak separatrix is an
irreducible component of $\mathcal{D}$.

We have the following definition given by Mattei-Salem [21, Définition 3.1.4]
to non-dicritical case and used by Genzmer [12] for arbitrary foliations:

######  Definition 2.2.

A foliation is in the second class or is of second type if there are no
tangent saddle-nodes in its reduction process of singularities.

Let $B$ be a separatrix of $\mathcal{F}$ at $p$. Suppose that $\\{y=0\\}$ is
the tangent cone of $B$, then we may choose one of its primitive Puiseux
parametrizations $\gamma(t)=(t^{n},\phi(t))$ at $p$ such that $n=\nu_{p}(B)$,
where $\nu_{p}(B)$ denotes the algebraic multiplicity of $B$. The tangency
index of $\mathcal{F}$ along $B$ at $p$ (or weak index in [10, page 1114]) is

$\text{Ind}_{p}(\mathcal{F},B):=\textsf{ord}_{t}Q(\gamma(t)).$

The tangency index $\text{Ind}_{p}(\mathcal{F},B)$ does not depend on the
chosen parametrization of $B$ because by properties of the multiplicity number
we get the equality $\textsf{ord}_{t}Q(\gamma(t))=i_{p}(Q,B)$. The foliation
$\mathcal{F}$ given by the $1$-form defined in (2.4) verifies
$\text{Ind}_{p}(\mathcal{F},B)=k+1>1$, where $B:\\{y=0\\}$.

The tangency index was defined in [5, page 159], where the authors denomine it
multiplicity of $\mathcal{F}$ along $B$ at $p$ and denoted by
$\mu_{\mathcal{F}}(B,p)$. In the same paper the authors define a similar
notion, the index with respect to a vector field $Z$ (see page 152) and denote
it by $Ind_{p}(Z/B)$ which coincides with the multiplicity of $\mathcal{F}$
along $B$ at $p$ that we introduce in (4.7) and we denote by
$\mu_{p}(\mathcal{F},B)$. The reader should pay attention to it to avoid
confusion. We adopt the notation given by Genzmer [12], instead of the
original given in [5] since $\mu_{p}(\mathcal{F},B)$ resembles a Milnor
number, which will be studied in Section 4.

Given a component $D\subset\mathcal{D}$, we denote by $\nu(D)$ its
multiplicity, which coincides with the algebraic multiplicity of a curve $E$
at $(\mathbb{C}^{2},p)$ whose strict transform $\pi^{*}E$ meets $D$
transversally outside a corner of $\mathcal{D}$. The following invariant is a
measure of the existence of tangent saddle-nodes in the reduction process of
singularities of a foliation:

######  Definition 2.3.

The _tangency excess_ of the foliation $\mathcal{F}$ is defined as
$\xi_{p}(\mathcal{F})=0$, when $p$ is a reduced singularity, and, in the non-
reduced case, as the number

$\xi_{p}(\mathcal{F})=\sum_{q\in\textsl{SN}(\mathcal{F})}\nu(D_{q})(\text{Ind}_{q}(\tilde{\mathcal{F}},\tilde{B})-1),$

where $\textsl{SN}(\mathcal{F})$ stands for the set of tangent saddle-nodes on
$\mathcal{D}$, $\tilde{B}$ is the weak separatrix passing by
$q\in\textsl{SN}(\mathcal{F})$, and $D_{q}$ is the component of $\mathcal{D}$
containing $\tilde{B}$. By (2.4), we observe that
$\text{Ind}_{q}(\tilde{\mathcal{F}},\tilde{B})=k+1>1$.

Remark that $\xi_{p}(\mathcal{F})\geq 0$ and, by definition,
$\xi_{p}(\mathcal{F})=0$ if and only if $\textsl{SN}(\mathcal{F})=\emptyset$,
that is, if and only if $\mathcal{F}$ is of second type. In several papers
(see for example [10], [4]) the tangency excess of $\mathcal{F}$ is denoted by
$\tau_{p}(\mathcal{F})$. In this paper, we denote it by $\xi_{p}(\mathcal{F})$
since we keep the letter $\tau$ for the Tjurina number of a curve or a
foliation.

The following proposition proved by Genzmer (see [12, Proposition 2.4]) will
be very useful in this paper:

######  Proposition 2.4.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and
$\mathcal{B}$ a balanced divisor of separatrices for $\mathcal{F}$. Denote by
$\nu_{p}(\mathcal{F})$ and $\nu_{p}(\mathcal{B})$ their algebraic
multiplicities respectively. Then

(2.6) $\nu_{p}(\mathcal{F})=\nu_{p}(\mathcal{B})-1+\xi_{p}(\mathcal{F}).$

Moreover,

$\nu_{p}(\mathcal{F})=\nu_{p}(\mathcal{B})-1$

if, and only if, $\mathcal{F}$ is a second type foliation.

Take a primitive parametrization $\gamma:(\mathbb{C},0)\to(\mathbb{C}^{2},p)$,
$\gamma(t)=(x(t),y(t))$, of a formal irreducible curve $B:f(x,y)=0$ at
$(\mathbb{C}^{2},p)$. Note that $B$ is a separatrix of the foliation
$\mathcal{F}:\omega=0$ if and only if $\gamma^{*}(\omega)=0$. If $B$ is not an
$\mathcal{F}$-invariant curve, we define the tangency order of $\mathcal{F}$
along $B$ at $p$ as

(2.7) $\textsf{tang}_{p}(\mathcal{F},B)=\textsf{ord}_{t}a(t),$

where $\gamma^{*}(\omega)=a(t)dt$. The tangency order does not depend on the
choosen parametrization of $B$, since
$\textsf{tang}_{p}(\mathcal{F},B)=i_{p}(B,v(f))$ where $v$ is from (2.2). The
tangency order was introduced in this last way in [3, page 22].

The behavior, under blow-up, of the tangency order, in the non-dicritical
case, was studied in [7, equality (4)]. The dicritical case is similar.
Indeed, if $\mathcal{F}:\omega=0$ is a singular foliation at
$(\mathbb{C}^{2},p)$, $\tilde{\mathcal{F}}:\tilde{\omega}=0$ is its strict
transform by the blow-up $\sigma$ at $p$ and $B$ is not an
$\mathcal{F}$-invariant curve then we have

$\tilde{\omega}=\left\\{\begin{array}[]{ll}x^{-\nu_{p}(\mathcal{F})}\sigma^{*}(\omega)&\text{if
$\sigma$ is non-dicritical;}\\\
x^{-(\nu_{p}(\mathcal{F})+1)}\sigma^{*}(\omega)&\text{if $\sigma$ is
dicritical.}\\\ \end{array}\right.$

Evaluating $\tilde{\omega}$ in a parametrization of the strict transform (by
$\sigma$) $\tilde{B}$ of $B$ and taking orders we get

(2.8)
$\textsf{tang}_{p}(\mathcal{F},B)=\begin{cases}\nu_{p}(\mathcal{F})\nu_{p}(B)+\textsf{tang}_{q}(\tilde{\mathcal{F}},\tilde{B})&\text{if
$\sigma$ is non-dicritical};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
(\nu_{p}(\mathcal{F})+1)\nu_{p}(B)+\textsf{tang}_{q}(\tilde{\mathcal{F}},\tilde{B})&\text{if
$\sigma$ is dicritical},\end{cases}$

where $q\in\tilde{B}\cap\sigma^{-1}(p).$

In [7, Corollary 1] it was stablished that
$i_{p}(\mathcal{B},B)\leq\textsf{tang}_{p}(\mathcal{F},B)+1$ and the equality
holds if and only if $\mathcal{F}$ is of second type. In [4, Lemma 4.2], for
complex analytic foliations, was improved [7, Corollary 1] determining
explicitly the difference
$(\textsf{tang}_{p}(\mathcal{F},B)+1)-i_{p}(\mathcal{B},B),$ in the next way:

$i_{p}(\mathcal{B},B)=\textsf{tang}_{p}(\mathcal{F},B)-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(B)\xi_{q}(\mathcal{F})+1,$

where $\mathcal{F}$ is a singular foliation at $(\mathbb{C}^{2},p)$,
$\mathcal{B}$ is a balanced divisor of separatrices for $\mathcal{F}$ and $B$
is a branch which is not $\mathcal{F}$-invariant. A proof, similar to the one
given in [4], holds for formal and dicritical foliations.

##  3\. The $\chi$-number of a foliation

For a singular foliation $\mathcal{F}$ at $(\mathbb{C}^{2},p)$ we introduce a
new number

$\chi_{p}(\mathcal{F}):=\left(\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})\right)-\xi_{p}(\mathcal{F}).$

Observe that

(3.1)
$\chi_{p}(\mathcal{F})=\displaystyle\sum_{q\in\mathcal{I}_{p}(\mathcal{F})\setminus\\{p\\}}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})+(\nu_{p}(\mathcal{F})-1)\xi_{p}(\mathcal{F}).$

In [23, Proposition 9.5] the authors prove that the tangency excess is a
$C^{\infty}$ invariant, and after [24] the algebraic multiplicity of a
holomorphic foliation is a $C^{1}$ invariant. Hence the $\chi$-number of a
holomorphic foliation is a $C^{1}$ invariant.

We enclosed some properties of $\chi_{p}(\mathcal{F})$ in the following
proposition:

######  Proposition 3.1.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$, then we
get:

  1. (1)

$\chi_{p}(\mathcal{F})\geq 0$;

  2. (2)

if $\mathcal{F}$ is of second type then $\chi_{p}(\mathcal{F})=0$;

  3. (3)

if $\chi_{p}(\mathcal{F})=0$, then either $\mathcal{F}$ has algebraic
multiplicity 1 at $p$ or $\mathcal{F}$ is of second type;

  4. (4)

if $\nu_{p}(\mathcal{F})>1$, then $\chi_{p}(\mathcal{F})=0$ if and only if
$\mathcal{F}$ is of second type.

###### Proof.

By (3.1) we have
$\chi_{p}(\mathcal{F})=\beta+(\nu_{p}(\mathcal{F})-1)\xi_{p}(\mathcal{F})$,
where
$\beta=\sum_{q\in\mathcal{I}_{p}(\mathcal{F})\setminus\\{p\\}}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F}).$
Clearly, $\chi_{p}(\mathcal{F})\geq 0$, since it is the sum of two nonnegative
numbers. Now, if $\mathcal{F}$ is of second type, we get
$\xi_{q}(\mathcal{F})=0$, for all $q\in\mathcal{I}_{p}(\mathcal{F})$, which
implies that $\chi_{p}(\mathcal{F})=0$. On the other hand, if
$\chi_{p}(\mathcal{F})=0$ then $\beta=0$ and
$(\nu_{p}(\mathcal{F})-1)\xi_{p}(\mathcal{F})=0$. This finishes the proof of
(3). Item (4) is an immediately consequence of (2) and (3). ∎

######  Remark 3.2.

Let $\omega=4xydx+(y-2x^{2})dy$ be a $1$-form. Observe that the foliation
$\mathcal{F}:\omega=0$ at $(\mathbb{C}^{2},0)$ is not of second type, its
algebraic multiplicity is one but $\chi_{0}(\mathcal{F})=1\neq 0$.

##  4\. Polar intersection, polar excess and Milnor numbers

Let $\omega=P(x,y)dx+Q(x,y)dy$ be a 1-form, where
$P(x,y),Q(x,y)\in\mathbb{C}[[x,y]]$. If $\mathcal{F}:\omega=0$ is a singular
(analytic or formal) foliation then the polar curve of $\mathcal{F}$ at
$(\mathbb{C}^{2},p)$ with respect to a point $(a:b)$ of the complex projective
line $\mathbb{P}^{1}(\mathbb{C})$ is the (analytic or formal) curve
$\mathcal{P}^{\mathcal{F}}_{(a:b)}:aP(x,y)+bQ(x,y)=0$. There exists an open
Zariski set $U$ of $\mathbb{P}^{1}(\mathbb{C})$ such that
$\\{aP(x,y)+bQ(x,y)=0\;:\;(a:b)\in U\\}$ is an equisingular family of plane
curves. Any element of this set is called generic polar curve of the foliation
$\mathcal{F}$ and we will denote it by $\mathcal{P}^{\mathcal{F}}$.

We borrow from [13, Section 4] the notion of polar curve of a meromorphic
1-form: let $\eta=\frac{\omega}{H(x,y)}$ be a meromorphic 1-form, where
$\omega=P(x,y)dx+Q(x,y)dy$ with $P(x,y),Q(x,y),H(x,y)\in\mathbb{C}[[x,y]]$.
The polar curve of $\eta$ at $(\mathbb{C}^{2},p)$ with respect to
$(a:b)\in\mathbb{P}^{1}(\mathbb{C})$ is the divisor
$\mathcal{P}^{\eta}_{(a:b)}$ with formal meromorphic equation

$\frac{aP(x,y)+bQ(x,y)}{H}=0.$

A polar curve $\frac{aP(x,y)+bQ(x,y)}{H}=0$ of a meromorphic 1-form
$\frac{\omega}{H(x,y)}$ will be generic if the polar curve $aP(x,y)+bQ(x,y)=0$
is a generic polar curve of the foliation defined by the $1$-form $\omega$.

Let $B:h(x,y)=0$ be a separatrix of a singular foliation $\mathcal{F}$. The
polar intersection number of $\mathcal{F}$ with respect to $B$ is the
intersection number $i_{p}({\mathcal{P}}^{\mathcal{F}},B)$.

######  Lemma 4.1.

Let $B:h(x,y)=0$ be a separatrix of a singular foliation $\mathcal{F}$ at
$(\mathbb{C}^{2},p)$ and consider ${F}_{B}$ and ${G}_{B}$ two balanced
divisors of separatrices for $\mathcal{F}$ adapted to $B$. Then

$i_{p}(\mathcal{P}^{d{F}_{B}},B)=i_{p}(\mathcal{P}^{d{G}_{B}},B).$

###### Proof.

Put $F_{B}=\frac{h\cdot\hat{f}\cdot g_{1}\cdots
g_{l}}{\phi_{1}\cdots\phi_{m}}$ and $G_{B}=\frac{h\cdot\hat{f}\cdot
h_{1}\cdots h_{s}}{\psi_{1}\cdots\psi_{r}}$, where $g_{i}(x,y)=0$,
$h_{i}(x,y)=0$, $\phi_{i}(x,y)=0$ and $\psi_{i}(x,y)=0$ are dicritical
separatrices of $\mathcal{F}$, $\hat{f}(x,y)=0$ defines the reduced curve
which is the union of all isolated separatrices of $\mathcal{F}$ except
perhaps $h(x,y)=0$ when this is also isolated. We get

$dF_{B}=\frac{\phi(\mathcal{P}\cdot h+\hat{f}\cdot g_{1}\cdots g_{l}\cdot
dh)-h\cdot\hat{f}\cdot g_{1}\cdots g_{l}d(\phi)}{\phi^{2}}$

and

$dG_{B}=\frac{\psi(\mathcal{Q}\cdot h+\hat{f}\cdot h_{1}\cdots h_{s}\cdot
dh)-h\cdot\hat{f}\cdot h_{1}\cdots h_{s}d(\psi)}{\psi^{2}},$

where $\phi=\phi_{1}\cdots\phi_{m}$, $\psi=\psi_{1}\cdots\psi_{r}$,
$\mathcal{P}=d(\hat{f}\cdot g_{1}\cdots
g_{l})=:\mathcal{P}_{1}dx+\mathcal{P}_{2}dy$ and $\mathcal{Q}=d(\hat{f}\cdot
h_{1}\cdots h_{s})=:\mathcal{Q}_{1}dx+\mathcal{Q}_{2}dy$. Put $u=\hat{f}\cdot
g_{1}\cdots g_{l}$ and $v=\hat{f}\cdot h_{1}\cdots h_{s}$. Hence

$\mathcal{P}^{d\hat{F}_{B}}=\frac{[\phi(\mathcal{P}_{1}\cdot
h+u\cdot\partial_{x}{h})-hu\cdot\partial_{x}{\phi}]a+[\phi(\mathcal{P}_{2}\cdot
h+u\cdot\partial_{y}{h})-hu\cdot\partial_{y}{\phi}]b}{\phi^{2}},$

and

$\mathcal{P}^{d\hat{G}_{B}}=\frac{[\psi(\mathcal{Q}_{1}\cdot
h+v\cdot\partial_{x}{h})-hv\cdot\partial_{x}{\psi}]a+[\psi(\mathcal{Q}_{2}\cdot
h+v\cdot\partial_{y}{h})-hv\cdot\partial_{y}{\psi}]b}{\psi^{2}}.$

So

(4.1) $\displaystyle i_{p}(\mathcal{P}^{d\hat{F}_{B}},B)$ $\displaystyle=$
$\displaystyle i_{p}\left(\hat{f}\cdot g_{1}\cdots
g_{l}\cdot\left(a\partial_{x}{h}+b\partial_{y}{h}\right),h\right)-i_{p}(\phi_{1}\cdots\phi_{m},h)$
$\displaystyle=$ $\displaystyle
i_{p}\left(\hat{f}\cdot\left(a\partial_{x}{h}+b\partial_{y}{h}\right),h\right)+i_{p}(g_{1}\cdots
g_{l},h)-i_{p}(\phi_{1}\cdots\phi_{m},h),$

and

(4.2) $\displaystyle i_{p}(\mathcal{P}^{d\hat{G}_{B}},B)$ $\displaystyle=$
$\displaystyle i_{p}\left(\hat{f}\cdot h_{1}\cdots
h_{s}\cdot\left(a\partial_{x}{h}+b\partial_{y}{h}\right),h\right)-i_{p}(\psi_{1}\cdots\psi_{r},h)$
$\displaystyle=$ $\displaystyle
i_{p}\left(\hat{f}\cdot\left(a\partial_{x}{h}+b\partial_{y}{h}\right),h\right)+i_{p}(h_{1}\cdots
h_{s},h)-i_{p}(\psi_{1}\cdots\psi_{r},h).$

We claim that $i_{p}(g_{1}\cdots
g_{l},h)-i_{p}(\phi_{1}\cdots\phi_{m},h)=i_{p}(h_{1}\cdots
h_{s},h)-i_{p}(\psi_{1}\cdots\psi_{r},h)$. Indeed, if every dicritical
separatrix of $\mathcal{F}$ is smooth and transversal to any isolated
separatrix, then using properties of the intersection multiplicity we have

$\displaystyle i_{p}(g_{1}\cdots g_{l},h)-i_{p}(\phi_{1}\cdots\phi_{m},h)$
$\displaystyle=$
$\displaystyle\nu_{p}(h)\left[\sum_{j=1}^{l}\nu_{p}(g_{j})-\sum_{j=1}^{m}\nu_{p}(\phi_{j})\right]$
$\displaystyle=$
$\displaystyle\nu_{p}(h)\left[\sum_{j=1}^{s}\nu_{p}(h_{j})-\sum_{j=1}^{r}\nu_{p}(\psi_{j})\right]$
$\displaystyle=$ $\displaystyle i_{p}(h_{1}\cdots
h_{s},h)-i_{p}(\psi_{1}\cdots\psi_{r},h),$

where the equality (4) holds since ${F}_{B}$ and ${G}_{B}$ are two balanced
divisors of separatrices for $\mathcal{F}$.

In the general case, after the reduction of singularities of the foliation we
can suppose that the strict transform of every dicritical separatrix of
$\mathcal{F}$ is smooth and transversal to any strict transform of every
isolated separatrix. We finish the proof using Noether formula. ∎

Lemma 4.1 allows us to define the polar excess number of a singular foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$ with respect to a separatrix $B$ of
$\mathcal{F}$ as

(4.4)
$\Delta_{p}(\mathcal{F},B):=i_{p}(\mathcal{P}^{\mathcal{F}},B)-i_{p}(\mathcal{P}^{d{F}_{B}},B),$

where ${F}_{B}$ is any balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$. On the other hand, if the foliation $\mathcal{F}$ is non-
dicritical then it is enough to consider ${F}_{B}$ as the total union of the
separatrices of $\mathcal{F}$.

Using properties of the intersection number we extend the definitions of polar
intersection and polar excess numbers to any divisor
${\mathcal{B}}:=\sum_{B}a_{B}B$ of separatrices of $\mathcal{F}$ in the
following way:

$i_{p}({\mathcal{P}}^{\mathcal{F}},{\mathcal{B}})=\sum_{B}a_{B}i_{p}({\mathcal{P}}^{\mathcal{F}},B)$

and

(4.5)
$\Delta_{p}(\mathcal{F},{\mathcal{B}})=\sum_{B}a_{B}\Delta_{p}(\mathcal{F},B)=\Delta_{p}(\mathcal{F},{\mathcal{B}}_{0})-\Delta_{p}(\mathcal{F},{\mathcal{B}}_{\infty}).$

If $\mathcal{B}$ is a primitive divisor then the difference
$\Delta_{p}(\mathcal{F},{\mathcal{B}}_{0})-\Delta_{p}(\mathcal{F},{\mathcal{B}}_{\infty})$
is independent of the choosen primitive balanced divisor of separatrices for
$\mathcal{F}$ (see [10, Section 3.6, page 1123]).

By [13, Proposition 4.6], $\Delta_{p}(\mathcal{F},B)$ is a non-negative
integer number for any irreducible component $B$ of $\mathcal{B}_{0}$. As a
consequence $\Delta_{p}(\mathcal{F},\mathcal{B})$ is also a non-negative
integer number for any effective divisor of separatrices $\mathcal{B}$.

The Milnor number $\mu_{p}(\mathcal{F})$ of the foliation $\mathcal{F}$ at $p$
given by the $1$-form $\omega=P(x,y)dx+Q(x,y)dy$ is defined by

$\mu_{p}(\mathcal{F})=i_{p}(P,Q).$

Remember that we consider $P$ and $Q$ coprime, so $\mu_{p}(\mathcal{F})$ is a
non negative integer. In [5, Theorem A] it was proved that the Milnor number
of a foliation is a topological invariant.

On the other hand, the Milnor number $\mu_{p}(C)$ at $p$ of a plane curve $C$
(non necessary irreducible) of equation $f(x,y)=0$ is

$\mu_{p}(C)=i_{p}\left(\partial_{x}f,\partial_{y}f\right).$

Observe that $\mu_{p}(C)$ is finite if and only if $f$ has not multiple
factors, that is, the curve $C$ is reduced.

Generalized curve foliations have a property of minimization of Milnor numbers
and are characterized, in the non-dicritical case, by several authors, see for
instance [2, Proposition 7] and [8, Théorème 3.3]. Recently, in [13, Theorem
A], the authors have characterized singular generalized curve foliations in
terms of its polar excess number as follows: let $\mathcal{F}$ be a singular
foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ be a balanced divisor of
separatrices for $\mathcal{F}$, then
$\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=0$ if and only if $\mathcal{F}$ is a
generalized curve foliation.

The following proposition provides a formula to compute the polar excess
number of a foliation with respect to the zero divisor of a reduced balanced
divisor of separatrices.

######  Proposition 4.2.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ be a reduced balanced
divisor of separatrices for $\mathcal{F}$. Then

$\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{0})+1.$

Moreover, $\mathcal{F}$ is a generalized curve foliation if and only if

$i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})=\mu_{p}(\mathcal{B}_{0})+\nu_{p}(\mathcal{B}_{0})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-1.$

###### Proof.

Let $\omega=P(x,y)dx+Q(x,y)dy$ be a 1-form inducing $\mathcal{F}$ and
$f(x,y)=0$ and $g(x,y)=0$ be the reduced equations of $\mathcal{B}_{0}$ and
$\mathcal{B}_{\infty}$ respectively. Since $\mathcal{B}$ is a set of
separatrices of $\mathcal{F}$ adapted to $\mathcal{B}_{0}$, by (4.4) and the
definition of the polar curve of $d(f/g)$ we have

$\displaystyle\Delta_{p}(\mathcal{F},\mathcal{B}_{0})$ $\displaystyle=$
$\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})-i_{p}(\mathcal{P}^{d(f/g)},\mathcal{B}_{0})$
$\displaystyle=$ $\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})-i_{p}\left(\frac{(g\partial_{x}f-f\partial_{x}g)a+(g\partial_{y}f-f\partial_{y}g)b}{g^{2}},f\right)$
$\displaystyle=$ $\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})-i_{p}\left(\frac{g(a\partial_{x}f+b\partial_{y}f)-f(a\partial_{x}g+b\partial_{y}g)}{g^{2}},f\right)$

where $(a:b)\in\mathbb{P}^{1}$. By applying properties on intersection numbers
and Teissier’s Proposition [28, Chapter II, Proposition 1.2], we get

$\displaystyle\Delta_{p}(\mathcal{F},\mathcal{B}_{0})$ $\displaystyle=$
$\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})-i_{p}(a\partial_{x}f+b\partial_{y}f,f)+i_{p}(g,f)$
$\displaystyle=$ $\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{0})+1.$

The second part of the proposition follows from the first part and the
characterization of generalized curve foliations given in [13, Theorem A]. ∎

The second part of Proposition 4.2 generalizes, to dicritical context, [7,
Proposition 2] stated in the non-dicritical case. We give a numerical
illustration of Proposition 4.2.

######  Example 4.3.

Let $\mathcal{F}$ be the foliation at $(\mathbb{C}^{2},0)$ defined by
$\omega=xdy-ydx.$ Observe that $\mathcal{F}$ is a dicritical generalized curve
foliation called the radial foliation. It has only one dicritical component
whose valence is $0$ in its reduction process of singularities. Thus
$\mathcal{B}=(x)+(y)+(x-y)-(x+y)$ is a reduced balanced divisor of
separatrices for $\mathcal{F}$, where $\mathcal{B}_{0}:xy(x-y)=0$ and
$\mathcal{B}_{\infty}:x+y=0$. We get $\mu_{0}(\mathcal{B}_{0})=4$,
$\nu_{0}(\mathcal{B}_{0})=3$, $i_{0}(\mathcal{B}_{0},\mathcal{B}_{\infty})=3$
and $i_{0}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})=3.$ Observe that if we
consider the reduced balanced divisor of separatrices $\mathcal{B}=(x)+(y)$
for $\mathcal{F}$ then $\mathcal{B}_{\infty}$ is a unit, so
$i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})=0$ and now
$\mu_{0}(\mathcal{B}_{0})=1$, $\nu_{0}(\mathcal{B}_{0})=2$. On the other hand,
if we consider the foliation $\mathcal{F}$ with a saddle-node (so
$\mathcal{F}$ is not a generalized curve foliation) and equation as in (2.4)
we get again $\mathcal{B}=(x)+(y)$ but
$i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})=k+2\neq 2$.

######  Lemma 4.4.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}$ be a balanced divisor of separatrices for $\mathcal{F}$. Then

$i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B})=\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F}),$

where the summation runs over all infinitely near points of $\mathcal{F}$ at
$p$.

###### Proof.

Let $\mathcal{P}^{\mathcal{F}}$ be a generic polar. Denote by
$\Gamma(\mathcal{P}^{\mathcal{F}})$ the set of irreducible components of
$\mathcal{P}^{\mathcal{F}}$. It follows from [4, Lemma 4.2] that

(4.6) $\displaystyle i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}i_{p}(A,\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}\left(\textsf{tang}_{p}(\mathcal{F},A)-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(A)\xi_{q}(\mathcal{F})+1\right)$
$\displaystyle=$
$\displaystyle\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}(\textsf{tang}_{p}(\mathcal{F},A)+1)-\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}\left(\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(A)\xi_{q}(\mathcal{F})\right)$
$\displaystyle=$
$\displaystyle\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}(\textsf{tang}_{p}(\mathcal{F},A)+1)-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\left(\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}\nu_{q}(A)\right)\xi_{q}(\mathcal{F})$
$\displaystyle=$
$\displaystyle\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}(\textsf{tang}_{p}(\mathcal{F},A)+1)-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{P}^{\mathcal{F}})\xi_{q}(\mathcal{F}).$

According to the proof of [7, Proposition 2], we have

$\sum_{A\in\Gamma(\mathcal{P}^{\mathcal{F}})}(\textsf{tang}_{p}(\mathcal{F},A)+1)=\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F}),$

and by [7, Remark 1]
$\nu_{q}(\mathcal{P}^{\mathcal{F}})=\nu_{q}(\mathcal{F})$. Substituting these
terms in the equation (5.5), we obtain

$i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B})=\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F}).$

∎

Lemma 4.4 improves [7, Proposition 2] determining explicitly the difference
between the polar intersection number with respect to a balanced divisor of
separatrices $\mathcal{B}$ and the sum of the Milnor number and the algebraic
multiplicity of the foliation $\mathcal{F}$ and generalizing the result to
dicritical foliations. On the other hand comparing Lemma 4.4 and [4,
Proposition 4.3] (proved for complex analytic foliations, but it also holds
for formal foliations) we conclude that

$\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})=\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(B)\xi_{q}(\mathcal{F}),$

for any $B$ which is not an $\mathcal{F}$-invariant curve. Hence the sum
$\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})$
coincides with the tangency excess of $\mathcal{F}$ along any irreducible
curve which is not an $\mathcal{F}$-invariant curve, introduced in [4,
equality (8)]. In particular, after the definition of the $\chi$-number, this
tangency excess equals $\chi_{p}(\mathcal{F})+\xi_{p}(\mathcal{F})$.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ induced by
the vector field $v$ and $B$ be a separatrix of $\mathcal{F}$. Let
$\gamma:(\mathbb{C},0)\to(\mathbb{C}^{2},p)$ be a primitive parametrization of
$B$, we can consider the multiplicity of $\mathcal{F}$ along $B$ at $p$
defined by

(4.7) $\mu_{p}(\mathcal{F},B)=\textsf{ord}_{t}\theta(t),$

where $\theta(t)$ is the unique vector field at $(\mathbb{C},0)$ such that
$\gamma_{*}\theta(t)=v\circ\gamma(t)$, see for instance [5, page 159]. If
$\omega=P(x,y)dx+Q(x,y)dy$ is a 1-form inducing $\mathcal{F}$ and
$\gamma(t)=(x(t),y(t))$, we get

(4.8) $\theta(t)=\begin{cases}-\frac{Q(\gamma(t))}{x^{\prime}(t)}&\text{if
$x(t)\neq 0$}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\frac{P(\gamma(t))}{y^{\prime}(t)}&\text{if $y(t)\neq 0$}.\end{cases}$

Hence, by taking order, we obtain

(4.9)
$\mu_{p}(\mathcal{F},B)=\begin{cases}\textsf{ord}_{t}Q(\gamma(t))-\textsf{ord}_{t}x(t)+1&\text{if
$x(t)\neq 0$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\textsf{ord}_{t}P(\gamma(t))-\textsf{ord}_{t}y(t)+1&\text{if $y(t)\neq
0$}.\end{cases}$

The following proposition has been proved in [7, Proposition 1] for non-
dicritical foliations, but we may check that it is valid to dicritical
foliations.

######  Proposition 4.5.

Consider a separatrix $B$ of a singular (holomorphic or formal) foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$. We have

$i_{p}(\mathcal{P}^{\mathcal{F}},B)=\mu_{p}(\mathcal{F},B)+\nu_{p}(B)-1.$

######  Remark 4.6.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C=\cup_{j=1}^{\ell}C_{j}$ is the total
union of separatrices of $\mathcal{F}$. Applying Proposition 4.5 to
$\mathcal{F}$ and $df$, where $C:f(x,y)=0$, we get
$\Delta(\mathcal{F},C_{j})=\mu_{p}(\mathcal{F},C_{j})-\mu_{p}(df,C_{j}),$ for
$j=1,\ldots,\ell$. Since $\Delta(\mathcal{F},C_{j})\geq 0$ we have
$\mu_{p}(\mathcal{F},C_{j})\geq\mu_{p}(df,C_{j})$ for any separatrix $C_{j}$
of $\mathcal{F}$.

As consequence of Lemma 4.4 and Proposition 4.5 we obtain a generalization of
[7, Corollary 2].

######  Proposition 4.7.

Let $\mathcal{F}$ be a singular (holomorphic or formal) foliation at
$(\mathbb{C}^{2},p)$ and let $\mathcal{B}=\sum_{B}a_{B}B$ be a balanced
divisor for separatrices of $\mathcal{F}$. We have

$\mu_{p}(\mathcal{F})=\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\chi_{p}(\mathcal{F})-\deg(\mathcal{B})+1.$

###### Proof.

By summing up polar intersection numbers over all irreducible components of
$\mathcal{B}$ and applying Proposition 4.5, we get

$\displaystyle i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B})$ $\displaystyle=$
$\displaystyle\sum_{B}a_{B}i_{p}(\mathcal{P}^{\mathcal{F}},B)=\sum_{B}a_{B}(\mu_{p}(\mathcal{F},B)+\nu_{p}(B)-1)$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\left(\sum_{B}a_{B}\nu_{p}(B)\right)-\deg(\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\nu_{p}(\mathcal{B})-\deg(\mathcal{B}).$

From Lemma 4.4, Proposition 2.4 and the definition of the $\chi$-number of
$\mathcal{F}$ we get

$\displaystyle\mu_{p}(\mathcal{F})$ $\displaystyle=$
$\displaystyle\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-\nu_{p}(\mathcal{F})+\nu_{p}(\mathcal{B})-\deg(\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-\xi_{p}(\mathcal{F})-\deg(\mathcal{B})+1$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}\mu_{p}(\mathcal{F},B)+\chi_{p}(\mathcal{F})-\deg(\mathcal{B})+1.$

∎

Let $f(x_{1},\ldots,x_{n})\in\mathbb{C}[x_{1},\ldots,x_{n}]$ be a polynomial,
where the origin is an isolated singular point of the hypersurface
$f^{-1}(0)$. The notion of the Milnor number $\mu(f)$ was introduced in [22,
Section 7] as the degree of the mapping $z\to\frac{\nabla(f)}{||\nabla(f)||}$,
where $\nabla$ denotes the gradient function. In particular, for complex plane
curves, Milnor proved, using topological tools, the purely algebraic equality
$\mu(f)=2\delta(f)-r(f)+1$, where $\delta(f)$ is the number of double points
and $r(f)$ is the number of irreducible factors of $f$ (see [22, Theorem
10.5]). The reader can find further formulae for the Milnor number of a plane
curve in [31, Section 6.5]). In particular in [31, Theorem 6.5.1]) it was
established the relationship between the Milnor number of a reduced plane
curve and the Milnor numbers of its irreducible components. The ingredients of
the proof of Wall are Milnor fibrations and the Euler characteristic. We give
another proof of this relationship, using foliations:

######  Proposition 4.8.

Let $C:f(x,y)=0$ be a germ of reduced singular curve at $(\mathbb{C}^{2},p)$.
Assume that $C=\cup_{j=1}^{\ell}C_{j}$ is the decomposition of $C$ in
irreducible components $C_{j}:f_{j}(x,y)=0$, where $f(x,y)=f_{1}(x,y)\cdots
f_{\ell}(x,y)$. Then

$\mu_{p}(C)+\ell-1=\sum_{j=1}^{\ell}\mu_{p}(C_{j})+2\sum_{1\leq
i<j\leq\ell}i_{p}(C_{i},C_{j}).$

###### Proof.

Applying Proposition 4.7 to the foliation defined by $\omega=df$ and to the
balanced divisor of separatrices $C=\sum_{j=1}^{\ell}C_{j}$ we have

(4.10) $\mu_{p}(C)+\ell-1=\sum_{j=1}^{\ell}\mu_{p}(df,C_{j}).$

It follows from Proposition 4.5 that

$\mu_{p}(df,C_{j})=i_{p}(\mathcal{P}^{df},C_{j})-\nu_{p}(C_{j})+1,\,\,\,\,\,\,\hbox{\rm
for}\,\,j=1,\ldots,\ell.$

Using properties on intersection numbers, we have

$i_{p}(\mathcal{P}^{df},C_{j})=i_{p}(\mathcal{P}^{df_{j}},C_{j})+\sum_{i\neq
j}i_{p}(C_{i},C_{j}).$

From Teissier’s Proposition [28, Chapter II, Proposition 1.2], we get

$i_{p}(\mathcal{P}^{df_{j}},C_{j})=\mu_{p}(C_{j})+\nu_{p}(C_{j})-1.$

Thus

(4.11) $\mu_{p}(df,C_{j})=\mu_{p}(C_{j})+\sum_{i\neq j}i_{p}(C_{i},C_{j}).$

The proof ends, by substituting (4.11) into (4.10). ∎

######  Theorem A.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\Delta_{p}(\mathcal{F},\mathcal{B})=\mu_{p}(\mathcal{F})-\sum_{B}a_{B}\mu_{p}(d{F}_{B},B)+\deg(\mathcal{B})-1-\chi_{p}(\mathcal{F}),$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$. Moreover, if $\mathcal{F}$ of a second type foliation, then

$\Delta_{p}(\mathcal{F},\mathcal{B})=\mu_{p}(\mathcal{F})-\sum_{B}a_{B}\mu_{p}(d{F}_{B},B)+\deg(\mathcal{B})-1.$

###### Proof.

By (4.5) and (4.4)

$\displaystyle\Delta_{p}(\mathcal{F},{\mathcal{B}})$ $\displaystyle=$
$\displaystyle\sum_{B}a_{B}\Delta_{p}(\mathcal{F},B)=\sum_{B}a_{B}\left(i_{p}(\mathcal{P}^{\mathcal{F}},B)-i_{p}(\mathcal{P}^{d{F}_{B}},B)\right)$
$\displaystyle=$ $\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B})-\sum_{B}a_{B}i_{p}(\mathcal{P}^{d{F}_{B}},B).$

Hence, after Lemma 4.4 and Proposition 4.5 we have

$\displaystyle\Delta_{p}(\mathcal{F},{\mathcal{B}})$ $\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-\sum_{B}a_{B}\left(\mu_{p}(d{F}_{B},B)+\nu_{p}(B)-1\right)$
$\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-\left(\sum_{B}a_{B}\mu_{p}(d{F}_{B},B)\right)-\nu_{p}(\mathcal{B})+\deg(\mathcal{B}).$

We finish the proof after Proposition 2.4 and the definition of the
$\chi$-number of $\mathcal{F}$.

On the other hand if $\mathcal{F}$ is a second type foliation then
$\chi_{p}(\mathcal{F})=0$ and the second part of the theorem follows. ∎

From Theorem A and Proposition 4.7 we get:

######  Corollary 4.9.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\Delta_{p}(\mathcal{F},\mathcal{B})=\sum_{B}a_{B}(\mu_{p}(\mathcal{F},B)-\mu_{p}(d{F}_{B},B)),$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$.

Corollary 4.9 restricted to non-dicritical singular foliations provides us a
new characterization of non-dicritical generalized curve foliations.

######  Corollary 4.10.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C=\cup_{j=1}^{\ell}C_{j}$ is the total
union of separatrices of $\mathcal{F}$. Then $\mathcal{F}$ is a generalized
curve foliation if and only if

$\mu_{p}(\mathcal{F},C_{j})=\mu_{p}(df,C_{j}),\,\,\,\,\,\,\,\,\,\,\hbox{\rm
for all}\,\,\,\,j=1,\ldots,\ell,$

where $f(x,y)=0$ is a reduced equation of $C$ at $p$.

###### Proof.

According to [13, Theorem A], $\mathcal{F}$ is a generalized curve foliation
at $(\mathbb{C}^{2},p)$ if and only if $\Delta_{p}(\mathcal{F},C)=0$, where
$C$ is the total union of separatrices of $\mathcal{F}$. It follows from
Corollary 4.9 that
$\Delta_{p}(\mathcal{F},C)=\sum_{j=1}^{\ell}(\mu_{p}(\mathcal{F},C_{j})-\mu_{p}(df,C_{j})),$
where $f(x,y)=0$ is a reduced equation of $C$ at $p$. Thus, by Remark 4.6,
$\Delta_{p}(\mathcal{F},C)=0$ if and only if
$\mu_{p}(\mathcal{F},C_{j})=\mu_{p}(df,C_{j})$ for all $j=1,\ldots,\ell.$ ∎

######  Remark 4.11.

Observe that if, in Corollary 4.10, the curve $C:f(x,y)=0$ is irreducible then
we rediscover the classic characterization of generalized curve foliations,
that is, $\mu_{p}(\mathcal{F})=\mu_{p}(df)=\mu_{p}(C)$.

##  5\. The Gómez-Mont-Seade-Verjovsky index

Let $\mathcal{F}:\omega=0$ be a singular foliation at $(\mathbb{C}^{2},p)$.
Let $C:f(x,y)=0$ be an $\mathcal{F}$-invariant curve, where
$f(x,y)\in\mathbb{C}[[x,y]]$ is reduced. Then, as in the convergent case,
there are $g,h\in\mathbb{C}[[x,y]]$ (depending on $f$ and $\omega$), with $f$
and $g$ and $f$ and $h$ relatively prime and a $1$-form $\eta$ (see [26, Lemma
1.1 and its proof]) such that

(5.1) $g\omega=hdf+f\eta.$

The Gómez-Mont-Seade-Verjovsky index of the foliation $\mathcal{F}$ at
$(\mathbb{C}^{2},p)$ (GSV-index) with respect to an analytic
$\mathcal{F}$-invariant curve $C$ is

(5.2) $GSV_{p}(\mathcal{F},C)=\frac{1}{2\pi i}\int_{\partial
C}\frac{g}{h}d\left(\frac{h}{g}\right),$

where $g,h\in\mathbb{C}\\{x,y\\}$ are from (5.1). This index was introduced in
[15] but here we follow the presentation of [2]. If $C$ is irreducible then
equality (5.2) becomes

(5.3)
$GSV_{p}(\mathcal{F},C)=\textsf{ord}_{t}\left(\frac{h}{g}\circ\gamma\right)(t)=i_{p}(f,h)-i_{p}(f,g),$

where $\gamma(t)$ is a Puiseux parametrization of $C$. By [2, page 532], we
get the adjunction formula

(5.4) $GSV_{p}(\mathcal{F},C_{1}\cup
C_{2})=GSV_{p}(\mathcal{F},C_{1})+GSV_{p}(\mathcal{F},C_{2})-2i_{p}(C_{1},C_{2}),$

for any two analytic $\mathcal{F}$-invariant curves, $C_{1}$ and $C_{2}$,
without common irreducible components.

The equality (5.3) allows us to extend the definition of the GSV-index to a
purely formal (non-analytic) irreducible $\mathcal{F}$-invariant curve; and
the equality (5.4) allows us to extend the definition of the GSV-index to
$\mathcal{F}$-invariant curves containing purely formal branches.

The following lemma generalizes the equality (5.3) to any reduced
$\mathcal{F}$-invariant curve (containing perhaps purely formal branches):

######  Lemma 5.1.

Let $C:f(x,y)=0$ be any reduced invariant curve of a singular foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$. Then

$GSV_{p}(\mathcal{F},C)=i_{p}(f,h)-i_{p}(f,g),$

where $g,h\in\mathbb{C}[[x,y]]$ are from (5.1).

###### Proof.

Suppose, without lost of generality, that $f(x,y)=f_{1}(x,y)f_{2}(x,y),$ where
$f_{1},f_{2}\in\mathbb{C}[[x,y]]$ are irreducible and put $C_{i}:f_{i}(x,y)=0$
for $1\leq i\leq 2$. By (5.1) we get

$g\omega=hf_{2}df_{1}+f_{1}(hdf_{2}+f_{2}\eta),$

for some $g,h\in\mathbb{C}[[x,y]]$ relative prime with $f$ and a $1$-form
$\eta$. Hence, if $\gamma_{1}(t)$ is a Puiseux parametrization of $C_{1}$,
then after (5.3) we have

$\displaystyle GSV_{p}(\mathcal{F},C_{1})$ $\displaystyle=$
$\displaystyle\textsf{ord}_{t}\left(\frac{hf_{2}}{g}\circ\gamma_{1}\right)(t)=\textsf{ord}_{t}\left(\frac{h}{g}\circ\gamma_{1}\right)(t)+\,\textsf{ord}_{t}\left({f_{2}}\circ\gamma_{1}\right)(t)$
$\displaystyle=$
$\displaystyle\textsf{ord}_{t}\left(\frac{h}{g}\circ\gamma_{1}\right)(t)+i_{p}(C_{1},C_{2}).$

Similarly, if $\gamma_{2}(t)$ denotes a Puiseux parametrization of $C_{2}$
then we have

$GSV_{p}(\mathcal{F},C_{2})=\textsf{ord}_{t}\left(\frac{h}{g}\circ\gamma_{2}\right)(t)+i_{p}(C_{1},C_{2}).$

The proof follows after equality (5.4) and properties of the intersection
number. ∎

In this section, we will use the following result due to Genzmer-Mol [12,
Theorem B] that establishes a relationship between the GSV-index and the polar
excess number of a foliation with respect to a set of separatrices.

######  Theorem 5.2.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Let $C$ be
a reduced curve of separatrices and
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ be a balanced divisor of
separatrices for $\mathcal{F}$ adapted to $C$. Then

$GSV_{p}(\mathcal{F},C)=\Delta_{p}(\mathcal{F},C)+i_{p}(C,\mathcal{B}_{0}\setminus
C)-i_{p}(C,\mathcal{B}_{\infty}).$

We note that the above theorem implies that if $\mathcal{B}$ is an effective
balanced divisor of separatrices for $\mathcal{F}$, then
$GSV_{p}(\mathcal{F},\mathcal{B})=\Delta_{p}(\mathcal{F},\mathcal{B})$.

On the other hand, as a consequence of Proposition 4.2 and Theorem 5.2 we get

######  Corollary 5.3.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Let
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ be a reduced balanced
divisor of separatrices for $\mathcal{F}$. Then

$GSV_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{0})+1.$

###### Proof.

By applying Theorem 5.2 to $C=\mathcal{B}_{0}$, we have

(5.5)
$GSV_{p}(\mathcal{F},\mathcal{B}_{0})=\Delta_{p}(\mathcal{F},\mathcal{B}_{0})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty}),$

and it follows from Proposition 4.2 that

(5.6)
$\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{0})+1.$

The proof ends, by substituting (5.6) in (5.5). ∎

######  Proposition 5.4.

Let $C:f(x,y)=0$ be any reduced invariant curve of a singular foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$. Then

$\mbox{\rm
GSV}_{p}(\mathcal{F},C)=i_{p}(\mathcal{P}^{\mathcal{F}},C)-i_{p}(\mathcal{P}^{df},C).$

###### Proof.

Let $\omega=P(x,y)dx+Q(x,y)dy$ be a 1-form inducing $\mathcal{F}$. By equality
(5.1) we get $g\omega=hdf+f\eta,$ where $\eta$ is a formal 1-form and
$g,h\in\mathbb{C}[[x,y]]$ with $g$ and $f$ relatively prime and $h$ and $f$
also relatively prime. Let $(a:b)\in\mathbb{P}^{1}$ such that the polar curves
$aP(x,y)+bQ(x,y)=0$ and $a\partial_{x}{f}+b\partial_{y}{f}=0$ of $\mathcal{F}$
and $df$ respectively, are generic. We have
$g\cdot(aP+bQ)=h\cdot(a\partial_{x}{f}+b\partial_{y}{f})+fk,$ for some
$k\in\mathbb{C}[[x,y]]$. Then
$i_{p}(f,g\cdot(aP+aQ))=i_{p}(f,h\cdot(a\partial_{x}{f}+b\partial_{y}{f})+fk)=i_{p}(f,h\cdot(a\partial_{x}{f}+b\partial_{y}{f}))$.
So
$i_{p}(f,g\cdot(aP+aQ))=i_{p}(f,h)+i_{p}(f,a\partial_{x}{f}+b\partial_{y}{f})$.

On the other hand $i_{p}(f,g\cdot(aP+bQ))=i_{p}(f,g)+i_{p}(f,aP+bQ)$, hence

(5.7)
$i_{p}(f,g)+i_{p}(f,aP+bQ)=i_{p}(f,h)+i_{p}(f,a\partial_{x}{f}+b\partial_{y}{f}).$

We finish the proof using Lemma 5.1 and equality (5.7).

∎

######  Remark 5.5.

If $\mathcal{F}$ is a non-dicritical singular foliation at
$(\mathbb{C}^{2},p)$, where $C$ is the total union of separatrices of
$\mathcal{F}$ then
$i_{p}(\mathcal{P}^{\mathcal{F}},f)-i_{p}(\mathcal{P}^{df},f)=\Delta_{p}(\mathcal{F},C)$,
but in general this two values are different as the following example shows:
consider the foliation $\mathcal{F}$ defined by $\omega=2xdy-3ydx$ and the
curve $C:$ $y^{2}-x^{3}=0$. Note that $\mathcal{F}$ admits the meromorphic
first integral $y^{2}/x^{3}$ and so that $C$ is $\mathcal{F}$-invariant. We
get $i_{0}(\mathcal{P}^{\mathcal{F}},C)-i_{0}(\mathcal{P}^{df},C)=-1,$ and
$\Delta_{0}(\mathcal{F},C)=0$, since $\mathcal{F}$ is a generalized curve
foliation.

We obtain the following corollary.

######  Corollary 5.6.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C$ is the total union of separatrices of
$\mathcal{F}$. Then

$\mbox{\rm
GSV}_{p}(\mathcal{F},C)=\mu_{p}(\mathcal{F})-\mu_{p}(C)-\chi_{p}(\mathcal{F}).$

###### Proof.

By Proposition 5.4 we have $\mbox{\rm
GSV}_{p}(\mathcal{F},C)=i_{p}(\mathcal{P}^{\mathcal{F}},C)-i_{p}(\mathcal{P}^{df},C)$
and applying Lemma 4.4 to $\mathcal{F}$ and $C$, we get
$i_{p}(\mathcal{P}^{\mathcal{F}},C)=\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})$.
Teissier’s Proposition [28, Chapter II, Proposition 1.2] implies that

$i_{p}(\mathcal{P}^{df},C)=\mu_{p}(C)+\nu_{p}(C)-1.$

Thus

$\displaystyle\mbox{\rm GSV}_{p}(\mathcal{F},C)$ $\displaystyle=$
$\displaystyle i_{p}(\mathcal{P}^{\mathcal{F}},C)-i_{p}(\mathcal{P}^{df},C)$
$\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F})-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-(\mu_{p}(C)+\nu_{p}(C)-1)$
$\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})-\mu_{p}(C)+\underbrace{(\nu_{p}(\mathcal{F})-\nu_{p}(C)+1)}_{\xi_{p}(\mathcal{F})}-\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})$
$\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})-\mu_{p}(C)-\left(\underbrace{\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\nu_{q}(\mathcal{F})\xi_{q}(\mathcal{F})-\xi_{p}(\mathcal{F})}_{\chi_{p}(\mathcal{F})}\right)$
$\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})-\mu_{p}(C)-\chi_{p}(\mathcal{F}).$

∎

To finish this section we state a relationship between the GSV-index and the
multiplicity of $\mathcal{F}$ along a fixed separatrix.

######  Proposition 5.7.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and
$B:f(x,y)=0$ be a separatrix of $\mathcal{F}$. Then

(5.8) $\mu_{p}(\mathcal{F},B)=\begin{cases}\mbox{\rm
GSV}_{p}(\mathcal{F},B)+\textsf{ord}_{t}\partial_{y}f(\gamma(t))-\textsf{ord}_{t}x(t)+1&\text{if
$x(t)\neq 0$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \mbox{\rm
GSV}_{p}(\mathcal{F},B)+\textsf{ord}_{t}\partial_{x}f(\gamma(t))-\textsf{ord}_{t}y(t)+1&\text{if
$y(t)\neq 0$},\end{cases}$

where $\gamma(t)=(x(t),y(t))$ is a Puiseux parametrization of $B$. In
particular, if $B$ is a non-singular separatrix, then
$\mu_{p}(\mathcal{F},B)=\mbox{\rm GSV}_{p}(\mathcal{F},B)$.

###### Proof.

Let $\omega=P(x,y)dx+Q(x,y)dy$ be a 1-form inducing $\mathcal{F}$ and
$f(x,y)=0$ be a reduced equation of $B$. By equality (5.1) we get

$g\omega=hdf+f\eta,$

where $\eta$ is a formal 1-form and $g,h\in\mathbb{C}[[x,y]]$, where $g$ and
$f$ are relatively prime and $h$ and $f$ are relatively prime. From equality
(4.8), we have that the unique vector field $\theta(t)$ such that
$\gamma_{*}\theta(t)=v(\gamma(t))$, where
$v=-Q(x,y)\frac{\partial}{\partial{x}}+P(x,y)\frac{\partial}{\partial{y}}$, is
given by

(5.9)
$\theta(t)=\begin{cases}\dfrac{-\left(\frac{h}{g}\right)(\gamma(t))\partial_{y}f(\gamma(t))}{x^{\prime}(t)}&\text{if
$x(t)\neq 0$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{\left(\frac{h}{g}\right)(\gamma(t))\partial_{x}f(\gamma(t))}{y^{\prime}(t)}&\text{if
$y(t)\neq 0$}.\end{cases}$

Therefore, the proof follows taking orders. ∎

##  6\. Tjurina number

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ defined by
the 1-form $\omega=P(x,y)dx+Q(x,y)dy$ and $C:f(x,y)=0$ be a
$\mathcal{F}$-invariant reduced curve. The Tjurina number of $\mathcal{F}$
with respect to $C$ is

$\tau_{p}(\mathcal{F},C)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,P,Q),$

The Tjurina number of any germ of reduced curve $C:f(x,y)=0$, with
$f(x,y)\in\mathbb{C}[[x,y]]$ is by definition

$\
\tau_{p}(C)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,\partial_{x}f,\partial_{y}f).$

In this section we will study the Tjurina number of a foliation with respect
to a balanced divisor of separatrices. First of all we present a lemma on
Commutative Algebra which we need on the next, but we do not know a precise
reference:

######  Lemma 6.1.

Let $f,g,p,q\in\mathbb{C}[[x,y]],$ where $f$ and $g$ are relatively prime.
Then

$\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,gp,gq)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,p,q)+\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,g).$

###### Proof.

Observe that
$\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,r_{1},\ldots,r_{n})=\dim_{\mathbb{C}}\mathcal{O}/(r^{\prime}_{1},\ldots,r^{\prime}_{n}),$
where $\mathcal{O}=\mathbb{C}[[x,y]]/(f)$ and $r^{\prime}_{i}=r_{i}+(f)$ for
any $i\in\\{1,\ldots,n\\}$ and any $r_{i}\in\mathbb{C}[[x,y]]$. We finish the
proof using the following exact sequence:

$0\longrightarrow\mathcal{O}/(p^{\prime},q^{\prime})\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\mathcal{O}/(g^{\prime}p^{\prime},g^{\prime}q^{\prime})\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\mathcal{O}/(g^{\prime})\longrightarrow
0,$

where
$\sigma(z^{\prime}+(p^{\prime},q^{\prime}))=g^{\prime}z^{\prime}+(g^{\prime}p^{\prime},g^{\prime}q^{\prime})$
and
$\delta(z^{\prime}+(g^{\prime}p^{\prime},g^{\prime}q^{\prime}))=z^{\prime}+(g^{\prime})$,
for any $z^{\prime}\in\mathcal{O}$. ∎

The following proposition has been proved by X. Gómez-Mont [14, Theorem 1] for
a foliation with a set of convergent separatrices. We show that the same
result holds in the formal context for effective reduced balanced divisor of
separatrices.

######  Proposition 6.2.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and $C$ be a
reduced curve of separatrices of $\mathcal{F}$. Then

$\tau_{p}(\mathcal{F},C)-\tau_{p}(C)=\mbox{\rm GSV}_{p}(\mathcal{F},C).$

###### Proof.

Let $\omega=P(x,y)dx+Q(x,y)dy$ be a 1-form inducing $\mathcal{F}$ and
$f(x,y)=0$ be the reduced equation of $C$. By equality (5.1) we get
$g\omega=hdf+f\eta,$ where $\eta$ is a formal 1-form and
$g,h\in\mathbb{C}[[x,y]]$ with $g$ and $f$ relatively prime and $h$ and $f$
also relatively prime.

Hence
$gPdx+gQdy=(h\partial_{x}{f}+f\eta_{x})dx+(h\partial_{y}{f}+f\eta_{y})dy,$
where $\eta=\eta_{x}dx+\eta_{y}dy$. We get

(6.1)
$gP=h\partial_{x}{f}+f\eta_{x},\,\,\,\,\,\text{and}\,\,\,\,\,gQ=h\partial_{y}{f}+f\eta_{y}.$

After equalities (6.1), properties of the intersection number and Lemma 6.1,
we have

$\displaystyle\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,gP,gQ)$ $\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h\partial_{x}{f}+f\eta_{x},h\partial_{y}{f}+f\eta_{y})$
$\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h\partial_{x}{f},h\partial_{y}{f})$
$\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h)+\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,\partial_{x}{f},\partial_{y}{f})$
$\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h)+\tau_{p}(C).$

Again, by Lemma 6.1 we get

$\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,P,Q)+\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,g)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h)+\tau_{p}(C).$

Hence
$\tau_{p}(\mathcal{F},C)-\tau_{p}(C)=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,h)-\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/(f,g)=i_{p}(f,h)-i_{p}(f,g)$.
The proof follows from Lemma 5.1. ∎

######  Example 6.3.

Let $\mathcal{F}$ be the foliation defined by the formal normal form of a
saddle-node (see (2.4)) $\omega=x^{k+1}dy-y(1+\lambda
x^{k})dx,\,\,\,\,\,\,k\geq 1,\,\,\,\,\,\lambda\in\mathbb{C}.$ The total union
of separatrices of $\mathcal{F}$ is $C=C_{1}\cup C_{2}$, where $C_{1}:x=0$
(strong separatrix) and $C_{2}:y=0$ (weak separatrix). An equality (5.1) for
$C_{1}$ is given for $g=1$, $h=-y(1+\lambda x^{k})$ and $\eta=x^{k}dy$, hence
by Lemma 5.1 we get $\mbox{\rm
GSV}_{0}(\mathcal{F},C_{1})=i_{0}(x,h)-i_{0}(x,g)=1$. Similarly, an equality
(5.1) for $C_{2}$ is given for $g=1$, $h=x^{k+1}$ and $\eta=-(1+\lambda
x^{k})dx$, thus $\mbox{\rm
GSV}_{0}(\mathcal{F},C_{2})=i_{0}(y,h)-i_{0}(y,g)=k+1$. Therefore, one finds

$\mbox{\rm GSV}_{0}(\mathcal{F},C)=\mbox{\rm
GSV}_{0}(\mathcal{F},C_{1})+\mbox{\rm
GSV}_{0}(\mathcal{F},C_{2})-2i_{0}(C_{1},C_{2})=1+(k+1)-2=k.$

On the other hand, we get
$\tau_{0}(\mathcal{F},C)-\tau_{0}(C)=(k+1)-1=k=\mbox{\rm
GSV}_{0}(\mathcal{F},C).$

######  Corollary B.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C$ is the total union of separatrices of
$\mathcal{F}$. Then

$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},C)=\mu_{p}(C)-\tau_{p}(C)+\chi_{p}(\mathcal{F}).$

Moreover, if $\mathcal{F}$ is of second type then
$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},C)=\mu_{p}(C)-\tau_{p}(C)$.

###### Proof.

It is a consequence of Proposition 6.2 and Corollary 5.6. ∎

Now, we characterize generalized curve foliations at $(\mathbb{C}^{2},p)$ in
terms of its Tjurina numbers.

######  Corollary 6.4.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ be a reduced balanced
divisor of separatrices for $\mathcal{F}$. Then $\mathcal{F}$ is a generalized
curve if and only if
$\tau_{p}(\mathcal{B}_{0})-\tau_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty}).$

###### Proof.

It follows from [13, Theorem A] that $\mathcal{F}$ is a generalized curve
foliation if and only if $\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=0$. Applying
Theorem 5.2 to $C=\mathcal{B}_{0}$, we get $\mbox{\rm
GSV}_{p}(\mathcal{F},\mathcal{B}_{0})=\Delta_{p}(\mathcal{F},\mathcal{B}_{0})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})$.
Hence $\mathcal{F}$ is a generalized curve foliation if and only if $\mbox{\rm
GSV}_{p}(\mathcal{F},\mathcal{B}_{0})=-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})$.
The proof ends, by applying Proposition 6.2 to $C=\mathcal{B}_{0}$. ∎

If ${\mathcal{B}}=\sum_{B}a_{B}B$ is a divisor of separatrices for
$\mathcal{F}$ then we put

$T_{p}(\mathcal{F},\mathcal{B})=\sum_{B}a_{B}\tau_{p}(\mathcal{F},B).$

The following theorem gives a relationship between the Milnor and Tjurina
numbers and the $\chi$-number, studied in Section 3.

######  Theorem C.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\displaystyle\mu_{p}(\mathcal{F})-T_{p}(\mathcal{F},\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}[\mu_{p}(dF_{B},B)-\tau_{p}(B)]-\deg(\mathcal{B})+1+\chi_{p}(\mathcal{F})$
$\displaystyle-\sum_{B}a_{B}[i_{p}(B,(F_{B})_{0}\setminus
B)-i_{p}(B,(F_{B})_{\infty})],$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$.

###### Proof.

By Proposition 6.2 we get
$T_{p}(\mathcal{F},\mathcal{B})=\sum_{B}a_{B}(GSV_{p}(\mathcal{F},B)+\tau_{p}(B)).$
Then
$T_{p}(\mathcal{F},\mathcal{B})-\sum_{B}a_{B}\tau_{p}(B)=\sum_{B}a_{B}GSV_{p}(\mathcal{F},B)$.
From Theorem 5.2, we have

$\displaystyle T_{p}(\mathcal{F},\mathcal{B})-\sum_{B}a_{B}\tau_{p}(B)$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}[\Delta_{p}(\mathcal{F},B)+i_{p}(B,(F_{B})_{0}\setminus
B)-i_{p}(B,(F_{B})_{\infty})]$ $\displaystyle=$
$\displaystyle\Delta_{p}(\mathcal{F},\mathcal{B})+\sum_{B}a_{B}[i_{p}(B,(F_{B})_{0}\setminus
B)-i_{p}(B,(F_{B})_{\infty})],$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$. We finish the proof using Theorem A. ∎

In order to illustrate Theorem C we present a family of dicritical foliations
that are not of second type.

######  Example 6.5.

Let $\lambda\in\mathbb{C}$ and $k\geq 3$ integer. Let $\mathcal{F}_{k}$ be the
singular foliation at $(\mathbb{C}^{2},0)$ defined by

$\omega_{k}=y(2x^{2k-2}+2(\lambda+1)x^{2}y^{k-2}-y^{k-1})dx+x(y^{k-1}-(\lambda+1)x^{2}y^{k-2}-x^{2k-2})dy.$

The foliation $\mathcal{F}_{k}$ is dicritical.

$\bullet$$0$$y$$x$Saddle-node singularityStrong separatrixWeak separatrix

Figure 1. Dicritical foliation $\mathcal{F}_{k}$

After one blow-up, the foliation has a unique non-reduced singularity $q$. A
further blow-up applied to $q$ produces a reduction of singularities of the
foliation with a dicritical component and a tangent saddle-node with strong
separatrix transversal to exceptional divisor. Therefore $\mathcal{F}_{k}$ is
not of second type. Let $B_{1}:y=0$ and $B_{2}:x=0$, then
$\mathcal{B}=B_{1}+B_{2}$ is an effective balanced divisor of separatrices for
$\mathcal{F}_{k}$. A simple calculation leads to:

$\nu_{0}(\mathcal{F}_{k})=k,\,\,\,\nu_{q}(\mathcal{F}_{k})=k-1,\,\,\,\,\xi_{0}(\mathcal{F}_{k})=k-1,\,\,\,\xi_{q}(\mathcal{F}_{k})=k-1,$

thus $\chi_{p}(\mathcal{F}_{k})=2(k-1)^{2}$. Moreover
$\mu_{0}(\mathcal{F}_{k})=(k-2)(2k-2)+5k-4$,
$T_{0}(\mathcal{F}_{k},\mathcal{B})=3k-1$,
$\tau_{0}(B_{1})=\tau_{0}(B_{2})=0$. Since $F(x,y)=xy$ defines a balanced
divisor of separatrices for $\mathcal{F}_{k}$ adapted to $B_{1}$ and $B_{2}$,
we get $\mu_{0}(dF,B_{1})+\mu_{0}(dF,B_{2})=2.$ Hence we have
$T_{0}(\mathcal{F}_{k},\mathcal{B})=3k-1$ and Theorem C is verified.

######  Corollary 6.6.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. If $\mathcal{F}$ is of second type, then

$\displaystyle\mu_{p}(\mathcal{F})-T_{p}(\mathcal{F},\mathcal{B})$
$\displaystyle=$
$\displaystyle\sum_{B}a_{B}[\mu_{p}(dF_{B},B)-\tau_{p}(B)]-\deg(\mathcal{B})+1$
$\displaystyle-\sum_{B}a_{B}[i_{p}(B,(F_{B})_{0}\setminus
B)-i_{p}(B,(F_{B})_{\infty})],$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$.

###### Proof.

By Lemma 3.1 item (2), $\chi_{p}(\mathcal{F})=0$ and the proof follows from
Theorem C. ∎

The following example shows that, in general, the reciprocal of Corollary 6.6
is not true.

######  Example 6.7 (Dulac’s foliation).

The foliation ${\mathcal{F}}$ defined by the $1$-form ${\omega}=(ny+x^{n})dx-
xdy$, with $n\geq 2$, admits a unique separatrix $C:$ $x=0$. Since
$\nu_{0}(\mathcal{F})=1\not=0=\nu_{0}(C)-1$, so $\mathcal{F}$ is not of second
type. Moreover $T_{0}(\mathcal{F},C)=1$, $\tau_{0}(C)=0$,
$\mu_{0}(\mathcal{F})=1$, $\mu_{0}(dx,C)=0$. Hence, $\mathcal{F}$ verifies the
equality of Corollary 6.6 but it is not a foliation of second type at
$0\in\mathbb{C}^{2}$.

$\bullet$$0$$y$$\longleftarrow$Isolated separatrix$x$Strong separatrixSaddle-
node singularityWeak separatrix

Figure 2. Dulac foliation $\mathcal{F}:\omega=(ny+x^{n})dx-xdy$

Now, we apply Theorem C to non-dicritical singular foliations at
$(\mathbb{C}^{2},p)$.

######  Corollary D.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C=\cup_{j=1}^{\ell}C_{j}$ is the total
union of separatrices of $\mathcal{F}$. Then

$\mu_{p}(\mathcal{F})-T_{p}(\mathcal{F},C)=\mu_{p}(C)-\sum_{j=1}^{\ell}\tau_{p}(C_{j})+\chi_{p}(\mathcal{F})-\sum_{j=1}^{\ell}i_{p}(C_{j},C\setminus
C_{j}).$

###### Proof.

By taking the effective divisor $\mathcal{B}=(f)$, where $f(x,y)=0$ is an
equation of $C$, and applying Proposition 4.7 to the foliation $df$, we get

(6.2) $\mu_{p}(C)=\sum_{j=1}^{\ell}\mu_{p}(df,C_{j})-\ell+1.$

The proof follows from Theorem C. ∎

######  Corollary 6.8.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$. Assume that
$\mathcal{F}$ is non-dicritical and $C=\cup_{j=1}^{\ell}C_{j}$ is the total
union of separatrices of $\mathcal{F}$. Then

$T_{p}(\mathcal{F},C)-\tau_{p}(\mathcal{F},C)=\sum_{j=1}^{\ell}\tau(C_{j})-\tau(C)+2\sum_{1\leq
i<j\leq\ell}i_{p}(C_{i},C_{j}).$

Here is an example to illustrate Corollary D.

######  Example 6.9.

Let $\omega=4xydx+(y-2x^{2})dy$ be a $1$-form defining a singular foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},0)$. The unique separatrix of $\mathcal{F}$
is the curve $C:y=0$. Since $\nu_{0}(\mathcal{F})=1$ and $\nu_{0}(C)=1$, the
foliation $\mathcal{F}$ is not of second type at the origin (see Proposition
2.4). Moreover, we get $T_{0}(\mathcal{F},C)=2$, $\tau_{0}(C)=0$,
$\mu_{0}(\mathcal{F})=3$, $\mu_{0}(C)=0$, and $\chi_{0}(\mathcal{F})=1$.

$\bullet$$0$$y$$x$Strong separatrixSaddle-node singularityWeak separatrix

Figure 3. Foliation ${\mathcal{F}}:\omega=4xydx+(y-2x^{2})dy$

###  6.1. Milnor and Tjurina numbers and some residue-type indices

We finish this section stating numerical relationships between some classic
indices, as Baum-Bott, Camacho-Sad and variational indices, of a singular
foliation at $(\mathbb{C}^{2},p)$ and the Milnor and Tjurina numbers.

Let $\mathcal{F}$ a singular foliation defined by a $1$-form $\omega$ as in
(2.2). Let $J(x,y)$ be the Jacobian matrix of $(Q(x,y),-P(x,y))$. The Baum-
Bott index (see [1]) of $\mathcal{F}$ at $p$ is

$\text{BB}_{p}(\mathcal{F})={\rm Res}_{p}\left\\{\frac{({\rm
tr}J)^{2}}{-P\cdot Q}dx\wedge dy\right\\},$

where ${\rm tr}J$ denotes the trace of $J$.

The Camacho-Sad index of $\mathcal{F}$ (CS index) with respect to an analytic
$\mathcal{F}$-invariant curve $C$ is

(6.3) $\text{CS}_{p}(\mathcal{F},C)=-\frac{1}{2\pi i}\int_{\partial
C}\frac{1}{h}\eta,$

where $g,h$ are from (5.1). The Camacho-Sad index was introduced by these
authors in [6] for a non-singular $\mathcal{F}$-invariant curve $C$. Later,
Lins Neto in [19] and Suwa in [26] generalize this index to singular
$\mathcal{F}$-invariant curves. If $C$ is irreducible then equality (6.3)
becomes

(6.4) $\text{CS}_{p}(\mathcal{F},C)=-{\rm
Res}_{t=0}\left(\gamma^{*}\frac{1}{h}\eta\right),$

where $\gamma(t)$ is a Puiseux parametrization of $C$. By [3, page 38] (see
also [27]) we get the adjunction formula

(6.5) $\text{CS}_{p}(\mathcal{F},C_{1}\cup
C_{2})=\text{CS}_{p}(\mathcal{F},C_{1})+\text{CS}_{p}(\mathcal{F},C_{2})+2i_{p}(C_{1},C_{2}),$

for any two analytic $\mathcal{F}$-invariant curves, $C_{1}$ and $C_{2}$,
without common irreducible components. The equality (6.4) allows us to extend
the definition of the CS index to a purely formal (non-analytic) irreducible
$\mathcal{F}$-invariant curve; and the equality (6.5) allows us to extend the
definition of the CS index to $\mathcal{F}$-invariant curves containing purely
formal branches.

In a neighborhood of a non-singular point of the foliation $\mathcal{F}$,
there is a $1$-form $\alpha$ such that $d\omega=\alpha\wedge\omega$. If
$\alpha^{\prime}$ is other such $1$-form, then $\alpha$ and $\alpha^{\prime}$
coincide over every leaf of $\mathcal{F}$. Hence, in a neighborhood of $0$
(away $0$) there exists a holomorphic multi-valued $1$-form $\alpha$ such that
$d\omega=\alpha\wedge\omega$ and that its restriction to each leaf of
$\mathcal{F}$ is single-valued. We say that $\alpha$ is an exponent form for
$\omega$. The variational index or variation of $\mathcal{F}$ relative to $C$
at $p$ is

$\mbox{\rm Var}_{p}(\mathcal{F},C)={\rm
Res}_{t=0}\left(\alpha_{\big{|}_{C}}\right)=\frac{1}{2\pi i}\int_{\partial
C}\alpha.$

The variational index was introduced in [17]. It is additive:

$\mbox{\rm Var}_{p}(\mathcal{F},C_{1}\cup C_{2})=\mbox{\rm
Var}_{p}(\mathcal{F},C_{1})+\mbox{\rm Var}_{p}(\mathcal{F},C_{2}),$

where $C_{1}$ and $C_{2}$ are $\mathcal{F}$-invariant curves without common
factors. For any divisor $\mathcal{B}=\sum_{B}a_{B}B$ of separatrices for
$\mathcal{F}$ we put

$\mbox{\rm Var}_{p}(\mathcal{F},\mathcal{B})=\sum_{B}a_{B}\mbox{\rm
Var}_{p}(\mathcal{F},B).$

For any analytic $\mathcal{F}$-invariant curve $C$ we have, after [2,
Proposition 5],

(6.6) $\mbox{\rm
Var}_{p}(\mathcal{F},C)=\text{CS}_{p}(\mathcal{F},C)+\mbox{\rm
GSV}_{p}(\mathcal{F},C).$

######  Proposition 6.10.

Let $\mathcal{F}$ be a singular foliation at $(\mathbb{C}^{2},p)$ and let
$\mathcal{B}=\sum_{B}a_{B}B$ be a balanced divisor of separatrices for
$\mathcal{F}$. Then

$\text{BB}_{p}(\mathcal{F})=\mbox{\rm
Var}_{p}(\mathcal{F},\mathcal{B})+\mu_{p}(\mathcal{F})-\sum_{B}a_{B}\mu_{p}(dF_{B},B)+\deg(\mathcal{B})-1-\chi_{p}(\mathcal{F})+\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\xi^{2}_{q}(\mathcal{F}),$

where ${F}_{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
adapted to $B$.

Moreover, if $\mathcal{F}$ is non-dicritical and $C$ is the total union of
separatrices of $\mathcal{F}$ then

$\text{BB}_{p}(\mathcal{F})=\text{CS}_{p}(\mathcal{F},C)+\tau_{p}(\mathcal{F},C)-\tau_{p}(C)+\mu_{p}(\mathcal{F})-\mu_{p}(C)-\chi_{p}(\mathcal{F})+\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\xi^{2}_{q}(\mathcal{F}).$

###### Proof.

Suppose that $\mathcal{F}$ is a singular foliation. By [10, Theorem 5.2] we
get

$\text{BB}_{p}(\mathcal{F})=\mbox{\rm
Var}_{p}(\mathcal{F},\mathcal{B})+\Delta_{p}(\mathcal{F},\mathcal{B})+\sum_{q\in\mathcal{I}_{p}(\mathcal{F})}\xi^{2}_{q}(\mathcal{F}),$

where the summation runs over all infinitely near points of $\mathcal{F}$ at
$p$. The proof of the first equality follows applying Theorem A to
$\Delta_{p}(\mathcal{F},\mathcal{B})$.

The proof of the non-dicritical case follows from the first part of the
proposition (for $\mathcal{B}=C$ and $F_{C_{i}}=f$, for any $1\leq i\leq\ell$,
where $f(x,y)=0$ is an equation of $C$), equality (6.6), Proposition 6.2 and
equality (6.2). ∎

##  7\. Bound for the Milnor sum of an algebraic curve

Let $\mathcal{F}$ be a holomorphic foliation on the complex projective plane
$\mathbb{P}^{2}$. The degree of $\mathcal{F}$ is the number of tangencies
between $\mathcal{F}$ and a generic line. Let $C$ be an algebraic
$\mathcal{F}$-invariant curve in $\mathbb{P}^{2}$. We say that $C$ is non-
dicritical if every singular point of $\mathcal{F}$ on $C$ is non-dicritical.

As an application of our previous results, we propose the following theorem:

######  Theorem 7.1.

Let $\mathcal{F}$ be a holomorphic foliation on $\mathbb{P}^{2}$ of degree $d$
leaving invariant a non-dicritical algebraic curve $C$ of degree $d_{0}$ such
that for each $p\in\textsf{Sing}(\mathcal{F})\cap C$, all local branches of
$\mbox{\rm Sep}_{p}(\mathcal{F})$ are contained in $C$. Then

$\sum_{p\in C}\mu_{p}(C)\leq\left(\sum_{p\in\textsf{Sing}(\mathcal{F})\cap
C}\mu_{p}(\mathcal{F})\right)+d^{2}_{0}-(d+2)d_{0}.$

Moreover, if $\textsf{Sing}(\mathcal{F})\subset C$, then

$\sum_{p\in C}\mu_{p}(C)\leq d^{2}-d(d_{0}-1)+(d_{0}-1)^{2}.$

###### Proof.

Fix an arbitrary point $p\in\textsf{Sing}(\mathcal{F})\cap C$. Then, by
Corollary 5.6, we have

$\displaystyle\mu_{p}(C)$ $\displaystyle=$
$\displaystyle\mu_{p}(\mathcal{F})-\mbox{\rm
GSV}_{p}(\mathcal{F},C)-\chi_{p}(\mathcal{F}).$

Since $\chi_{p}(\mathcal{F})\geq 0$ by Proposition 3.1 item (1), we get

$\mu_{p}(C)\leq\mu_{p}(\mathcal{F})-\mbox{\rm GSV}_{p}(\mathcal{F},C).$

The equality $\displaystyle\sum_{p\in\textsf{Sing}(\mathcal{F})\cap
C}\mbox{\rm GSV}_{p}(\mathcal{F},C)=(d+2)d_{0}-d_{0}^{2}$ (cf. [2, Proposition
4]) implies the inequality

$\sum_{p\in C}\mu_{p}(C)\leq\left(\sum_{p\in\textsf{Sing}(\mathcal{F})\cap
C}\mu_{p}(\mathcal{F})\right)+d^{2}_{0}-(d+2)d_{0}.$

On the other hand, if $\textsf{Sing}(\mathcal{F})\subset C$, we use the
equality
$\displaystyle\sum_{p\in\textsf{Sing}(\mathcal{F})}\mu_{p}(\mathcal{F})=d^{2}+d+1$
(cf. [3, page 28]) to finish the proof. ∎

######  Remark 7.2.

Observe that the hypothesis all local branches of $\mbox{\rm
Sep}_{p}(\mathcal{F})$ and all singularities of $\mathcal{F}$ are contained in
$C$ in Theorem 7.1 implies, by [10, Proposition 6.1], that $\deg
C=\deg\mathcal{F}+2$ and $\mathcal{F}$ is a logarithmic foliation.

Acknowledgment: The first-named author thanks Universidad de La Laguna for the
hospitality during his visit.

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