# On characterizations of nondicritical generalized curve foliations
00footnotetext:  2000 Mathematics Subject Classification: Primary32S65;
Secondary 32S05.
Key words and phrases: Generalized curve foliation, Weierstrass form, Plane
singular analytic curve, Second type foliation.
The first-named author was partially supported by the Spanish Projects
PID2019-105896GB-I00 and the second-named author was partially supported by
CNPq-Brazil.

Evelia R. García Barroso, Marcelo E. Hernandes and M. Fernando Hernández
Iglesias

###### Abstract

We characterize nondicrital generalized curve foliations with fixed reduced
separatrix. Moreover, we give suficient conditions when a plane analytic curve
is its reduced separatrix. For that, we introduce a distinguished expression
for a given 1-form, called Weierstrass form. Then, using Weierstrass forms, we
characterize the nondicritical generalized curve foliations: first, for
foliations with monomial separatrix using toric resolution; second, for
foliations with reduced separatrix, using the $GSV$-index. In this last case
the characterization, which is our main result, could be interpreted in
function of a polar of the foliation and a polar of its reduced separatrix.

##  1 Introduction

In the study of holomorphic foliations to determine topological information is
an important and a non trivial question. A special class of foliations gives
some information about this problem: the generalized curve foliations. A
foliation is said generalized curve if no saddle-nodes appear in its
desingularization process. Moreover, a nondicrital generalized curve foliation
and the union of its separatrices have the same process of reduction of
singularities (see [Cam-LN-S], Theorem 2).

In this paper, we consider this class of foliations with the aim to present
some new characterizations of nondicritical generalized curve foliation by
means the $1$-form that defines it.

First, we characterize, in Theorem 3.4, the nondicrital generalized curved
foliations with monomial separatrix using toric resolution, the prenormal form
given by Loray in [Lo, page 157] and the notion of _weighted order_ associated
with a foliation and a curve (see Definition 3.1). This allows us to give, in
Corollary 3.6, the condition in order that a foliation of second type becomes
a generalized curve foliation following [FS-GB-SM, Theorem 1.2 (b)]. Second,
using the Weierstrass division of power series, we introduce, in Definition
4.2, the notion of _Weierstrass form_ of any $1$-form with respect to any
polynomial in one variable and coefficients in the complex power ring in one
variable with a unit as principal coefficient. We can consider the Weierstrass
form as a generalization, to any 1-form, of the prenormal form given by Loray
for foliations with monomial separatrix.

In [B], Brunella stablished relations among the Baum-Bott, Camacho-Sad and
Gómez-Mont- Seade-Verjovsky indices associated to one foliation. He proved
that if the foliation is generalized curve then the Baum-Bott and Camacho-Sad
indices are equal and the Gómez-Mont-Seade-Verjovsky index is zero. This paper
goes in this direction.

Our main result is Theorem 5.3, where we characterize the nondicritical
generalized curve foliations using the _GSV-index_ and the notion of
Weierstrass form associated with a $1$-form defining a foliation. This
characterization is given in terms of intersection numbers and it could be
interpreted in the language of polar curves of the foliation and its union of
separatrices.

We precise, in Proposition 5.10, this characterization for foliations with a
single separatrix and for foliations with a single separatrix of genus 1 in
Corollary 5.11, this last one in terms of the weighted order.

The Weierstrass form has been very useful throughout the development of this
work and we consider that it may be interesting in the future for foliation
studies. Example of this is Proposition 4.3, where given
$f\in\mathbb{C}\\{x\\}[y]$ with $\hbox{\rm ord}f=\deg_{y}(f)$ and principal
coefficient a unit in $\mathbb{C}\\{x\\}$, and using the notion of Weierstrass
form of a 1-form ${W}$ (defining a non-dicritical foliation) with respect to
$f$, we give suficient conditions when $f$ is the equation of the union of
separatrices of the foliation determined by $W$.

##  2 Preliminaries

Let $f(x,y)\in\mathbb{C}\\{x,y\\}$ be a non unit power series, where
$\mathbb{C}\\{x,y\\}$ is the set of convergent power series. A plane curve
$C:\\{f(x,y)=0\\}$ is by definition the zeroes set determined by
$f(x,y)\in\mathbb{C}\\{x,y\\}$. The curve $C$ is irreducible (respectively
reduced) if $f$ is irreducible in $\mathbb{C}\\{x,y\\}$ (respectively $f$ has
no multiple factors). An irreducible plane curve is called branch. The
multiplicity of $C$, denoted by $\hbox{\rm mult$\,$}C$, is by definition the
order of the power series $f(x,y)$, that is $\hbox{\rm mult$\,$}C=\hbox{\rm
ord}f$. Remember that the order of $f$ is the minimum of degrees of terms of
$f$.

Consider a branch $C:\\{f(x,y)=0\\}$ of multiplicity $n$. After a change of
coordinates we can suppose that $x=0$ is transversal (not tangent) to $C$ at
$0$. From Newton’s Theorem, $C$ admits an expansion with rational exponents
$y(x^{1/n})$, such that $f(x,y(x^{1/n}))=0$. According to Puiseux, the branch
$C$ admits $n$ different expansions $\\{y_{i}(x^{1/n})\\}_{1\leq i\leq n}$,
where $y_{i}(x^{1/n})=y(\varepsilon_{i}x^{1/n})$ and
$\\{\varepsilon_{i}\\}_{1\leq i\leq n}$ are the $n$th roots of unity in
$\mathbb{C}$. We can write $f$, up to product by a unit in
$\mathbb{C}\\{x,y\\}$, as the product

$f(x,y)=\prod_{i=1}^{n}\left(y-y_{i}(x^{1/n})\right).$

The expansions $\\{y_{i}(x^{1/n})\\}_{1\leq i\leq n}$ are called Newton-
Puiseux roots of the branch $C$ (or equivalently of $f$). Any Newton-Puiseux
root $y_{i}(x^{1/n})$ is of the form

$y_{i}(x^{1/n})=\sum_{j\geq n}a_{j}^{(i)}x^{j/n},$

where $j\in\mathbb{N}$, $a_{j}^{(i)}\in\mathbb{C}$ and it determines a
parametrization of $C$ as follows

$(x_{i}(t),y_{i}(t))=\left(t^{n},\sum_{j\geq n}a_{j}^{(i)}t^{j}\right).$

It is well-known that if the multiplicity of a branch is bigger than one, then
there exist $g\in\mathbb{N}\backslash\\{0\\}$ and positive integers
$\beta_{0}=n$ and $\beta_{k}=\min\\{k\,:\,a^{(i)}_{k}\neq 0\
\mbox{and}\gcd(n,\beta_{1},\beta_{2},\ldots,\beta_{k-1})\;\hbox{\rm does not
divide }k\\}$ for $1\leq k\leq g$. In the sequel we put
$e_{k}=\gcd(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{k})$ for $0\leq k\leq
g$. We get $e_{0}=n>e_{1}>\cdots>e_{g}=1$. Set $n_{k}:=\frac{e_{k-1}}{e_{k}}$.
In particular, $n=\beta_{0}=n_{1}\cdots n_{g}$.

The sequence $(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{g})$ is called the
Puiseux characteristic exponents of the branch $C$ and the number $g$ is
called the genus of the branch $C$. We denote by
$K(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{g})$ the set of plane branches
with Puiseux characteristic exponents
$(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{g})$. It is well-known, after
Brauner-Zariski, that the characteristic exponents determine the topological
class of the branch $C$.

Let $h_{1}(x,y),h_{2}(x,y)\in\mathbb{C}\\{x,y\\}$ be two power series and
$I=(h_{1},h_{2})$ the ideal generated by $h_{1},h_{2}\in\mathbb{C}\\{x,y\\}$.
The intersection number of the curves $C_{i}:\\{h_{i}(x,y)=0\\}$, $1\leq i\leq
2$, is $i_{0}(h_{1},h_{2}):=\dim_{\mathbb{C}}\mathbb{C}\\{x,y\\}/I.$

Given $f\in K(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{g})$ the semigroup
$\Gamma(f)$ associated to $f$ is

$\Gamma(f)=\\{i_{0}(f,h)\,:\,\ h\in\mathbb{C}\\{x,y\\}\setminus(f)\\}.$

The semigroup $\Gamma(f)$ admits a unique minimal system of generators, given
by $\\{v_{0},v_{1},v_{2},\ldots,v_{g}\\}$, that is, $\Gamma(f)=\langle
v_{0},v_{1},v_{2},\ldots,v_{g}\rangle=\left\\{\sum_{i=0}^{g}\alpha_{i}v_{i}\,:\,\alpha_{i}\in\mathbb{N}\right\\}$.
It is a well know fact that the semigroup $\Gamma(f)$ and the characteristic
exponents are mutually determined. Moreover, we can obtain the minimal system
of generators by the Puiseux characteristic exponents using the relations (see
[Z], Theorem 3.9):

$v_{0}=n=\beta_{0},\ \ \ v_{1}=m=\beta_{1},\ \ \
v_{i+1}=n_{i}v_{i}+\beta_{i+1}-\beta_{i}\ \ \mbox{for}\ i=1,\ldots,g-1.$ (1)

Observe that $e_{i}=\gcd(v_{0},\ldots,v_{i})$ for $0\leq i\leq g$.

For any reduced plane curve (not necessary irreducible) $C:\\{f(x,y)=0\\}$, an
important topological invariant, useful in this paper, is the Milnor number,
that is the intersection number $\mu(f):=i_{0}\left(f_{x},f_{y}\right)$, where
$f_{y}$ (respectively $f_{x}$) denotes the partial derivative of $f$ with
respect to $y$ (respectively $x$).

By Teissier’s Lemma (see [Te, Chapter II, Proposition 1.2]) we have

$i_{0}(f,f_{y})=\mu(f)+i_{0}(f,x)-1.$ (2)

We also have

$i_{0}(f,f_{x})=\mu(f)+i_{0}(f,y)-1.$ (3)

For more details on plane curves see for example [Hef] or [W].

Let $\Omega^{1}_{\mathbb{C}^{2},0}:=\mathbb{C}\\{x,y\\}{\hbox{\rm
d}}x+\mathbb{C}\\{x,y\\}{\hbox{\rm d}}y$ be the $\mathbb{C}\\{x,y\\}$-module
of holomorphic 1-form.

A holomorphic foliation singular at the origin is defined, in a neighbourhood
of the origin, by an equation ${\mathcal{F}}_{W}:\\{W=0\\},$ where $W$ is a
1-form $W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y,$ with
$A,B\in\mathbb{C}\\{x,y\\}$ without common factors and such that
$A(0,0)=B(0,0)=0$. The polar of ${\mathcal{F}}_{W}:\\{W=0\\}$ with respect the
direction $(b:-a)\in\mathbb{P}^{1}$ is the curve $aA(x,y)+bB(x,y)=0$.

The multiplicity of $W$ is $\hbox{\rm mult$\,$}(W):=\min\\{\hbox{\rm
ord}(A(x,y)),\hbox{\rm ord}(B(x,y))\\}$. More precisely if $\hbox{\rm
mult$\,$}(W)=n_{0}$ then we can write $W=\sum_{i+j\geq
n_{0}}a_{i,j}x^{i}y^{j}{\hbox{\rm d}}x+\sum_{i+j\geq
n_{0}}b_{i,j}x^{i}y^{j}{\hbox{\rm d}}y$, and for some $i,j$ such that
$i+j=n_{0}$ we get $a_{i,j}\neq 0$ or $b_{i,j}\neq 0$. The multiplicity of the
foliation ${\mathcal{F}}_{W}:\\{W=0\\}$ is by definition the multiplicity of
the 1-form $W$.

Let ${\mathcal{F}}_{W}$ be a singular foliation at the origin. The Jacobian
matrix of the linear part of $W$ is the matrix

$\left(\begin{array}[]{rr}-B_{x}(0,0)&-B_{y}(0,0)\\\
A_{x}(0,0)&A_{y}(0,0)\end{array}\right).$

Denote by $\lambda_{1},\lambda_{2}$ its complex eigenvalues. We say that the
origin is an irreducible singularity of ${\mathcal{F}}_{W}$ if one of the
following conditions is satisfied:

  1. 1.

$\lambda_{1}\lambda_{2}\neq 0$ and
$\frac{\lambda_{1}}{\lambda_{2}}\notin\mathbb{Q}^{+}$,

  2. 2.

$\lambda_{1}=0$ and $\lambda_{2}\neq 0$; or $\lambda_{2}=0$ and
$\lambda_{1}\neq 0$.

If the first condition holds we say that the origin is a simple or non-
degenerate singularity of ${\mathcal{F}}_{W}$. Nevertheless if the second
condition holds then we say that the origin is a saddle-node singularity of
${\mathcal{F}}_{W}$.

######  Example 2.1.

If $W=y^{2}{\hbox{\rm d}}x+x{\hbox{\rm d}}y$ then the Jacobian matrix of the
linear part of $W$ is $\left(\begin{array}[]{rc}-1&0\\\
0&0\end{array}\right)$. Its eigenvalues are $-1$ y $0$, so the origin is a
saddle-node singularity of ${\mathcal{F}}_{W}\,$.

Let $\Pi:(M,D)\longrightarrow(\mathbb{C}^{2},0)$ the process of reduction of
singularities of the foliation ${\mathcal{F}}_{W}$ (see [Sei]), given by a
finite composition of quadratic transformations (blow-ups) having
$D=\Pi^{-1}(0)$ as the _exceptional divisor_ , which is a finite union of
projective lines with normal crossings; and where $M$ is a non-singular
analytic manifold. There is a 1-form $W^{\prime}$ such that $\Pi^{-1}(0)\cdot
W^{\prime}=\Pi^{*}(W):=\Pi^{*}(\mathcal{F}_{W})$. The foliation
${\mathcal{F}}_{W^{\prime}}$ is called the strict transform of
${\mathcal{F}}_{W}$.

A foliation is generalized curve if no saddle-nodes appear in its
desingularization.

A branch $C:\\{f(x,y)=0\\}$ is a separatrix of the foliation
${\mathcal{F}}_{W}:\\{W=0\\}$ if $f$ divides $W\wedge df$. This is equivalent
to the existence of a parametrization $(x(t),y(t))$ of the branch $C$ such
that $A(x(t),y(t)){\hbox{\rm d}}x(t)+B(x(t),y(t)){\hbox{\rm d}}y(t)=0$. In
[Cam-S], Camacho and Sad prove that any singular foliation in
$(\mathbb{C}^{2},0)$ admits a separatrix. Their proof is based on the
desingularization process of the foliation. In Example 2.1 the branches $x=0$
and $y=0$ are separatrices of ${\mathcal{F}}_{W}$. According to the number of
separatrices that pass through the singular point of the foliation we can
classify the foliations in dicritical in the case that we have infinite
separatrices or nondicritical when we have a finite number of separatrices. As
a consequence of [Cam-LN-S, Theorem 1] we get

######  Theorem 2.2.

Let ${\mathcal{F}}_{W}$ be a nondicritical foliation with union of
separatrices $\\{f(x,y)=0\\}$. Then $\hbox{\rm mult$\,$}(W)\geq\hbox{\rm
ord}(f)-1$.

######  Example 2.3.

If $W_{1}=nx{\hbox{\rm d}}y-my{\hbox{\rm d}}x$ with $n$ and $m$ coprime
positive integers, then the branches $\\{x=0\\},\\{y=0\\}$ and
$\\{y^{n}-cx^{m}=0\\},$ $c\neq 0$ are separatrices of $W_{1}=0,$ so the
foliation ${\mathcal{F}}_{W_{1}}$ is dicritical. Nevertheless if
$W_{2}=x{\hbox{\rm d}}y+y{\hbox{\rm d}}x$ then the foliation
${\mathcal{F}}_{W_{2}}$ is nondicritical since its only separatrices are
$\\{x=0\\}$ and $\\{y=0\\}$.

Let $C_{i}:\\{f_{i}(x,y)=0\\}$, $1\leq i\leq r$, be the set of different
separatrices of a nondicritical foliation ${\mathcal{F}}_{W}:\\{W=0\\}$ and
put $f(x,y):=f_{1}(x,y)\cdots f_{r}(x,y)$. The reduced curve
$C:\\{f(x,y)=0\\}$ will be called the union of the separatrices of the
nondicritical foliation ${\mathcal{F}}_{W}$.

Other important notion we will use is the Newton polygon that we introduce in
the sequel. Let $S\subset\mathbb{N}^{2}$. We consider the convex hull
$\mathtt{conv}(S)$ of the Minkowski sum $S+\mathbb{R}^{2}_{\geq 0}$, where
$\mathbb{R}_{\geq 0}$ denote the non-negative real numbers. By definition, the
Newton polygon of $S$, denoted by $\mathcal{NP}(S)$, is the union of the
compact edges of the boundary of $\mathtt{conv}(S)$.

The support of a power series $f(x,y)=\sum_{i,j}a_{i,j}x^{i}y^{j}$ is
$\hbox{\rm supp$\,$}f:=\\{(i,j)\in\mathbb{N}^{2}\;:\;a_{i,j}\neq 0\\}$. The
support of a foliation ${\mathcal{F}}_{W}:\\{W=0\\}$ , where
$W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$, is the union of the supports
of $x\cdot A(x.y)$ and $y\cdot B(x,y)$, and we will denote it $\hbox{\rm
supp$\,$}W$, that is $\hbox{\rm supp$\,$}W=\hbox{\rm supp$\,$}(x\cdot
A(x,y))\cup\hbox{\rm supp$\,$}(y\cdot B(x,y))$. The Newton polygon of a power
series $f(x,y)\in\mathbb{C}\\{x,y\\}$ is by definition the Newton polygon of
$\hbox{\rm supp$\,$}f$ and we will denote it $\mathcal{NP}(f)$. The Newton
polygon of a foliation ${\mathcal{F}}_{W}$ is by definition the Newton polygon
of $\hbox{\rm supp$\,$}W$, and we will denote it $\mathcal{NP}(W)$. Observe
that the Newton polygon depends on coordinates. Remark that
$\mathcal{NP}(u\cdot f)=\mathcal{NP}(f)$ for any $u,f\in\mathbb{C}\\{x,y\\}$,
where $u$ is a unit. Hence, we can define the Newton polygon of the curve
$C:\\{f(x,y)=0\\}$ as the Newton polygon of any of its equations.

######  Proposition 2.4.

([R, Proposition 3.8]) Let $W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$ be
a 1-form. If ${\mathcal{F}}_{W}$ is a nondicritical generalized curve
foliation and $C:\\{f(x,y)=0\\}$ is its union of separatrices then the Newton
polygons of ${\mathcal{F}}_{W}$ and $C$ are the same.

Saravia, in her PhD thesis [SM] (see also [FS-GB-SM, Example 3.3]), shows that
the foliation ${\mathcal{F}}_{W}$, where $W=((b-1)xy-y^{3}){\hbox{\rm
d}}x+(xy-bx^{2}+xy^{2}){\hbox{\rm d}}y$, with $b\not\in\mathbb{Q}$, is not a
generalized curve foliation but its Newton polygon equals to the Newton
polygon of its union of separatrices $f(x,y)=xy(x-y)$.

Milnor number can be also defined in the foliation context. Let
$A(x,y),B(x,y)\in\mathbb{C}\\{x,y\\}$ with $A(0,0)=B(0,0)=0$. Let
$W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$ be a 1-form and consider the
singular foliation at the origin ${\mathcal{F}}_{W}$. The Milnor number at the
origin of the foliation ${\mathcal{F}}_{W}$ is
$\mu({\mathcal{F}}_{W}):=i_{0}(A(x,y),B(x,y))$.

Following [Cam-LN-S, Theorem 4] we have the next characterization of
nondicritical generalized curve foliations using Milnor numbers:

######  Theorem 2.5.

Let $W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$ be a 1-form. Suppose that
${\mathcal{F}}_{W}$ is a nondicritical generalized curve foliation and
$C:\\{f(x,y)=0\\}$ is its union of separatrices. Then
$\mu({\mathcal{F}}_{W})\geq\mu(f)$. The equality holds if and only if
${\mathcal{F}}_{W}$ is a generalized curve foliation.

Let $C:\\{f(x,y)=0\\}$ be a reduced plane curve. In what follows we consider:

$\bullet\;$ $[Fol(f)]$ the set of all 1-forms $W$, defining a nondicritical
foliation ${\mathcal{F}}_{W}$, such that $f$ divides $W\wedge{\hbox{\rm d}}f$
.

$\bullet\;$ $Fol(f)$ the set of nondicrital foliations defined by elements of
$[Fol(f)]$ which union of separatrices is $C:\\{f(x,y)=0\\}$.

In this paper we will present new characterizations for generalized curve
foliations.

##  3 Characterization of generalized curve foliations with monomial
separatrix

In this section we characterize the nondicritical generalized curve foliations
with monomial separatrix, by means of the weighted order defined as following

######  Definition 3.1.

Let $p,q\in\mathbb{Z}^{+}$. The weighted order $\upsilon_{p,q}$ for power
series and differential forms is:

$\upsilon_{p,q}\big{(}\sum_{i,j}a_{i,j}x^{i}y^{j})=\min\\{ip+jq\;:\;a_{i,j}\neq
0\\},$

and

$\upsilon_{p,q}\big{(}\sum_{i,j}A_{i,j}x^{i-1}y^{j}{\hbox{\rm
d}}x+\sum_{i,j}B_{i,j}x^{i}y^{j-1}{\hbox{\rm
d}}y)=\min\\{ip+jq\;:\;A_{i,j}\neq 0\,\,\textrm{or}\,\,B_{i,j}\neq 0\\}.$

Let $f\in\mathbb{C}\\{x,y\\}$ and $n,m$ positive integers not necessarily
coprime. For $f=y^{n}-x^{m}$, Frank Loray obtained a prenormal form of an
arbitrary element ${\mathcal{F}}_{W}\in{\mathcal{F}ol}(f)$.

######  Theorem 3.2.

([Lo, page 157]) If $f=y^{n}-x^{m}$ and ${\mathcal{F}}_{W}\in Fol(f)$ then

$W={\hbox{\rm d}}f+\sum_{{\begin{array}[]{c}0\leq i\leq m-2\\\ 0\leq j\leq
n-2\end{array}}}P_{i,j}(f)x^{i}y^{j}(nx{\hbox{\rm d}}y-my{\hbox{\rm d}}x),$
(4)

for some $P_{i,j}(z)\in\mathbb{C}\\{z\\}$.

For $f(x,y)=y^{n}-x^{m}$ with $0<n\leq m$ not necessary coprime, we will give
a characterization when the foliation ${\mathcal{F}}_{W}\in Fol(f)$ is
generalized curve.

The 1-form $W$ in $(\ref{eq:L})$ can be rewritten as

$W={\hbox{\rm d}}f+(\Delta(x,y)+f(x,y)g(x,y))(nx{\hbox{\rm d}}y-my{\hbox{\rm
d}}x),$ (5)

where $g\in\mathbb{C}\\{x,y\\}$ and
$\Delta(x,y)=\displaystyle\sum_{{\begin{array}[]{c}0\leq i\leq m-2\\\ 0\leq
j\leq n-2\end{array}}}a_{i,j}x^{i}y^{j}.$

Let $\gcd(n,m)=d$. Since $m\geq n$ there are integers $p,q\in\mathbb{Z}^{+}$
with $mp-nq=d$.

######  Lemma 3.3.

Let ${\mathcal{F}}_{W}\in Fol(f)$ be a foliation defined by $W$ as in (5). If
$\upsilon_{n,m}(\Delta(x,y))\geq mn-n-m$ then the toric morphism

$\Pi(u,v)=\left(u^{p}v^{\frac{n}{d}},u^{q}v^{\frac{m}{d}}\right)$

desingularizes ${\mathcal{F}}_{W}$ and the singular points of its strict
transform are $(0,0)$ and $(\xi,0)$, where $\xi^{d}=1$. Moreover the Jacobian
matrix at the point $(0,0)$ is

$\left(\begin{array}[]{cc}-\frac{mn}{d}&0\\\ 0&nq\end{array}\right),$ (6)

and the Jacobian matrix at the point $(\xi,0)$ is:

$\left(\begin{array}[]{cc}mn&0\\\
&-d(1+\displaystyle\sum_{r,s}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})\end{array}\right),$
(7)

where $r,s$ verify $m(r+1)+n(s+1)=nm$.

###### Proof.

Notice that $\Pi^{*}W=u^{qn-1}v^{\frac{mm}{d}-1}\cdot W^{\prime}$, where

$\displaystyle W^{\prime}$ $\displaystyle=$ $\displaystyle[(nq-mpu^{d})+(nq-
mp)S+h(u,v)(nqu^{p+q}v^{\frac{n+m}{d}}-pmu^{d+p+q}v^{\frac{n+m}{d}})]vdu$ (8)
$\displaystyle+$
$\displaystyle\left[\frac{mn}{d}(1-u^{d})u+\frac{mnh}{d}(u^{p+q+1}v^{\frac{m+n}{d}}-u^{p+q+d+2}v^{\frac{m+n}{d}})\right]dv,$

with $S=\sum a_{i,j}u^{p(i+1)+q(j+1)-qn}v^{\frac{m(j+1)+n(i+1)-mn}{d}}$ and
$h(u,v)\in\mathbb{C}\\{u,v\\}$. By hypothesis, if $(i,j)\in\hbox{\rm
supp$\,$}\Delta$ then $ni+jm\geq mn-m-n$, and we have $\textrm{ord}_{v}(S)\geq
0$. On the other hand we claim that $\hbox{\rm ord}_{u}(S)>0$, that is
$p(i+1)+q(j+1)>qn$. In effect, from $ni+jm\geq mn-n-m$ and $mp-nq=d$ we get
$p(i+1)+q(j+1)=\frac{pn(i+1)+pm(j+1)}{n}-\frac{d(j+1)}{n}\geq\frac{pmn-d(j+1)}{n}$.
Since $0\leq j\leq n-2$ we have $\frac{pmn-d(j+1)}{n}>\frac{pnm-dn}{n}$, so
$p(i+1)+q(j+1)>pm-d=nq$.

Therefore the singular points of ${\mathcal{F}}_{W^{\prime}}$ are $(0,0)$ and
$(\xi,0)$, with $\xi^{d}=1$. The point $(0,0)$ corresponds to the intersection
of the exceptional divisors, the other points correspond to the irreducible
components of $y^{n}-x^{m}$.

Rewrite (8) as $W^{\prime}=A(u,v){\hbox{\rm d}}u+B(u,v){\hbox{\rm d}}v$. We
get $B_{u}(0,0)=\frac{mn}{d}$, $B_{v}(0,0)=A_{u}(0,0)=0$ and $A_{v}(0,0)=nq$,
so the Jacobian matrix at $(0,0)$ is (6) and the origin is not a saddle-node.
On the other hand $B_{u}(\xi,0)=\frac{mn}{d}$, $B_{v}(\xi,0)=A_{u}(\xi,0)=0$
and
$A_{v}(\xi,0)=1+\displaystyle\sum_{m(r+1)+n(s+1)=mn}a_{r,s}\xi^{p(r+1)+q(s+1)-qn}$.
Hence the Jacobian matrix at $(\xi,0)$ is (7). ∎

Using Lemma 3.3 we get:

######  Theorem 3.4.

Let ${\mathcal{F}}_{W}\in Fol(f)$ be a foliation defined by $W$ as in (5).
Then ${\mathcal{F}}_{W}$ is a generalized curve foliation if and only if
$\upsilon_{n,m}(\Delta(x,y))\geq mn-n-m$, and
$(1+\sum_{m(r+1)+n(s+1)=mn}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})\notin\mathbb{Q}^{-}\cup\\{0\\}$,
where $m(r+1)+n(s+1)=nm$ and $\xi^{d}=1$.

###### Proof.

Suppose that the foliation ${\mathcal{F}}_{W}$ is generalized curve. Since its
union of separatrices is $\\{f(x,y)=0\\}$ and by Proposition 2.4, we get the
equality $\mathcal{NP}(W)=\mathcal{NP}(df)$ and we have $\upsilon_{n,m}(W)\geq
mn$. In addition, using the representation given in (5) we have
$\upsilon_{n,m}(y\sum a_{i,j}x^{i}y^{j}{\hbox{\rm d}}x)\geq mn$ and
$\upsilon_{n,m}(x\sum a_{i,j}x^{i}y^{j}{\hbox{\rm d}}y)\geq mn$; so
$\upsilon_{n,m}(\Delta(x,y))+n+m=n(i+1)+m(j+1)\geq mn$. Moreover the points
$(\xi,0)$ are not saddle-nodes, hence by (7) we get
$(1+\sum_{r,s}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})\notin\mathbb{Q}^{-}\cup\\{0\\}$.

On the other hand, the hypothesis $\upsilon_{n,m}(\Delta(x,y))\geq mn-n-m$
allows us to apply Lemma 3.3. Since
$(1+\sum_{r,s}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})\notin\mathbb{Q}^{-}\cup\\{0\\}$
we conclude that ${\mathcal{F}}_{W}\in Fol(f)$ is a generalized curve
foliation. ∎

######  Example 3.5.

Suppose that $W={\hbox{\rm d}}(y^{3}-x^{6})+axy(3x{\hbox{\rm
d}}y-6y){\hbox{\rm d}}x$, where $a\in\mathbb{C}$. In this case $p=q=1$, and
the toric morphism is $x=uv$, $y=uv^{2}$. The total transform of $W$ is

$\Pi^{*}W:\,\,\,u^{2}v^{5}[((3-6u^{3})-3au)v{\hbox{\rm
d}}u+6(1-u^{3})u{\hbox{\rm d}}v]=u^{2}v^{5}(A{\hbox{\rm d}}x+B{\hbox{\rm
d}}y),$

where $A=((3-6u^{3})-3au)v$ and $B=6(1-u^{3})u$. Then the singularities of the
strict transform of $W$ are $(0,0),(\xi,0)$ with $\xi^{3}=1$. Now
$6(r+1)+3(s+1)=18$ if and only if $(r,s)=(1,1)$. Hence, if
$(1+a\xi)\in\mathbb{Q}^{-}$, for some $\xi\,\textrm{with}\,\,\xi^{3}=1$, then
the foliation ${\mathcal{F}}_{W}$ is not generalized curve.

Mattei and Salem [Ma-Sal] consider a family of foliations, more general than
the generalized curved foliations, called foliations of the _second type_ ,
where saddle nodes are admitted in the reduction process, provided that they
lie in the regular part of the divisor, with their _weak separatrices_ (the
separatrices associated with the zero eingenvalue) transversal to the divisor.

In the next corollary we give the condition in order that a foliation of
second type becomes a generalized curve foliation following [FS-GB-SM, Theorem
1.2 (b)]:

######  Corollary 3.6.

Let ${\mathcal{F}}_{W}\in Fol(f)$ be a foliation defined by $W$ as in (5).
Suppose that ${\mathcal{F}}_{W}\in Fol(f)$ is of the second type. Then
${\mathcal{F}}_{W}\in Fol(f)$ is a generalized curve foliation if and only if

$(1+\sum_{m(r+1)+n(s+1)=mn}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})\notin\mathbb{Q}^{-}\cup\\{0\\}.$

###### Proof.

It is a consequence of [FS-GB-SM, Theorem 1.2] and Theorem 3.4. ∎

######  Remark 3.7.

From Theorem 3.4 we can deduce Proposition 5.7 of [FS-GB-SM]: let $n,m$ be two
positive integers which are not coprime. Consider $f(x,y)=y^{n}-x^{m}$ and
${\mathcal{F}}_{W}\in Fol(f)$, where $W={\hbox{\rm
d}}f+\Delta^{\prime}(x,y)(nx{\hbox{\rm d}}y-my{\hbox{\rm d}}x)$ with
$\Delta^{\prime}(x,y)\in\mathbb{C}\\{x,y\\}$. If we suppose that
$i_{0}(\Delta^{\prime},f)>mn-m-n$ then from the proof of Lemma 3.3 we get that
$\textrm{ord}_{v}(S)>0$. Hence
$-d(1+\displaystyle\sum_{r,s}a_{r,s}\xi^{p(r+1)+q(s+1)-qn})=-d$. In
particular, the foliation ${\mathcal{F}}_{W}$ is generalized curve.

In Example 3.5 we have a family of nondicritical generalized curve foliations
${\mathcal{F}}_{W}$ with $i_{0}(\Delta^{\prime},f)=mn-m-n$ when
$(1+a\xi)\not\in\mathbb{Q}^{-}$.

##  4 Weierstrass form of a 1-form

In this section we introduce, using the Weierstrass division of power series,
a distinguished equation for a given 1-form, with respect to any Weierstrass
polynomial, called Weierstrass form. The Weierstrass form is very useful for
computations and it can be considered as a generalization, to any 1-form of
the prenormal form given by Loray for foliations with monomial separatrix. The
Weierstrass form is well-defined for any $1$-form. Nevertheless, in this paper
we are using the Weierstrass forms associated with $1$-form $W$ defining a
nondicritical foliation.

Let $f=\sum_{i=0}^{n}a_{i}(x)y^{n-i}\in\mathbb{C}\\{x\\}[y]$ be a reducible
polynomial (not necessarily irreducible) of degree $\deg_{y}(f)=n$,
$a_{0}(0)\neq 0$ and $W\in\Omega^{1}_{\mathbb{C}^{2},0}$ be a 1-form. Here we
do not suppose that $C:\\{f(x,y)=0\\}$ is the union of separatrices of the
foliation $\mathcal{F}_{W}$. In that follows we will obtain an equation of $W$
in function of $f$ and ${\hbox{\rm d}}f$.

######  Lemma 4.1.

If $W\in\Omega^{1}_{\mathbb{C}^{2},0}$ and
$f(x,y)=\sum_{i=0}^{n}a_{i}(x)y^{n-i}\in\mathbb{C}\\{x\\}[y]$ with
$\deg_{y}(f)=n>1$ and $a_{0}(0)\neq 0$, then there exist unique
$h,p\in\mathbb{C}\\{x,y\\}$ and $A,B\in\mathbb{C}\\{x\\}[y]$ with $B=0$ or
$\deg_{y}(B)<n-1$ and $A=0$ or $\deg_{y}(A)<n$ such that

$W=h{\hbox{\rm d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y.$ (9)

###### Proof.

Put $W=R(x,y){\hbox{\rm d}}x+S(x,y){\hbox{\rm
d}}y\in\Omega^{1}_{\mathbb{C}^{2},0}$. After the Weierstrass division of $S$
by $f_{y}$, there exist $h\in\mathbb{C}\\{x,y\\}$ and
$B\in\mathbb{C}\\{x\\}[y]$ such that $S=hf_{y}+B$, where $B=0$ or
$\deg_{y}B<\deg_{y}f_{y}=\deg_{y}f-1=n-1$ and $W=R{\hbox{\rm
d}}x+hf_{y}{\hbox{\rm d}}y+B{\hbox{\rm d}}y.$ Since ${\hbox{\rm
d}}f=f_{x}{\hbox{\rm d}}x+f_{y}{\hbox{\rm d}}y$ then $W=(R-hf_{x}){\hbox{\rm
d}}x+h{\hbox{\rm d}}f+B{\hbox{\rm d}}y.$ Now, by the Weierstrass division of
$R-hf_{x}$ by $f$ there exist $p\in\mathbb{C}\\{x,y\\}$ and
$A\in\mathbb{C}\\{x\\}[y]$ such that $R-hf_{x}=pf+A$ with $A=0$ or
$\deg_{y}A<\deg_{y}f=n$ and

$W=h{\hbox{\rm d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y.$

Suppose that $h_{1}{\hbox{\rm d}}f+p_{1}f{\hbox{\rm d}}x+A_{1}{\hbox{\rm
d}}x+B_{1}{\hbox{\rm d}}y=h_{2}{\hbox{\rm d}}f+p_{2}f{\hbox{\rm
d}}x+A_{2}{\hbox{\rm d}}x+B_{2}{\hbox{\rm d}}y$ with
$h_{i},p_{i}\in\mathbb{C}\\{x,y\\}$ and $A_{i},B_{i}\in\mathbb{C}\\{x\\}[y]$
with $\deg_{y}(B_{i})<n-1$ and $\deg_{y}(A_{i})<n$ for $i=1,2$. Then
$(h_{1}-h_{2})f_{y}=B_{2}-B_{1}$ and
$(h_{1}-h_{2})f_{x}+(p_{1}-p_{2})f=A_{2}-A_{1}$.

As $\deg_{y}(B_{i})<n-1=\deg_{y}(f_{y})$, we get $h_{1}=h_{2}$ and
$B_{1}=B_{2}$. Similarly, $\deg_{y}(A_{i})<n=\deg_{y}(f)$ implies that
$p_{1}=p_{2}$ and $A_{1}=A_{2}$. ∎

######  Definition 4.2.

Let $W\in\Omega^{1}_{\mathbb{C}^{2},0}$ and
$f=\sum_{i=0}^{n}a_{i}(x)y^{n-i}\in\mathbb{C}\\{x\\}[y]$ with
$\deg_{y}(f)=n>1$ and $a_{0}(0)\neq 0$. The Weierstrass form of $W$ with
respect to $f$ is $h{\hbox{\rm d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm
d}}x+B{\hbox{\rm d}}y$, where $h,p\in\mathbb{C}\\{x,y\\}$ and
$A,B\in\mathbb{C}\\{x\\}[y]$ are unique and satisfying $B=0$ or
$\deg_{y}(B)<n-1$ and $A=0$ or $\deg_{y}(A)<n$.

We can rewrite (9) as

$W=h{\hbox{\rm d}}f+pf{\hbox{\rm d}}x+\omega,$ (10)

where $\omega=\sum A_{i,j}x^{i-1}y^{j}{\hbox{\rm d}}x+\sum
B_{i,j}x^{i}y^{j-1}{\hbox{\rm d}}y=A{\hbox{\rm d}}x+B{\hbox{\rm d}}y$ for some
$A_{i,j},B_{i,j}\in\mathbb{C}$. Since $B=0$ or $\deg_{y}B<n-1$ then
$B_{0,n}=0$.

Observe that, in (10), $h$, $p$ and $w$ depend on $f$. Moreover if
$W=R(x,y){\hbox{\rm d}}x+S(x,y){\hbox{\rm
d}}y\in\Omega^{1}_{\mathbb{C}^{2},0}$, where
$R(x,y),S(x,y)\in\mathbb{C}\\{x\\}[y]$ and $f\in\mathbb{C}\\{x\\}[y]$ with
$\deg_{y}(f)>\max\\{\deg_{y}(R),\deg_{y}(S)+1\\}$ then, in (10), $h=p=0$ and
the Weierstrass form of $W$ is $w$.

######  Proposition 4.3.

Let ${W}\in[Fol(f)]$ as (10), where $h$ is a unit of $\mathbb{C}\\{x,y\\}$ and
$f(x,y)=\sum_{i=0}^{n}a_{i}(x)y^{n-i}\in\mathbb{C}\\{x\\}[y]$ with
$n=\deg_{y}(f)=\hbox{\rm ord}f$ and $a_{0}(0)\neq 0$. Then
${\mathcal{F}}_{W}\in Fol(f)$, that is $C$ is the union of separatrices of
${\mathcal{F}}_{W}$.

###### Proof.

Given that $\deg_{y}(f)=\hbox{\rm ord}f$, the multiplicity of $W$ equals the
multiplicity of $h{\hbox{\rm d}}f+\omega$. Since $h$ is a unit we get
$\hbox{\rm mult$\,$}(W)=\min\\{\hbox{\rm ord}(f_{x}+A),\hbox{\rm
ord}(f_{y}+B)\\}\leq\hbox{\rm ord}(f_{y}+B)$. But $B_{0,n}=0$ and $\hbox{\rm
ord}f=\deg_{y}(f)$, so $\hbox{\rm ord}(f_{y}+B)\leq n-1$. Hence

$\hbox{\rm mult$\,$}(W)\leq n-1.$ (11)

Suppose that ${\mathcal{F}}_{W}$ has other separatrix $g(x,y)=0$. By Theorem
2.2 we have $\hbox{\rm mult$\,$}(W)\geq\hbox{\rm ord}(fg)-1>n-1,$ which is a
contradiction after the inequality (11).

∎

In Section 5 we will see how the Weierstrass forms allow us to give new
characterizations of generalized curve foliations. We can also characterize
the second type foliations using the Weierstrass forms. To do this, we first
remember the characterization given by Mattei and Salem:

######  Theorem 4.4.

([Ma-Sal, Théorème 3.1.9]) Let $W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$
be a 1-form. Suppose that ${\mathcal{F}}_{W}$ is a non dicritical foliation
and $C:\\{f(x,y)=0\\}$ is its union of separatrices. Then ${\mathcal{F}}_{W}$
is a second type foliation if and only if $\hbox{\rm mult$\,$}(W)=\hbox{\rm
mult$\,$}({\hbox{\rm d}}f)$.

Let $f(x,y)=\sum_{i=0}^{n}a_{i}(x)y^{n-i}\in\mathbb{C}\\{x\\}[y]$ with
$n=\deg_{y}(f)=\hbox{\rm ord}f$ and $a_{0}(0)\neq 0$. Consider a $1$-form
$W\in\Omega^{1}_{\mathbb{C}^{2},0}$ in the Weierstrass form as (10), where
$\omega=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$. Let us relate some
algebraic aspects of the $1$-forms $W$, $\omega$ and the parameters $h$ and
$p$ in the Weierstrass form.

Notice that $\frac{W\wedge{\hbox{\rm d}}f}{{\hbox{\rm d}}x\wedge{\hbox{\rm
d}}y}\in(f)$ if and only if $\frac{\omega\wedge{\hbox{\rm d}}f}{{\hbox{\rm
d}}x\wedge{\hbox{\rm d}}y}\in(f)$. More specifically, if $g$ is the cofactor
of $W$, that is, $W\wedge{\hbox{\rm d}}f=gf{\hbox{\rm d}}x\wedge{\hbox{\rm
d}}y$, then the cofactor of $\omega$ is $g-pf_{y}$.

In addition, remark that:

  * •

If $h(0,0)=0$ then $\hbox{\rm mult$\,$}(W)=\hbox{\rm mult$\,$}({\hbox{\rm
d}}f)$ if and only if $\hbox{\rm mult$\,$}(\omega)=\hbox{\rm
mult$\,$}({\hbox{\rm d}}f).$

  * •

If $h(0,0)\neq 0$ then $\hbox{\rm mult$\,$}(W)=\hbox{\rm mult$\,$}({\hbox{\rm
d}}f)$ if and only if $\hbox{\rm mult$\,$}(\omega)\geq\hbox{\rm
mult$\,$}({\hbox{\rm d}}f).$

As, by Theorem 4.4, the equality $\hbox{\rm mult$\,$}(W)=\hbox{\rm
mult$\,$}({\hbox{\rm d}}f)$ characterizes the $1$-forms that define second
type foliations, we can read this property using $\omega$.

In (10) we remarked that for any Weierstrass form $W=h{\hbox{\rm
d}}f+pf{\hbox{\rm d}}x+\omega,$ where $\omega=\sum
A_{i,j}x^{i-1}y^{j}{\hbox{\rm d}}x+\sum B_{i,j}x^{i}y^{j-1}{\hbox{\rm
d}}y=A{\hbox{\rm d}}x+B{\hbox{\rm d}}y$ for some
$A_{i,j},B_{i,j}\in\mathbb{C}$, we get $B_{0,n}=0$.

If $f\in K(n,m)$ and $W\in[Fol(f)]$ we can obtain additional information about
the coeficient $A$.

We can write, without lost of generality

$f(x,y)=y^{n}-x^{m}+\sum_{in+jm>nm}a_{i,j}x^{i}y^{j}.$ (12)

######  Lemma 4.5.

Let $f\in K(n,m)$. If $W\in[Fol(f)]$, with $W$ as in (10), then $A_{m,0}=0$.

###### Proof.

Consider $f(x,y)$ as in (12). Since $C:\\{f(x,y)=0\\}$ is the separatrix of
${\mathcal{F}}_{W}$ and $\omega=W-h{\hbox{\rm d}}f-pf{\hbox{\rm d}}x$ then
$f(x,y)=0$ is also a separatrix of $w$. Consider a parametrization of
$f(x,y)=0$ given by $(x(t),y(t))=(t^{n},\,t^{m}+\cdots)$, where $\cdots$ means
terms of greater degree. Therefore $\sum A_{i,j}x(t)^{i-1}y(t)^{j}{\hbox{\rm
d}}x(t)+\sum B_{i,j}x(t)^{i}y(t)^{j-1}{\hbox{\rm d}}y(t)=0$. Suppose that
$A_{m,0}\neq 0$. Since $A_{m,0}x(t)^{m-1}{\hbox{\rm d}}x(t)=nA_{m,0}t^{nm-1}$
then there is $(i_{0},j_{0})\in\hbox{\rm supp$\,$}{\mathcal{A}}\cup\hbox{\rm
supp$\,$}{\mathcal{B}}$ such that $ni_{0}+mj_{0}=nm$, where
${\mathcal{A}}=\sum A_{i,j}x(t)^{i-1}y(t)^{j}{\hbox{\rm
d}}x(t)-nA_{m,0}t^{nm-1}$ and ${\mathcal{B}}=\sum
B_{i,j}x(t)^{i}y(t)^{j-1}{\hbox{\rm d}}y(t)$. But $n$ and $m$ are coprime, so
$(i_{0},j_{0})\in\\{(m,0),(0,n)\\}$, which is a contradiction since these
points are not in $\hbox{\rm supp$\,$}{\mathcal{A}}\cup\hbox{\rm
supp$\,$}{\mathcal{B}}.$ ∎

##  5 Characterization of generalized curve foliations: general case

In this section we present our main result. For that we need the notion of
_GSV-index_.

Let $f(x,y)\in\mathbb{C}\\{x,y\\}$ and $W\in[Fol(f)]$. By [Li, page 198] in
the irreducible case and [Su, (1.1) Lemma] in the reduced case, there are
$g,k\in\mathbb{C}\\{x,y\\}$ and a 1-form $\eta$ such that $gW=k{\hbox{\rm
d}}f+f\eta$, with $f$ and $k$ coprime.

######  Definition 5.1.

With the above notations, the GSV-index of $W$ with respect to
$C:\\{f(x,y)=0\\}$ is

$GSV(W,C):=\frac{1}{2\pi i}\int_{\partial C}\frac{g}{k}{\hbox{\rm
d}}\left(\frac{k}{g}\right).$

Suppose that $f(x,y)$ is irreducible and consider a parametrization
$(x(t),y(t))$ of $C$. Write $W=A(x,y){\hbox{\rm d}}x+B(x,y){\hbox{\rm d}}y$
and $\eta=p(x,y){\hbox{\rm d}}x+q(x,y){\hbox{\rm d}}y$. We have

$W=\left(\frac{k}{g}f_{x}+\frac{f}{g}p\right){\hbox{\rm
d}}x+\left(\frac{k}{g}f_{y}+\frac{f}{g}q\right){\hbox{\rm d}}y.$

On the other hand,

$\frac{k}{g}(x(t),y(t))=\frac{\left(\frac{k}{g}f_{y}+\frac{f}{g}q\right)}{f_{y}}(x(t),y(t)),$

so the number of zeroes of $k$ restricted to $C$ minus the number of zeroes of
$g$ restricted to $C$ equals $\hbox{\rm ord}_{t}\frac{B}{f_{y}}(x(t),y(t))$.
Hence by Rouché-Hurwitz theorem we have

$GSV(W,C)=\hbox{\rm ord}_{t}\frac{B}{f_{y}}(x(t),y(t)).$ (13)

A similar calculation as before shows that

$GSV(W,C)=\hbox{\rm ord}_{t}\frac{A}{f_{x}}(x(t),y(t)).$

Now, consider $C:\\{f(x,y)=0\\}$ for $f=f_{1}\cdot f_{2}$ with $f_{1}$ and
$f_{2}$ irreducible. After [B, page 532] we have

$GSV(W,C)=GSV(W,C_{1})+GSV(W,C_{2})-2i_{0}(f_{1},f_{2}),$ (14)

where $C_{i}:\\{f_{i}(x,y)=0\\}.$ The equality (14) is also true when
$f_{1},f_{2}$ are reduced, not necessary irreducible.

Cavalier and Lehmann gave a characterization of generalized curve foliations
using the $GSV$-index:

######  Theorem 5.2.

([Cav-Le, Théorème 3.3]) Let $C:\\{f(x,y)=0\\}$ be a reduced curve and
${\mathcal{F}}_{W}\in Fol(f)$ a nondicritical foliation. Then
${\mathcal{F}}_{W}$ is generalized curve if and only if $GSV(W,C)=0$.

In the remainder of the section we will present our main result and
consequences of it.

Let $f\in\mathbb{C}\\{x\\}[y]$, where $f=f_{1}\cdots f_{r}$ is the
factorization of $f$ into irreducible factors. Consider $C:\\{f(x,y)=0\\}$.

######  Theorem 5.3.

Let ${\mathcal{F}}_{W}\in Fol(f)$ and $h{\hbox{\rm d}}f+pf{\hbox{\rm
d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y=\mathcal{A}{\hbox{\rm
d}}x+\mathcal{B}{\hbox{\rm d}}y$ the Weierstrass form of $W$ with respect to
$f$. Then ${\mathcal{F}}_{W}$ is a generalized curve foliation if and only if
$h\in\mathbb{C}\\{x,y\\}$ is a unit and
$i_{0}(\mathcal{B},f)=\mu(f)+i_{0}(f,x)-1.$

###### Proof.

First, observe that $\mathcal{B}=hf_{y}+B$. Hence,

$i_{0}(\mathcal{B},f)=i_{0}(hf_{y}+B,f)=\sum_{j=1}^{r}i_{0}\Big{(}h\prod_{i\neq
j}f_{i}(f_{j})_{y}+B,f_{j}\Big{)}.$

Suppose that $\sum_{j=1}^{r}i_{0}\big{(}h\prod_{i\neq
j}f_{i}(f_{j})_{y}+B,f_{j}\big{)}=\mu(f)+i_{0}(f,x)-1$ and $h(x,y)$ is a unit.
If $C_{i}:\\{f_{i}(x,y)=0\\}$ and $C:\\{f_{1}\cdots f_{r}=0\\}$ then, by (14),
the $GSV$\- index of $W$ with respect to $C$ is

$GSV(W,C)=\sum_{i=1}^{r}GSV(W,C_{i})-2\sum_{1\leq i<j\leq
r}i_{0}(f_{i},f_{j}).$

By hypothesis,

$W=\Big{(}h\sum_{i=1}^{r}\prod_{j\neq i}f_{j}(f_{i})_{x}+fp+A\Big{)}{\hbox{\rm
d}}x+\Big{(}h\sum_{i=1}^{r}\prod_{j\neq i}f_{j}(f_{i})_{y}+B\Big{)}{\hbox{\rm
d}}y=\mathcal{A}{\hbox{\rm d}}x+\mathcal{B}{\hbox{\rm d}}y,$

hence $\mathcal{A}=h\sum_{i=1}^{r}\prod_{j\neq i}f_{j}(f_{i})_{x}+fp+A$ and
$\mathcal{B}=h\sum_{i=1}^{r}\prod_{j\neq i}f_{j}(f_{i})_{y}+B$. Then by (13)
the $GSV$-index of $W$ with respect to the separatrix
$C_{j}:\\{f_{j}(x,y)=0\\}$ is

$GSV(W,C_{j})=\hbox{\rm
ord}_{t}\left(\frac{{\mathcal{B}}(x_{j}(t),y_{j}(t))}{(f_{j})_{y}(x_{j}(t),y_{j}(t))}\right)=i_{0}({\mathcal{B}},f_{j})-i_{0}((f_{j})_{y},f_{j}),$

where $(x_{j}(t),y_{j}(t))$ is a parametrization of $C_{j}$. By definition of
${\mathcal{B}}$ and using (2) we have

$GSV(W,C_{j})=i_{0}\Big{(}h\prod_{i\neq
j}f_{i}(f_{j})_{y}+B,f_{j}\Big{)}-\Big{(}\mu(f_{j})+i_{0}(f_{j},x)-1\Big{)}.$

Hence

$GSV(W,C)=\sum_{j=1}^{r}i_{0}\Big{(}h\prod_{i\neq
j}f_{i}(f_{j})_{y}+B,f_{j}\Big{)}-\sum_{j=1}^{r}\Big{(}\mu(f_{j})+i_{0}(f_{j},x)-1\Big{)}-2\sum_{1\leq
i<j\leq r}i_{0}(f_{i},f_{j}).$

Now, by the formula of Milnor for a reduced curve (see for example [W, Theorem
6.5.1]) we obtain

$GSV(W,C)=\sum_{j=1}^{r}i_{0}\Big{(}h\prod_{i\neq
j}f_{i}(f_{j})_{y}+B,f_{j}\Big{)}-\Big{(}\mu(f)+i_{0}(f,x)-1\Big{)}.$ (15)

So, by Theorem 5.2 we conclude that ${\mathcal{F}}_{W}$ is a generalized curve
foliation.

Suppose now that ${\mathcal{F}}_{W}$ is a generalized curve foliation which
union of separatrices is $C:\\{f(x,y)=0\\}$. Hence
${\mathcal{N}P}(W)={\mathcal{N}P}(f)$. Moreover, since $h{\hbox{\rm
d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y$ is the Weierstrass
form of $W$ with respect to $f$ we get $B_{0,n}=0$ and we conclude that $h$ is
a unit. In particular, we have again the equality (15).

Finally, since ${\mathcal{F}}_{W}$ is a generalized curve foliation, again by
Theorem 5.2, we get $0=GSV(W,C)$. This finishes the proof. ∎

Theorem 5.3 gives a characterization of a generalized curve foliation
${\mathcal{F}}_{W}$, where $W=\mathcal{A}{\hbox{\rm d}}x+\mathcal{B}{\hbox{\rm
d}}y$, using the polar $\mathcal{B}$ of the foliation ${\mathcal{F}}_{W}$ and
the polar $f_{y}$ of the separatrix $C:\\{f(x,y)=0\\}$.

######  Corollary 5.4.

Let ${\mathcal{F}}_{W}\in Fol(f)$ and $h{\hbox{\rm d}}f+pf{\hbox{\rm
d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y=\mathcal{A}{\hbox{\rm
d}}x+\mathcal{B}{\hbox{\rm d}}y$ the Weierstrass form of $W$ with respect to
$f$. Then ${\mathcal{F}}_{W}$ is a generalized curve foliation if and only if
$h\in\mathbb{C}\\{x,y\\}$ is a unit and $i_{0}(\mathcal{B},f)=i_{0}(f_{y},f).$
In particular $i_{0}(B,f)\geq i_{0}(f_{y},f).$

Suppose now that $f\in\mathbb{C}\\{x\\}[y]$ is an irreducible monic polynomial
of degree $n=\hbox{\rm ord}f$ with semigroup $\Gamma(f)=\langle
v_{0},v_{1},v_{2},\ldots,v_{g}\rangle$ ($v_{0}=n$). Remember that we denote
$e_{i}=\gcd(v_{0},\ldots,v_{i})$ for $i\in\\{0,\ldots,g\\}$ and
$n_{i}=\frac{e_{i-1}}{e_{i}}$ for $i\in\\{1,\ldots,g\\}$. By convention we put
$n_{0}=1$. We say that $f_{k}\in\mathbb{C}\\{x\\}[y]$ is a $k$-semiroot of the
polynomial $f$ if $f_{k}$ is monic,
$\deg_{y}(f_{k})=\frac{v_{0}}{e_{k-1}}=n_{0}n_{1}\cdots n_{k-1}$ and
$i_{0}(f_{k},f)=v_{k}$. The notion of semi-root is a generalization of the
characteristic approximate roots introduced and studied by Abhyankar and Moh
in [A-M].

Applying [Po, Corollary 5.4] to the polynomial $B\in\mathbb{C}\\{x\\}[y]$ in
(9), it can be uniquely written as a finite sum of the form

$B=\sum_{\hbox{\rm finite}}a_{\alpha}(x)f_{1}^{\alpha_{1}}\cdots
f_{g}^{\alpha_{g}},$ (16)

where $a_{\alpha}(x)\in\mathbb{C}\\{x\\}$, $f_{k}$ are $k$-semiroots of $f$
and $0\leq\alpha_{k}<n_{k}$. Since $\deg_{y}B<\deg_{y}f-1$ the polynomial $f$
does not appear as a factor in the terms of the right-hand side of (16).

As a $k$-semiroot $f_{k}$ is irreducible and admits semigroup
$\langle\frac{v_{0}}{e_{k-1}}=\frac{n}{e_{k-1}},\frac{v_{1}}{e_{k-1}}=\frac{m}{e_{k-1}},\ldots,\frac{v_{k-1}}{e_{k-1}}\rangle$
it follows that its Newton polygon has a single compact face with vertices
$\left(0,\frac{m}{e_{k-1}}\right)$ and $\left(\frac{n}{e_{k-1}},0\right)$ and
consequently $v_{n,m}(f_{k})=\frac{nm}{e_{k-1}}$.

In this way, using the relations (1) we get

$i_{0}(f,f_{k})=v_{k}=\frac{nm}{e_{k-1}}+\sum_{i=2}^{k}\frac{n_{i}\cdots
n_{k}}{n_{k}}(\beta_{i}-\beta_{i-1})=v_{n,m}(f_{k})+\sum_{i=2}^{k}\frac{e_{i-1}}{e_{k-1}}(\beta_{i}-\beta_{i-1}).$
(17)

In particular, $i_{0}(f,f_{k})\geq v_{n,m}(f_{k})$ with equality if and only
if $k=1$.

######  Remark 5.5.

For any two distinct terms $T_{\alpha}:=a_{\alpha}(x)f_{1}^{\alpha_{1}}\cdots
f_{g}^{\alpha_{g}}$ and
$T_{\alpha^{\prime}}:=a_{\alpha^{\prime}}(x)f_{1}^{\alpha^{\prime}_{1}}\cdots
f_{g}^{\alpha^{\prime}_{g}}$ of (16) we get $i_{0}(T_{\alpha},f)\neq
i_{0}(T_{\alpha^{\prime}},f)$.

In addition, remark that we can not have $\alpha_{i}=n_{i}-1$ for all
$i=1,\ldots,g$. Indeed, if this is the case we get

$\displaystyle\deg_{y}(B)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{g}(n_{i}-1)\deg_{y}(f_{i})=\sum_{i=1}^{g}(n_{i}-1)n_{0}n_{1}\cdots
n_{i-1}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{g}n_{0}n_{1}\cdots
n_{i-1}n_{i}-\sum_{i=1}^{g}n_{0}n_{1}\cdots n_{i-1}=n_{0}n_{1}\cdots
n_{g}-n_{0}=v_{0}-1=n-1,$

which is a contradiction.

As a consequence of $\eqref{*}$ and Remark 5.5 we have that $i_{0}(f,B)\geq
v_{n,m}(f)$ with equality if and only if $f\in K(n,m)$.

######  Lemma 5.6.

With the above notations, if $f$ is irreducible then $i_{0}(B,f)\neq
i_{0}(f_{y},f)$.

###### Proof.

Suppose that $i_{0}(f_{y},f)=i_{0}(B,f)$. After Remark 5.5, there exists a
unique $g$-tuple $(\alpha_{1},\ldots,\alpha_{g})$ with $0\leq\alpha_{i}<n_{i}$
such that

$i_{0}(B,f)=i_{0}(a_{\alpha}(x)f_{1}^{\alpha_{1}}\cdot\ldots\cdot
f_{g}^{\alpha_{g}},f)=\sum_{i=1}^{g}\alpha_{i}v_{i}+\lambda_{0}v_{0},$

where $\lambda_{0}=\hbox{\rm ord}_{x}a_{\alpha}(x)$. Now, by (2) we get
$i_{0}(f_{y},f)=\mu(f)+v_{0}-1=\sum_{i=1}^{g}(n_{i}-1)v_{i}$ (see for example
[Hef, Proposition 7.5 (ii), page 102] for the last equality). In this way, we
have

$\sum_{i=1}^{g}(n_{i}-1)v_{i}=i_{0}(f_{y},f)=i_{0}(B,f)=\sum_{i=1}^{g}\alpha_{i}v_{i}+\lambda_{0}v_{0},$

that is $\sum_{i=1}^{g}(n_{i}-1-\alpha_{i})v_{i}-\lambda_{0}v_{0}=0$. But,
this implies that $\lambda_{0}=0$ and $\alpha_{i}=n_{i}-1$ for all
$i=1,\ldots,g$, which is a contradiction. Hence, $i_{0}(f_{y},f)\neq
i_{0}(B,f)$. ∎

######  Remark 5.7.

Observe that Lemma 5.6 is not true for $f$ reduced (non-irreducible): consider
$f(x,y)=y^{2}-x^{2}$ and $B=x^{e}$. We have $\deg_{y}(B)=0<1=\deg_{y}(f)-1$,
$i_{0}(f_{y},f)=2$ and $i_{0}(B,f)=2e$. So, for $e=1$ we get
$i_{0}(f_{y},f)=i_{0}(B,f)$ and $i_{0}(f_{y},f)\neq i_{0}(B,f)$ for $e\neq 1$.

######  Corollary 5.8.

With the above notations, for $f$ irreducible we have,
$i_{0}(B,f)>i_{0}(f_{y},f)$ if and only if
$i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$.

###### Proof.

Suppose that $i_{0}(B,f)>i_{0}(f_{y},f)$. Then

$i_{0}(f_{y}+B,f)=\min\\{i_{0}(f_{y},f),i_{0}(B,f)\\}=i_{0}(f_{y},f),$

and by Teissier’s Lemma (see (2)) we have
$i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$.

Now we suppose that $i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$, that is, by
Teissier’s Lemma, $i_{0}(f_{y}+B,f)=i_{0}(f,f_{y})$ and by Lemma 5.6 we
conclude $i_{0}(B,f)>i_{0}(f_{y},f)$. ∎

######  Remark 5.9.

If $W$ is written as (9) and $W\in[Fol(f)]$ then $A(x,y){\hbox{\rm
d}}x+B(x,y){\hbox{\rm d}}y\in[Fol(f)]$. We must have
$i_{0}(A,f)+v_{0}=i_{0}(B,f)+v_{1}$. Hence by (2) and (3) we get
$i_{0}(A,f)-i_{0}(f_{x},f)=i_{0}(B,f)-i_{0}(f_{y},f)$. So,
$i_{0}(B,f)>i_{0}(f_{y},f)$ if and only if, $i_{0}(A,f)>i_{0}(f_{x},f)$.
Moreover, by Lemma 5.6, we have $i_{0}(A,f)\neq i_{0}(f_{x},f)$. So,
$i_{0}(f_{x}+A,f)=\min\\{i_{0}(f_{x},f),i_{0}(A,f)\\}$. Consequently
$i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$ if and only if
$i_{0}(f_{x}+A,f)=\mu(f)+i_{0}(f,y)-1$.

Next proposition gives us a characterization of generalized curve foliations
with a single separatrix.

######  Proposition 5.10.

Let $f(x,y)\in\mathbb{C}\\{x\\}[y]$ be irreducible, ${\mathcal{F}}_{W}\in
Fol(f)$, where $h{\hbox{\rm d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm
d}}x+B{\hbox{\rm d}}y$ is the Weierstrass form of $W$ with respect to $f$.
Then ${\mathcal{F}}_{W}$ is a generalized curve foliation if and only if
$h\in\mathbb{C}\\{x,y\\}$ is a unit and $i_{0}(B,f)>i_{0}(f_{y},f)$.

###### Proof.

It is a consequence of Corollaries 5.4 and 5.8. ∎

Next corollary gives us a characterization of generalized curve foliations
with a single separatrix of genus $1$ in terms of the weighted order (see
Definition 3.1).

######  Corollary 5.11.

Let $f\in K(n,m)$, ${\mathcal{F}}_{W}\in Fol(f)$, where $h{\hbox{\rm
d}}f+pf{\hbox{\rm d}}x+A{\hbox{\rm d}}x+B{\hbox{\rm d}}y$ is the Weierstrass
form of $W$ with respect to $f$. Then ${\mathcal{F}}_{W}$ is a generalized
curve foliation if and only if $\upsilon_{n,m}(\omega)>nm$, where
$\omega=A{\hbox{\rm d}}x+B{\hbox{\rm d}}y$.

###### Proof.

By Remark 5.9 the equality $i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$ is
equivalent to $i_{0}(f_{x}+A,f)=\mu(f)+i_{0}(f,y)-1$. It follows, by Corollary
5.8, that this is equivalent to claim $i_{0}(f_{y},f)<i_{0}(B,f)$ and
$i_{0}(f_{x},f)<i_{0}(A,f)$. As $f\in K(n,m)$, we have $\mu(f)=nm-n-m+1$,
$i_{0}(f_{y},f)=nm-m$ and $i_{0}(f_{x},f)=nm-n$ (see (2) and (3)). So,
$nm<i_{0}(B,f)+m$ and $nm<i_{0}(A,f)+n$. Since $i_{0}(H,f)=v_{n,m}(H)$, for
any $H\in\mathbb{C}\\{x,y\\}$ then $i_{0}(f_{y}+B,f)=\mu(f)+i_{0}(f,x)-1$ is
equivalent to $nm<\min\\{v_{n,m}(B)+m,v_{n,m}(A)+n\\}=v_{n,m}(\omega)$.We
finish the proof using Proposition 5.10 and Corollary 5.8. ∎

######  Example 5.12.

Let $W_{c}={\hbox{\rm d}}(y^{3}-x^{4})+cx^{2}y(3x{\hbox{\rm d}}y-4y{\hbox{\rm
d}}x)$, $c\in\mathbb{C}^{*}.$ By Proposition 4.3 we get
${\mathcal{F}}_{W_{c}}\in Fol(y^{3}-x^{4})$. In this case
$B(x,y)=-4cx^{2}y^{2}$ and $i_{0}(B,f)=14>\mu(f)+i_{0}(f,x)-1=9$, then
${\mathcal{F}}_{W_{c}}$ is a generalized curve foliation.

## References

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Evelia Rosa García Barroso
Departamento de Matemáticas, Estadística e I.O.
Sección de Matemáticas, Universidad de La Laguna
Apartado de Correos 456
38200 La Laguna, Tenerife, España
e-mail<EMAIL_ADDRESS>
ORCID ID: 0000-0001-7575-2619

Marcelo Escudeiro Hernandes
Departamento de Matemática
Universidade Estadual de Maringá
Avenida Colombo 5790
Maringá, Maringá-PR 87020-900.
Brazil
e-mail<EMAIL_ADDRESS>
ORCID ID: 0000-0003-0441-8503

M. Fernando Hernández Iglesias
Facultad de Ciencias Matemáticas, Escuela de Matemática,
Universidad Nacional Mayor de San Marcos,
Cercado de Lima 15081, Peru
e-mail<EMAIL_ADDRESS>
ORCID ID: 0000-0003-0026-157X


Since $0<\xi_{-}<\xi_{+}<1$ and $\xi_{0}<0$ in this case, we find

$\displaystyle e_{\rm max}=\sqrt{1-\xi_{-}}\,,~{}~{}~{}e_{\rm
min}=\sqrt{1-\xi_{+}}.$

It occurs when

$\displaystyle-{3+\alpha_{\rm dS}\over 2}<C_{\rm KL}<0\,,~{}~{}~{}{\rm
and}~{}~{}~{}\vartheta<{(\sqrt{3+\alpha_{\rm dS}}-\sqrt{-2C_{\rm
KL}})\over\sqrt{5(1+\alpha_{\rm dS})}}\,.$

The KL time scale is given by

$\displaystyle T_{\rm KL}=\tau_{\rm KL}\,\mathfrak{T}_{\rm KL}^{\rm(lib)},$
(83)

where

$\displaystyle\mathfrak{T}_{\rm KL}^{\rm(lib)}\equiv{1\over
12\sqrt{2(1+2\alpha_{\rm dS})(3+\alpha_{\rm
dS})}}\times\int_{\xi_{-}}^{\xi_{+}}d\xi{1\over\sqrt{(\xi-\xi_{0})(\xi-\xi_{+})(\xi-\xi_{-})}}\,.~{}~{}~{}~{}~{}~{}~{}$
(84)

The maximum and minimum values of the eccentricity in the KL oscillations are
determined by two conserved parameters, $\vartheta$ and $C_{\rm KL}$. We
present one example in Fig. 11. We choose $\vartheta=0.5$ and show the maximum
value, $e_{\rm max}$ (the red curve), and the minimum value, $e_{\rm min}$
(the blue curve), in terms of $C_{\rm KL}$. The solid curves denote the case
of $\alpha_{\rm dS}=0.0794$ (Model I), while the dotted curves is the case of
$\alpha_{\rm dS}=0$. For Model Ic, we find $C_{\rm KL}=-0.0000359$, which is
consistent with our result in Table I. The relativistic effect with
$\alpha_{\rm dS}$ is small.

Figure 11: The maximum and minimum values of the eccentricity in KL
oscillations. We choose $\vartheta=0.5$. The red and blue curves depict the
maximum and minimum values, respectively. The solid curves denote the case of
$\alpha_{\rm dS}=0.0794$ (Model I). We also show the case of $\alpha_{\rm
dS}=0$ by the dotted curves.

The maximum eccentricity in KL oscillations is important, especially when we
discuss emission of GWs. We plot the range of the initial conditions ($e_{0}$
and $I_{0}$) which show the large maximum values of the eccentricity in Fig.
12. The light blue, light red and light green regions correspond to $0.9\leq
e_{\rm max}<1.0$, $0.8\leq e_{\rm max}<0.9$, and $0.7\leq e_{\rm max}<0.8$,
respectively. The red dots denote Models a, b, c, and d. Note that this figure
is valid for all models I, II, and III. Hence, when the initial inclination
angle is large, the maximum eccentricity may grow close to unity.

The time scale of the KL oscillations is important for observation of the
gravitational waves. From Eqs (82) and (84), assuming the integrals do not so
much depend on the relativistic parameters, we may roughly evaluate the
relativistic effect (including de Sitter precession), which is given by

$\displaystyle{T_{\rm KL}\over T_{\rm
KL}^{\rm(N)}}\approx{1+3\mathfrak{r}_{0}^{2}\mathfrak{w}_{0}^{2}\over\sqrt{(1+2\alpha_{\rm
dS})(1+\alpha_{\rm dS}/3)}}\,.$

where $T_{\rm KL}^{\rm(N)}$ is the Newtonian value. This ratio changes from
$0.4427$ to 1 as $\mathfrak{r}_{0}$ increases from the ISCO radius to
infinity. Hence, the KL time scale near ISCO radius may become smaller than
one-half of the Newtonian value.

Figure 12: The range of the initial conditions ($e_{0}$ and $I_{0}$) for the
large maximum values of the eccentricity. The light blue, light red and light
green regions correspond to the regions of $0.9\leq e_{\rm max}<1.0$, $0.8\leq
e_{\rm max}<0.9$, and $0.7\leq e_{\rm max}<0.8$, respectively. The red dots
denote Models a, b, c, and d. The critical angle is also shown by the green
star.

We can also evaluate a critical inclination angle, beyond which the KL
oscillation occurs even when the initial eccentricity is very small. It is
given by the condition for a bifurcation point with $C_{\rm KL}=0$ with
$\omega=90^{\circ}$. Setting

$\displaystyle(1+2\alpha_{\rm dS})-{5\over 2}(1+\alpha_{\rm dS})\sin^{2}I_{\rm
crit}=0\,,$

we obtain

$\displaystyle I_{\rm crit}=\sin^{-1}\sqrt{2(1+2\alpha_{\rm dS})\over
5(1+\alpha_{\rm dS})}\,.$

We find that the critical inclination angle changes from $41.6^{\circ}$ to the
Newtonian value $I_{\rm crit}^{\rm(N)}=\sin^{-1}\sqrt{2/5}\approx
39.2^{\circ}$ as $\mathfrak{r}_{0}$ increases from the ISCO radius to
infinity.

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