Center conditions: pull-back of differential equations 111 Keywords:
Holomorphic foliations, Picard-Lefschetz theory
Yadollah Zare

###### Abstract

The space of polynomial differential equations of a fixed degree with a center
singularity has many irreducible components. We prove that pull-back
differential equations form an irreducible component of such a space. The
method used in this article is inspired by Ilyashenko and Movasati’s method.
The main concepts are the Picard-Lefschetz theory of a polynomial in two
variables with complex coefficients, the Dynkin diagram of the polynomial and
the iterated integral.

##  0 Introduction

Let $\mathbb{C}[x,y]_{\leq d}$ be the set of polynomials in the two variables
$x,y$, and coefficients in $\mathbb{C}$ of degree less than or equal to
$d\in\mathbb{N}_{0}$. The space of algebraic foliations

$\mathcal{F}=\mathcal{F}(\omega)\ \ ,\ \ \ \omega\in\Omega_{d}^{1},$

where

$\Omega_{d}^{1}:=\\{P(x,y)dy-Q(x,y)dx|\ \ P,Q\in\mathbb{C}[x,y]_{\leq d}\\},$

is the projectivization of the vector space $\Omega_{d}^{1}$, and it is
denoted by $\mathcal{F}(d)$. The maximum degree of the polynomials $P$ and $Q$
is known as the (affine) degree of $\mathcal{F}$. The space $\mathcal{F}(d)$
is a rational variety by taking the coefficients of polynomials as the
coordinates of affine variety $A^{N}$ for some $N$. The set of singularities
of the foliation $\mathcal{F}$ is $V(P)\cap V(Q)$. If
$(P_{x}Q_{y}-P_{y}Q_{x})(p)\neq 0$, for an isolated singularity $p$ of
$\mathcal{F}$, then $p$ is called _reduced_ singularity. If there is a
holomorphic coordinate system $(\tilde{x},\tilde{y})$ in a neighborhood of a
reduced singularity $p$ with $\tilde{x}(p)=0$, $\tilde{y}(p)=0$ such that in
this coordinate system

$\omega\wedge d(\tilde{x}^{2}+\tilde{y}^{2})=0,$

then the point $p$ is called a _center singularity_. The closure of the set of
algebraic foliations of fixed degree $d$ with at least one center in
$\mathcal{F}(d)$, which is denoted by $\mathcal{M}(d)$, is an algebraic subset
of $\mathcal{F}(d)$ (see for instance, [12] and [10]). Identifying irreducible
components of $\mathcal{M}(d)$ is the center condition problem in the context
of polynomial differential equations on the real plane. The complete
classification of irreducible components of $\mathcal{M}(2)$ is done by H.
Dulac in [4] (see also [2] p.601). This classification gives applications on
the number of limit cycles in the context of polynomial differential equations
on the real plane. Ilyashenko in [9], by computing tangent space at some
smooth points of the space of Hamiltonian foliations $\mathcal{F}(df)$,
$f\in\mathbb{C}[x,y]_{\leq d+1}$, proved the following:

######  Theorem 0.1.

The space of Hamiltonian foliations of degree $d$ forms an irreducible
component of $\mathcal{M}(d)$.

H. Movasati in [14], by computing the tangent cone $\mathcal{M}(d)$ at a
special point proved the following:

######  Theorem 0.2.

The space of $\mathcal{L}(d_{1},d_{2},\cdots,d_{s})$ of logarithmic foliations

$\displaystyle\mathcal{F}(\omega)$ $\displaystyle\omega=f_{1}\cdots
f_{s}\sum_{i=1}^{s}\lambda_{i}\frac{df_{i}}{f_{i}}$
$f_{i}\in\mathbb{C}[x,y]_{\leq d_{i}}\ \ \ \lambda_{i}\in\mathbb{C}\ \ \
i=1,2,\cdots,s,\ \ \ d=\sum_{i=1}^{s}d_{i}-1$

is an irreducible component of $\mathcal{M}(d)$.

Let $\mathcal{P}(a,n)$ be the set of foliation

(1) $\mathcal{F}(F^{*}(\omega))\ \ \ where\ \ \ \omega\in\Omega_{a}^{1}\ ,$
$F:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}\ \ is\ defined\ by\
(x,y)\rightarrow(R,S)\ \ and\ R,S\in\mathbb{C}[x,y]_{\leq n}\ ,\ n\geq 2.$

For a generic morphism $F$ and foliation $\mathcal{F}$, there exist a leaf of
$\mathcal{F}$ such that it has an intersection with $F(D)$ at some points with
multiplicity 2, where $D$ is the curve $V(R_{x}S_{y}-R_{y}S_{x})$. Therefore,
$F^{*}(\mathcal{F})$ has a center singularity.

######  Theorem 0.3.

The space $\mathcal{P}(a,n)$ of pull-back differential equations

$\mathcal{F}(\omega)\ ,\ \omega=P(R,S)dS-Q(R,S)dR$

where

$R,S\in\mathbb{C}[x,y]_{\leq n}\ ,\ P,Q\in\mathbb{C}[x,y]_{\leq a}\ ,\
d=an+n-1\ ,\ n\geq 2$

forms an irreducible component of $\mathcal{M}(d)$.

This paper is inspired by Ilyashenko’s paper [9] and H. Movasati’s paper [14]
and a sketch of our proof is the following:
Consider a generic $F$ and a generic polynomial $f\in\mathbb{C}[x,y]$ of
degree $a+1$. It is clear that the point $\mathcal{F}(d(f\circ F))$ is in the
intersection of $\mathcal{H}(an+n-1)$ and $\mathcal{P}(a,n)$ of the algebraic
set $\mathcal{M}(an+n-1)$. It is needed to show that the tangent cone of
$\mathcal{M}(an+n-1)$ at the point $\mathcal{F}$ is equal to
$T_{\mathcal{F}}{\mathcal{H}}(an+n-1)\cup T_{\mathcal{F}}{\mathcal{P}}(a,n)$,
in order to prove Theorem 0.3. The proof will be explained in sections 1,2 and
4.

In §1, by taking the deformation $d(f\circ
F)+\epsilon^{k}\omega_{k}+\epsilon^{k+1}\omega_{k+1}+\cdots+\epsilon^{2k}\omega_{2k}+h.o.t$
where $\omega_{k}\neq 0$ of $d(f\circ F)$, and using Petrov module concept, we
show that there is a polynomial 1-form $\alpha\in\Omega^{1}$ with degree $a$
and a polynomial $K\in\mathbb{C}[x,y]$ such that $\omega_{k}$ is of the form
$F^{*}(\alpha)+dK$. This paper is organized as follows:

In §2, we are going to calculate the explicit form $dK$, by using the iterated
integral and Melnikov function $M_{2k}$. This gives us the proof of Theorem
0.3

In §3, we see some applications of theorem 0.3. We found a maximum lower bound
for the cyclicity of a tangency vanishing cycle in a deformation $\mathcal{F}$
inside $\mathcal{F}(d)$ which is dependent on a factorization of $d$ to two
natural numbers.

In §4, we study the action of the monodromy group on a tangency vanishing
cycle in a regular fiber $f\circ F$.

##  1 Pull-back of differential equations

Inspired by H. Movasati’s method (see [14]), we will calculate the tangent
cone of $\mathcal{M}(n(a+1)-1)$ at the point in the intersection of
Hamiltonian and pull-back algebraic differential equations. Similar to [14]
and [9] our methods are based on Picard-Lefschetz theory for the foliations
with a first integral.

Let $\mathcal{F}:=\mathcal{F}(\omega)\in\mathcal{F}(a)$ be a foliation of
degree $a$, and $F=(R,S):\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ be a
morphism, where $R,S\in\mathbb{C}[x,y]_{\leq n}$ and $n\geq 2$. If a point $q$
is the tangent point of $F(D)$ and a leaf of the foliation $\mathcal{F}$ then
a point in $F^{-1}(q)$ is called a _tangency_ critical point of the foliation
$F^{*}(\mathcal{F})$.

######  Theorem 1.1.

Consider the deformation

$\mathcal{F}_{\epsilon}:\mathcal{F}^{*}(\omega)+\epsilon\omega_{1}+\cdots\ ,\
deg(\mathcal{F}_{\epsilon})\leq d\ ,\ d=an+n-1,\ n\geq 2$

of the foliation $F^{*}(\mathcal{F})$. Let $p$ be one of the tangency critical
points of foliation $F^{*}(\mathcal{F}(\omega))$. For a generic 222By generic
we mean always a non-empty Zariski open subset of the ambient space.  choice
of $\omega$ and $F$, if the deformed foliation $\mathcal{F}_{\epsilon}$ for
all small $\epsilon$ has center singularity near $p$, then
$\mathcal{F}_{\epsilon}$ is also a pull-back foliation. More precisely, there
is a foliation $\tilde{\mathcal{F}}_{\epsilon}\in\mathcal{F}(a)$ and a
polynomial map
$F_{\epsilon}=(R_{\epsilon},S_{\epsilon}):\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$
$R_{\epsilon},S_{\epsilon}\in\mathbb{C}[x,y]_{\leq n}$ such that

$F_{\epsilon}^{*}\tilde{\mathcal{F}}_{\epsilon}=\mathcal{F}_{\epsilon}\ ,\
F_{0}=F\ \ ,\ \ \mathcal{F}_{0}=\mathcal{F}.$

Note that Theorem 1.1 is equivalent to Theorem 0.3.

###  1.1 Tangent space

The set $\mathcal{P}(a,n)$ is an irreducible algebraic subset of
$\mathcal{M}(an+n-1)$ (by taking the coefficient of the polynomials as
coordinates of the map from the space of polynomials with degree $an+n-1$ to
the projective space). We are going to show that $\mathcal{P}(a,n)$ is also a
component of $\mathcal{M}(an+n-1)$. Let us take a point $\mathcal{F}$ of
$\mathcal{P}(a,n)$, then make a deformation
$\mathcal{F}_{\epsilon}\in\mathcal{P}(a,n)$ and calculate the tangent vector
space of $\mathcal{P}(a,n)$ at $\mathcal{F}$:

$\begin{split}F_{\epsilon}^{*}(\omega_{\epsilon})&=(F+\epsilon
F_{1})_{\epsilon}^{*}(\omega+\epsilon\alpha_{1})+O(\epsilon^{2})\\\
&=F^{*}(\omega)+\epsilon W+O(\epsilon^{2}),\end{split}$

where

$W=P(R,S)dS_{1}-Q(R,S)dR_{1}+$ $R_{1}(\frac{\partial P}{\partial
x}(R,S)dS-\frac{\partial Q}{\partial x}(R,S)dR)+S_{1}(\frac{\partial
P}{\partial y}(R,S)dS-\frac{\partial Q}{\partial
y}(R,S)dR)+F^{*}(\alpha_{1}).$

For a smooth point $\mathcal{F}$ of $\mathcal{P}(a,n)$, the tangent space of
$\mathcal{P}(a,n)$ at $\mathcal{F}$ is just the set of all vectors $W$, which
is contained in the tangent space of $\mathcal{M}(d)$ at $\mathcal{F}$, and in
order to prove our main theorem it is enough to prove that the equality
happens. Now, we are going to compute the tangent cone of
$\mathcal{M}(an+n-1)$ at the point in the intersection of Hamiltonian
component and the set $\mathcal{P}(a,n)$.

###  1.2 A foliation in the intersection of two algebraic sets

Let $F:=\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ be defined by

(2) $\displaystyle(x,y)\rightarrow(R(x),S(y)):=(\prod_{i=1}^{n}(x-t_{i})\ ,\
\prod_{j=1}^{n}(y-t_{j}^{\prime}))$

where $t_{i}\ ,\ t_{j}^{\prime}\in\mathbb{R}^{\geq 0}$ , $R$ and $S$ are Morse
functions. Let $g,\ h$ be two polynomials of degree $a+1$ defined by

(3) $\displaystyle g(x):=\prod_{i=1}^{a+1}(x-s_{i}),$ $\displaystyle and$
$\displaystyle h(y):=\prod_{i=1}^{a+1}(y-s_{j}^{\prime}),$

and meet the following conditions:

  1. 1.

All $s_{i}\ ,s_{j}^{\prime}$ are positive real numbers,

  2. 2.

Both equations $R(x)=s_{i}$ and $S(y)=s_{j}^{\prime}$ have $n$ real roots ,

  3. 3.

The functions $g,h,g\circ R$ and $h\circ S$ are _Morse_ , which is a
holomorphic function with no degenerate critical points.

  4. 4.

If $p$ is a critical point of $R$ (resp. $S$) and $q\in R^{-1}(q_{1})$ (resp.
$q\in S^{-1}(q_{1})$) where $q_{1}$ is a critical point of $g$ (resp. $h$),
then $|g\circ R(p)|>|g\circ R(q)|$ (resp. $|h\circ S(p)|>|h\circ S(q)|$). In
fact, by moving the roots of $g$ and $h$ on the real line this is the
assumable definition.

Let $f\in\mathbb{C}[x,y]_{\leq a+1}$ be defined by

(4) $\displaystyle f(x,y):=g(x)+h(y).$

We can suppose that the intersection of the set of the critical values of
$g\circ R$ and $h\circ S$ is empty. The foliation $\mathcal{F}_{0}$ has three
kinds of singularities :

  1. 1.

Pull-back of centers of $\mathcal{F}(df)$,

  2. 2.

Tangency critical points of the foliation $F^{*}(df)$,

  3. 3.

The points in $V(R_{x})\cap V(S_{y})$.

Let $X(a,n)$ be the irreducible component of $\mathcal{M}(an+n-1)$ containing
$\mathcal{P}(a,n)$.

Consider the deformation

(5) $\displaystyle\mathcal{F}_{\epsilon}:d(f\circ
F)+\omega_{k}\epsilon^{k}+\omega_{k+1}\epsilon^{k+1}+\cdots\ ,\
deg(\omega_{i})\leq d,$

of $\mathcal{F}_{0}=\mathcal{F}(d(f\circ F))$.

Assume that $\mathcal{F}_{\epsilon}$ belongs to $X(a,n)$. This implies that
$\mathcal{F}_{\epsilon}$ always has a center singularity near a fixed tangency
center $p$ of $\mathcal{F}_{0}$. The set of all differential forms
$\omega_{k}$ is the tangent cone of $M(an+n-1)$ at $\mathcal{F}$. Note that
taking $k=1$ is not sufficient for calculating the tangent cone.

Let $\delta_{t}$ be a continuous family of the vanishing cycles around a
tangency critical $p$ and $\Sigma$ be a transverse section to $\mathcal{F}$ at
some point of $\delta_{t}$. We are able to write the Taylor expansion of the
deformed holonomy $h_{\epsilon}(t)$

$h_{\epsilon}(t)-t=M_{1}(t)\epsilon+M_{2}\epsilon^{2}+\cdots+M_{i}(t)\epsilon^{i}+\cdots.$

Here $M_{i}(t)$ is the i-th Melnikov function of the deformation. Since
$\omega_{i}=0$ for $0\leq i\leq k-1$, then

$M_{1}=M_{2}=\dots=M_{k-1}=0.$

If $\Sigma$ is parametrized by the image of $f$,i,e. $t=f(z)$ , $z\in\Sigma$
then

(6) $M_{k}(t)=-\int_{\delta_{t}}\omega_{k}\ .$

See for instance [5].

######  Theorem 1.2.

The morphism $F_{*}:H_{1}((f\circ F)^{-1}(b),\mathbb{Z})\rightarrow
H_{1}(f^{-1}(b),\mathbb{Z})$ is surjective and $ker(F_{*})$ is a group
generated by the action monodromy group $\pi_{1}(\mathbb{C}\setminus C,b)$ on
a vanishing cycle around a tangency point.

We will prove this theorem at the end of §4, see Theorem 4.10.

###  1.3 Brieskorn lattice/Petrov Modules

Consider the Brieskorn lattice/Petrov module

$\displaystyle
H_{f}:=\frac{\Omega_{\mathbb{C}^{2}}^{1}}{df\wedge\Omega_{\mathbb{C}^{2}}^{0}+d\Omega_{\mathbb{C}^{2}}^{0}}$
$\displaystyle and$ $\displaystyle H_{f\circ
F}:=\frac{\Omega_{\mathbb{C}^{2}}^{1}}{d(f\circ
F)\wedge\Omega_{\mathbb{C}^{2}}^{0}+d\Omega_{\mathbb{C}^{2}}^{0}},$

where $H_{f}$ and $H_{f\circ F}$ are $\mathbb{C}[s]$-module and
$\mathbb{C}[s^{\prime}]$-module respectively, (here $s=f$ and
$s^{\prime}=f\circ F$ , and also $\Omega_{\mathbb{C}^{2}}^{i}$ , $i=0,1$ are
the set of polynomial differential forms in $\mathbb{C}^{2}$.).

######  Definition 1.1.

A polynomial $l\in\mathbb{C}[x,y]$ of degree $d$ with homogeneous leading part
$l_{d}$ is called _transversal_ to infinity, if $l_{d}$ factors out as the
product of d pairwise different linear forms.

Consider the Milnor module

$\displaystyle V_{l_{d}}:=\frac{\mathbb{C}[x,y]}{<(l_{d})_{x},(l_{d})_{y}>},$

with the basis $\\{x^{i}y^{j}|0\leq i,j\leq d-2\\}$ (see e.g. ([13] chapter
10)). We define

$\begin{split}A_{ij}&:=\frac{i+1}{d}+\frac{j+1}{d},\\\ \eta&:=xdy-ydx,\\\
\eta_{ij}&:=x^{i}y^{j}\eta,\end{split}$

where $0\leq i,j\leq d-2$ .

######  Theorem 1.3.

Let $l(x,y)\in\mathbb{C}[x,y]$ be a polynomial transversal to infinity of
degree $d$ . The $\mathbb{C}[l]$-module $H_{l}$ is free and $\eta_{ij}$, where
$0\leq i,j\leq d-2$, forms a basis of $H_{l}$. Furthermore, every
$\omega\in\Omega_{\mathbb{C}^{2}}^{1}$ can be written

(7) $\displaystyle\omega=\sum_{0\leq i,j\leq
d-2}{h_{ij}(l)\eta_{ij}}+dl\wedge\zeta_{1}+d\zeta_{2},$ $\displaystyle
h_{ij}\in\mathbb{C}[l]\ ,\ \zeta_{1},\zeta_{2}\in\mathbb{C}[x,y]\ \ ,\ \
deg(h_{ij})\leq\frac{deg(\omega)}{d}-A_{ij}.$

See e.g. [13] Theorem 10.9.1. and [8].

######  Proposition 1.1.

Pull-backs of $\eta_{ij}$ for all $i$ and $j$ are independent in $H_{f\circ
F}$ under the map $F^{*}$ and can be extended to a basis for $H_{f\circ F}$.

###### Proof.

The map $F^{*}:H_{f}\rightarrow H_{f\circ F}$ is injective, and
$F^{*}(\eta_{ij})$ are linear independent. We have
$F^{*}(\eta_{ij}=x^{i}y^{j}(ydx-xdy))=R^{i}S^{j}(RdS-SdR)$, and the
coefficients of $R^{i}S^{j}$ are in $\mathbb{C}$. By Theorem 1.3 we can write

$\displaystyle RdS-SdR=\sum_{0\leq i,j\leq
an+n-1}P_{ij}(s^{\prime})\bar{\eta}_{ij}.$

We are going to show that the functions
$P_{ij}(s^{\prime})\in\mathbb{C}[s^{\prime}=f\circ F]$ are constant. We know
that $xdy=\frac{1}{2}(xdy-ydx)=\frac{1}{2}(\bar{\eta})$ and
$ydx=-\frac{1}{2}(\bar{\eta})$ in $H_{f\circ F}$. By changing the coordinate
assume that $R(0)=S(0)=0$, therefore

$\displaystyle RdS-
SdR=(\prod_{i=2}^{n}(x-t_{i})S_{y})xdy-(\prod_{i=2}^{n}(y-t^{\prime}_{i})R_{x})ydx$
$\displaystyle=(\prod_{i=2}^{n}(x-t_{i})S_{y})(\frac{1}{2}\bar{\eta})+(\prod_{i=2}^{n}(y-t^{\prime}_{i})R_{x})(\frac{1}{2}\bar{\eta}),$

since $deg(RdS-SdR)<an+n-1$ then the functions $P_{ij}$’s are constant. It
means that if we write
$F^{*}(\eta_{\beta})=\sum_{\beta^{\prime}}P_{\beta^{\prime}}(s^{\prime})\bar{\eta}_{\beta^{\prime}}$,
then all of the coefficients $P_{\beta^{\prime}}(s^{\prime})$ are in
$\mathbb{C}$. In other words, $F^{*}(\eta_{\beta})$ for all $\beta=ij$ can
extend to a basis of $H_{f\circ F}$. ∎

###  1.4 Relatively Exact 1-form

######  Definition 1.2.

Let $\mathcal{F}$ be a foliation in $\mathbb{C}^{2}$. If the restriction of a
meromorphic 1-form $\omega_{1}$ on $\mathbb{C}^{2}$ to each leaf $L$ of
$\mathcal{F}$ is exact, then it is called relatively exact modulo
$\mathcal{F}$, i.e. there is a meromorphic function $f$ on L so that
$\omega_{1}|_{L}=df$.

Note that a meromorphic 1-form $\omega_{1}$ is relatively exact modulo
$\mathcal{F}$ if and only if

$\int_{\delta}\omega_{1}=0,$

for all closed cycles in the leaves of $\mathcal{F}$.

######  Proposition 1.2.

Every relatively exact polynomial 1-form $\omega$ in $\mathbb{C}^{2}$ of
degree $d$ modulo a Hamiltonian foliation $\mathcal{F}(df)$ has the form

$\omega=dg+p.df$

where $g$ and $p$ are polynomials so that $deg(p)$ divides $d$, and
$deg(g)\leq d$.

See e.g. ([12] Theorem 4.1.).

###  1.5 Computing the Tangent Cone

Let $F$ be a morphism from $\mathbb{C}^{2}$ into itself and $f$ be a
polynomial of degree $a+1$ that are defined in (2) and (4) respectively. Let
also consider the deformation
$\mathcal{F}_{\epsilon}=\mathcal{\omega_{\epsilon}}$ of $\mathcal{F}(d(f\circ
F))$, where

$\omega_{\epsilon}=d(f\circ
F)+\epsilon^{k}\omega_{k}+\epsilon^{k+1}\omega_{k+1}+\dots\ \ \ ,\ \
deg(\omega_{j})\leq an+n-1.$

It is not necessary to start $k$ from one. Then from the equality (6) we have
the following :

######  Theorem 1.4.

There is a polynomial differential 1-form $\alpha_{1}$ with
$deg(\alpha_{1})\leq deg(\omega_{k})$ and a polynomial
$K\in\mathbb{C}[x,y]_{\leq{a+1}}$ such that

$\omega_{k}=F^{*}(\alpha_{1})+dK,$

where $F:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ is defined by
$(x,y)\rightarrow(R,S)$ as in (2).

###### Proof.

For a regular value $b$ of the function $f\circ F$, it is clear that the liner
map

$\displaystyle F_{*}:H_{1}((f\circ F)^{-1}(b),\mathbb{Z})\rightarrow
H_{1}((f)^{-1}(b),\mathbb{Z}),$

is surjective. Then

$\displaystyle F_{b}^{*}=H_{dR}^{1}(f^{-1}(b))\rightarrow H_{dR}^{1}((f\circ
F)^{-1}(b))$

is injective. According to Theorem 1.2, $\int_{\delta}\omega_{k}=0$ for all
$\delta\in ker(F_{*})$, this implies that the linear map

$\displaystyle H_{1}(f^{-1}(b),\mathbb{Z})\rightarrow\mathbb{C}$
$\displaystyle defined\ \ by$
$\displaystyle\delta\rightarrow\int_{\gamma}\omega_{k},$

for an element $\gamma\in F^{-1}(\delta)$, is well defined. By duality of de
Rham cohomology and singular homology there is a differential form
$\alpha_{b}$ in $f^{-1}(b)$ such that

$\displaystyle\int_{\gamma}\omega_{k}=\int_{\delta}\alpha_{b}\ \ for\ a\
\gamma\in F^{-1}(\delta).$

By using Atiyah-Hodge theorem (see e.g. [13]) the form $\alpha_{b}$ can be
taken algebraically. All these $\alpha_{b}$’s give us a holomorphic global
section $\alpha$ of cohomology bundle of $f$ outside the critical values of
$f$;

$\displaystyle\alpha_{t}\in H_{dR}^{1}(f^{-1}(t))$ $\displaystyle where$
$\displaystyle t\in\mathbb{C}\setminus\\{c_{1},\dots,c_{a^{2}}\\}.$

We are going to show that it is a holomorphic global section in the whole
$\mathbb{C}$.
By the Theorem 1.3 we can write

$\displaystyle\alpha=\sum_{\beta}h_{\beta}\eta_{\beta}$ $\displaystyle where$
$\displaystyle h_{\beta}\ s\ \ are\ \ holomorphic\ \ in\ \
\mathbb{C}\setminus\\{c_{1},\dots,c_{a^{2}}\\}\ ,\ \beta=(i,j).$

The Periodic matrix

$\begin{bmatrix}\int_{\delta_{c_{k}}}\eta_{\beta}\end{bmatrix}_{\mu\times\mu}$

is invertible, where $\mu$ is the rank of $H_{1}((f\circ
F)^{-1}(b),\mathbb{Z})$, (see e.g. [9] Proposition 26.44). Therefore, the
$h_{\beta}$’s coefficients are meromorphic functions on $t$, because

$\begin{bmatrix}h_{\beta_{1}}\\\ \vdots\\\
h_{\beta_{\mu}}\end{bmatrix}=\begin{bmatrix}\int_{\delta_{c_{k}}}\eta_{\beta}\end{bmatrix}_{\mu\times\mu}^{-1}\begin{bmatrix}\int_{\delta_{c_{1}}}\alpha\\\
\vdots\\\ \int_{\delta_{c_{\mu}}}\alpha\end{bmatrix},$

and by Theorem 10.7 in Chapter 10 of [1] each integral
$||\int_{\delta_{c_{k}}}\eta_{\beta}||\leq\ const\ ||t-c_{k}||^{-N}$ for a
natural number $N$ and $t$ close to singular value $c_{k}$. Thus, all the
elements of the matrices on the right side of the equality have finite growth
at critical values. This implies that, there is a polynomial
$P(s)\in\mathbb{C}[s]$ such that $P.\alpha$ is a holomorphic form. We can
write $P.\alpha=\sum_{\beta}{h^{\prime}}_{\beta}\eta_{\beta}$ , then
$F^{*}(P)\omega_{k}-F^{*}(P.\alpha)=0$ in $H_{f\circ F}$. According to
Proposition 1.1 the set of $F^{*}(\eta_{\beta})$ for all $\beta$ can be
extended to a basis of $H_{f\circ F}$. Therefore, we have

(8) $\displaystyle
F^{*}(P)\omega_{k}=\sum_{\beta}F^{*}(P).{h}_{\beta}(s)\eta_{\beta}+\sum_{\sigma}F^{*}(P)a_{\sigma}\tilde{\eta}_{\sigma}.$

Since each element of $H_{f\circ F}$ can be written uniquely as a linear
combination of the elements on this basis, then $a_{\sigma}=0$ for all
$\sigma$. In other words, $F^{*}(P).{h}_{\beta}=F^{*}({h^{\prime}}_{\beta})$,
hence $P|{h^{\prime}}_{\beta}$. This implies that $\alpha$ is a holomorphic
1-form. By Theorem 4.5, the degree of $h_{\beta}$ in the equation (8) is less
than or equal to $\frac{deg(\omega_{k})}{as+s}-A_{\beta}<1$, hence $h_{\beta}$
are constant for all $\beta$. To find the form of $\omega_{k}$ we use the
Proposition 1.2 and we conclude that
$\int_{F^{*}(\delta)}\alpha=\int_{\delta}\omega_{k}$ for all cycles $\delta$
in the fibers of $f\circ F$. This implies that $\omega-F^{*}(\alpha)$ is
relatively exact modulo $\mathcal{F}(d(f\circ F))$, then by Proposition (1.2)
there are polynomials $K$ and $A$ such that $\omega-F^{*}(\alpha)=dK+Ad(f\circ
F)$. The fact that $deg(\omega-F^{*}(\alpha))\leq deg(f\circ F)-1$ implies
$A\equiv 0$, so we get our desired equality. ∎

The proof of the main theorem is still not finished. We have to prove that the
polynomial $K$ in the Theorem 1.4 is of the form (9). For this goal, we need
to compute higher order Melnikov functions. This will be done in the next
section.

##  2 Higher order Milnikov function

L.Gavrilov in [7] has shown that the higher order Melnikov functions can be
expressed in terms of iterated integrals. Basic properties of iterated
integrals are established by A. N. Parsin in 1969 and a systematic approach
for de Rham cohomology type theorems for iterated integrals was made by K. T.
Chen around 1977. H. Movasati and I. Nakai in [16] used the concept of higher
order Melnikov functions by iterated integrals.

Let $\gamma:[0,1]\rightarrow\mathbb{C}^{2}$ be a piecewise smooth path on
$\mathbb{C}^{2}$. Let $\omega_{1},\omega_{2},\dots,\omega_{n}$ be smooth
1-forms on $\mathbb{C}^{2}$, $\gamma^{*}(\omega_{i})=f_{i}(t)dt$ for the
pulled-back of the forms $\omega_{i}$ to the interval [0,1]. Recall that the
ordinary line integral given by

$\int_{\gamma}\omega_{1}=\int_{[0,1]}\gamma^{*}(\omega_{1})=\int_{0}^{1}f_{1}(t_{1})dt$

does not depend on the choice of parametrization of $\gamma$.

######  Definition 2.1.

Iterated integral of $\omega_{1},\omega_{2},\dots,\omega_{n}$ along the path
$\gamma$ is defined by

$\int_{\gamma}\omega_{1}.\omega_{2}\dots\omega_{n}=\int_{0\leq
t_{1}\leq\dots\leq t_{n}\leq 1}(f_{1}(t)dt_{1}\dots f_{n}(t_{n})dt_{n}).$

Let us consider the deformation

$\mathcal{F}_{\epsilon}:d(f\circ
F)+\omega_{k}\epsilon^{k}+\omega_{k+1}\epsilon^{k+1}+\dots\ \ \
deg(\omega_{i})\leq n(a+1)-1.$

of $d(f\circ F)$. The deformed holonomy along the path $\delta_{t}$ in
$\Sigma$ is

$h_{\epsilon}(t)-t=M_{1}(t)\epsilon+\dots+M_{k}(t)\epsilon^{k}+\dots+M_{2k}(t)\epsilon^{2k}+\dots.$

Since $\omega_{i}=0$ where $0<i<k-1$, then $M_{1}=M_{2}=\dots=M_{k-1}=0$. By
using Theorem 3.2 in [15] (Higher order approximation), we conclude that
$M_{i}(t)=-\int_{\delta_{t}}\omega_{i}$ where $k\leq i<2k$, and also

$M_{2k}(t):=-\int_{\delta_{t}}(\omega_{k}.(\frac{d\omega_{k}}{d(f\circ
F)})+\omega_{2k}).$

Note that the vector $W$ in the case $\omega=d(f\circ F)$ is of the form

(9) $\displaystyle W=d(R_{1}\frac{\partial f}{\partial
x}(R,S)+S_{1}\frac{\partial f}{\partial y}(R,S))+F^{*}(\alpha_{1}).$

######  Lemma 2.1.

The polynomial $K$ in the Theorem 1.4 is of the form

$K=R_{1}\frac{\partial f}{\partial x}(R,S)+S_{1}\frac{\partial f}{\partial
y}(R,S),$

where $R_{1},S_{1}\in\mathbb{C}[x,y]_{\leq n}$.

###### Proof.

$\displaystyle\int_{\delta_{t}}\omega_{k}.(\frac{d\omega_{k}}{d(f\circ F)})$
$\displaystyle=$
$\displaystyle\int_{\delta_{t}}(F^{*}\alpha_{1}+dK).F^{*}(\frac{d\alpha_{1}}{df})$
$\displaystyle=$
$\displaystyle\int_{\delta_{t}}F^{*}(\alpha_{1}.\frac{d\alpha_{1}}{df})+\int_{\delta_{t}}(dK).F^{*}(\frac{d\alpha_{1}}{df})$
$\displaystyle=$
$\displaystyle\int_{F_{*}(\delta_{t})}\alpha_{1}.\frac{d\alpha_{1}}{df}+\int_{t_{1}\leq
t_{2}}(K(t_{1})-K(p_{t_{1}})).F^{*}(\frac{d\alpha_{1}}{df})$ $\displaystyle=$
$\displaystyle\int_{F_{*}(\delta_{t})}\alpha_{1}.\frac{d\alpha_{1}}{df}+\int_{\delta_{t}}KF^{*}(\frac{d\alpha_{1}}{df})-K(p_{t})\int_{\delta_{t}}F^{*}(\frac{d\alpha_{1}}{df})$
$\displaystyle=$
$\displaystyle\int_{\delta_{t}}KF^{*}(\frac{d\alpha_{1}}{df}).$

Here $p_{t}$ is a point in the cycle $\delta_{t}$. Now the equality
$M_{2k}(t)=0$ and a similar argument as in the last lemma implies that

$\omega_{2k}+K.F^{*}(\frac{d\alpha_{1}}{df})=F^{*}(\alpha_{2})+dK_{2}+A_{j}d(f\circ
F),$

and therefore,

$K.F^{*}(d\alpha_{1})=-\omega_{2k}\wedge d(f\circ F)+F^{*}(\alpha_{2})\wedge
d(f\circ F)+dK_{2}\wedge d(f\circ F).$

Since $d\alpha_{1}$ is a 2-form like $h(x,y)dx\wedge dy$, then

$F^{*}(h(x,y)dx\wedge dy)=F^{*}(h).F^{*}(dx\wedge dy)=(h\circ
F).(R_{x}S_{y}-R_{y}S_{x})dx\wedge dy.$

This implies that $K.h(F).(R_{x}S_{y}-R_{y}S_{x})$ is in the ideal

$I=<R_{x}f_{x}(R,S)+R_{y}f_{y}(R,S)\ ,\ S_{x}f_{x}(R,S)+S_{y}f_{y}(R,S)>.$

Now consider the radical ideal $I_{1}=<f_{x}(R,S)\ ,\ f_{y}(R,S)>$ and
$J=<h(F).(R_{x}S_{y}-R_{y}S_{x})>$, then it is clear that $I\subset I_{1}$ and
$I_{1}.J\subset I$, so

$I_{1}.J\subseteq I\cap J\subseteq I_{1}\cap J.$

We can assume that the curve $V(h)$ does not pass any critical point of $f$.
By our hypothesis $F$ and $f$ are generic so we have $V(J)\cap
V(I_{1})=\emptyset$. This means that $J+I_{1}=\mathbb{C}[x,y]$ thus we have

$I_{1}.J=I_{1}\cap J\ \ \ \ \Rightarrow\ \ \ I\cap J=I_{1}.J,$

which states that $K\in<f_{x}(R,S)\ ,\ f_{y}(R,S)>$ therefore,we get the
result

$K=R_{1}\frac{\partial f}{\partial x}(R,S)+S_{1}\frac{\partial f}{\partial
y}(R,S)\ \ \ \ where\ \ \ R_{1}\,\ S_{1}\in\mathbb{C}[x,y]_{\leq n}.$

∎

######  Corollary 2.1.

The point $\mathcal{F}_{0}:=\mathcal{F}(d(f\circ F))$ is in
$\mathcal{M}(as+s-1)$, so tangent cone of $\mathcal{M}(as+s-1)$ at the point
$\mathcal{F}_{0}$ is

$TC_{\mathcal{F}_{0}}\mathcal{M}(as+s-1)=T_{\mathcal{F}_{0}}\mathcal{P}(a,s)\cup
T_{\mathcal{F}_{0}}\mathcal{H}(as+s-1).$

###  2.1 Proof of Main theorem

Consider a germ of an analytic variety ($X$,0) in ($\mathbb{C}^{n}$,0). The
analytic path $\gamma:(\mathbb{C},0)\rightarrow(X,0)$ has the Taylor expansion
$\gamma=\omega\epsilon^{l}+\omega^{\prime}\epsilon^{l+1}+\dots$ ,
$\omega,\omega^{\prime},\dots\in\mathbb{C}$. Let $T_{l}$ be the set of all
$\omega$. The tangent cone $TC_{0}X$ of $X$ at 0 is
$TC_{0}X=\cup_{l=1}^{\infty}T_{l}$.
The tangent cone $TC_{0}X$ is an algebraic set with _pure dimension_ $dim(X)$,
i.e. each irreducible component of $TC_{0}X$ is of dimension $dim(X)$. If 0 is
a smooth point of $X$ then $TC_{0}X$ is the usual tangent space of $X$ at 0.

The variety $\mathcal{P}(n,a)$ is parametrized by

$\tau:\mathcal{P}_{n}\times\mathcal{P}_{n}\times\mathcal{P}_{a}\times\mathcal{P}_{a}\rightarrow\mathcal{F}(d),\
\ \ \ \ d=na+n-1$ $\tau(R,S,P,Q)=P(R,S)dS-Q(R,S)dR,n\geq 2,$

and so it is irreducible.

###### Proof.

_of Theorem 0.3:_ Let $\mathcal{F}_{0}:=\mathcal{F}(d(f\circ F))$ where $f$,
$F$ defined in (2),(4) respectively. For the proof of our main theorem, it is
enough to show that $X:=(\mathcal{P}(n,a),\mathcal{F}_{0})$ is an irreducible
component of $(\mathcal{M}(d),\mathcal{F}_{0})$. According to Corollary 2.1 we
have:

(10)
$TC_{\mathcal{F}_{0}}\mathcal{M}(d)=TC_{\mathcal{F}_{0}}\mathcal{P}(n,a)\cup
TC_{\mathcal{F}_{0}}H(d).$

Let $X^{\prime}$ be an irreducible component of
$(\mathcal{M}(d),\mathcal{F}_{0})$ such that $X\subset X^{\prime}$. If
$TC_{\mathcal{F}_{0}}X\subset Y$, where $Y$ is the irreducible component of
$TC_{\mathcal{F}_{0}}X^{\prime}$, then it must be a subset of
$TC_{\mathcal{F}_{0}}X$, because the equality (10) is union decomposition of
$TC_{\mathcal{F}_{0}}\mathcal{M}(d)$ to irreducible component. This implies
that $Y=TC_{\mathcal{F}_{0}}X$. Dimension of $Y\subset
TC_{\mathcal{F}_{0}}X^{\prime}$ is equal to dimension $X$, so
$dim(X^{\prime})=dim(X)$. Therefore, $X=X^{\prime}$ because $X\subset
X^{\prime}$ and $X,\ X^{\prime}$ are irreducible algebraic sets and they have
the same dimension. ∎

##  3 Limit cycles

Consider a real planer 1-form $\omega=P(x,y)dy-Q(x,y)dx$ where $P$ and $Q$ are
polynomials of degree less than or equal to $d$. Let the foliation
$\mathcal{F}$ induced by the 1-form $\omega$.

######  Definition 3.1.

A closed trajectory which is limit set of some trajectories of a real
foliation $\mathcal{F}$ is called limit cycle.

The _Hilbert number_ , which denotes by $H_{d}$, is the maximum possible
number of limit cycles of a real foliation $\mathcal{F}(\omega)$. It is still
unsolved whether $H_{d}$ is finite, even for the simple case $d=2$. It is
known that $H_{d}\geq k.d^{2}$ for some constant $k$, but in 1995, C.J
Christopher and N.G. Lloyd found a strong lower bound $d^{2}log\ d$ for the
Hilbert numbers, see [3].

Let $X$ be an irreducible component of $\mathcal{M}(d)$. Let $p$ be a real
center singularity of a real foliation $\mathcal{F}\in
X-sing(\mathcal{M}(d))$. By real foliation we mean the equation of the
foliation has real coefficient. Let $\delta_{t},t\in(\mathbb{R},0)$ be a
family of real vanishing cycles around $p$. Roughly speaking, the _cyclicity_
of $\delta_{0}$ is the maximum number of limit cycles appearing near
$\delta_{0}$ after a deformation of $\mathcal{F}$ in $\mathcal{F}(d)$. The
cyclicity of $\delta_{0}$ in a deformation of $\mathcal{F}$ inside
$\mathcal{F}(d)$ is greater than $codim_{\mathcal{F}(d)}(X)-1$. The reader can
find the exact definition of cyclicity and the proof of this fact in [15]. Yu.
Ilyashenko in [9] shows that
$codim_{\mathcal{F}(d)}(\mathcal{H}(d))-1=\frac{(d+2)(d-1)}{2}-1$. The best
upper bound for the cyclicity of a vanishing cycle of a Hamiltonian equation
is the P.Mardesic’s result $\frac{d^{4}+d^{2}-2}{2}$ in [11]. H. Movasati in
[14], shows that the cyclicity of $\delta_{0}$ of a logarithmic foliation
$\mathcal{F}(f\sum_{i=1}^{s}\lambda_{i}\frac{df_{i}}{f_{i}})\in\mathcal{L}(d_{1},\dots,d_{s})$
is not less that

$(d+1)(d+2)-\sum_{i=1}^{s}(\frac{(d_{i}+1)(d_{i}+2)}{2})-1.$

This lower bound reaches to maximum when $d_{i}=1,s=d+1,i=1,\dots,s$. In this
case the cyclicity of $\delta$ is not less than $d^{2}-1$.

######  Proposition 3.1.

Suppose that $n>1$ and $d:=an+n-1$. The cyclicity of $\delta_{0}$ in a
deformation of $\mathcal{F}$ in $\mathcal{F}(d)$ is not less than

$C:=(d+1)(d+2)-((n+1)(n+2)+(\frac{d+1}{n})(\frac{d+1}{n}+1))-1.$

By considering $d+1=(a+1)n$ as a fixed value, when $n\leq a+1$ and in addition
the distance of $a+1$ and $n$ is minimum, then
$(n+1)(n+2)+(\frac{d+1}{n})(\frac{d+1}{n}+1)$ will be minimum. This
minimization will be led to maximizing of the cyclicity. If $n$ and $(a+1)$
are near to $\sqrt{d+1}$ then the cyclicity $C$ close to
$d^{2}+d-4\sqrt{d+1}-3$. If $a+1=p$ and $n=q$ , where $p,q$ are primes and
$p>q$, then $C=:(pq)^{2}+pq-q^{2}-3q-p^{2}-p-3$, for instance, when $q=2$ we
have $3p^{2}+p-13$.

##  4 Picard-Lefschetz Theory

In this section, we intend to study the topology of a regular fiber of a
polynomial function with one and two variables. The main idea of this section
is to understand the intersection number between two vanishing cycles and the
action of monodromy group on a vanishing cycle in the case of pull-back of
cycles under a morphism.

Let $f$ be a Morse function with the finite set of critical values $C$ labeled
by $c_{1},c_{2},\dots,c_{s}$.

######  Definition 4.1.

A distinguished system of paths related to $f$ is the system of smooth paths
in $\mathbb{C}$, starting at the regular point $b\in\mathbb{C}\setminus C$ and
ending at a point in $C$ such that

  1. 1.

The paths have no self-intersections;

  2. 2.

Different paths meet only at their common point $b$.

Consider a small ball $U_{p}$ in $\mathbb{C}^{m}$ with center at the Morse
critical point $p$. Let the value $b$ be very close to $c:=f(p)$, but not
equal to it. Let $\alpha:=[0,1]\rightarrow f(U_{p})$ be a path that starts at
$b$, ends at $c$ and does not pass through any other critical value of $f$. By
the Morse lemma, there is a local coordinate system $x_{1},\dots,x_{m}$ in a
neighborhood of $p$ such that the function $f$ can be written in the form
$f(x_{1},\dots,x_{m})=c+\sum_{j}x_{j}^{2}$. Consider the sphere
$S_{t}:=\\{x\in f^{-1}(\alpha(t))|Im(x_{j}(x))=0\\}\cap U_{p}$. Whenever $t$
tends to 1, then $S_{t}$ tends to $p$.

######  Definition 4.2.

If $S_{0}:=\delta$ in the nonsingular fiber $f^{-1}(b)$, then the homology
class $\delta\in H_{m-1}(f^{-1}(b),\mathbb{Z})$ is called a vanishing cycle
along the path $\alpha$ or just a vanishing cycle.

######  Theorem 4.1.

(see e.g [1]) The collection of the vanishing cycles along all paths of a
distinguished system of paths forms a basis of the group
$H_{m-1}(f^{-1}(b),\mathbb{Z})$.

######  Definition 4.3.

Let $\lambda_{c}$ be a path of a distinguished system, and $\lambda$ be a loop
in $\pi_{1}(\mathbb{C}\setminus C,b)$ such that  a) it turns once anti-
clockwise around the critical point $c$, and  b) the closure of the interior
of $\lambda$ contains the path $\lambda_{c}$ and does not contain any other
point of $C$.  Then the loop $\lambda$ is called a simple loop, corresponding
to $\lambda_{c}$.

The paths that are homotopic to a simple loop $\lambda$ give a class of
homotopic homeomorphism maps $\\{h_{\lambda}:f^{-1}(b)\rightarrow
f^{-1}(b)\\}$. This class defines a unique well-defined map

$h_{\lambda}:H_{m-1}(f^{-1}(b),\mathbb{Z})\rightarrow
H_{m-1}(f^{-1}(b),\mathbb{Z}).$

######  Definition 4.4.

For a regular value $b$ of $f$, we have

$\displaystyle h:\pi_{1}(\mathbb{C}\setminus C,b)\times
H_{m-1}(f^{-1}(b),\mathbb{Z})\rightarrow H_{m-1}(f^{-1}(b),\mathbb{Z})$
$\displaystyle h(\lambda,\cdot)=h_{\lambda}(\cdot).$

The image of $\pi_{1}(\mathbb{C}\setminus C,b)$ in
$Aut(H_{m-1}(f^{-1}(b),\mathbb{Z}))$ is called the monodromy group and its
action $h$ is called the action of the monodromy group on the homology group
of $f^{-1}(b)$.

Picard-Lefschetz formula: Let $\lambda$ be a monodromy (simple loop) around
the critical value $c$, the action of monodromy $h_{\lambda}$ on a cycle
$\delta\in H_{m-1}(f^{-1}(b),\mathbb{Z})$ is given by

$h_{\lambda}(\delta)=\delta+\sum_{j}(-1)^{\frac{m(m-1)}{2}}<\delta,\delta_{j}>\delta_{j},$

where $j$ runs through all the vanishing cycles around the singularities with
value $c$, and $<\cdot,\cdot>$ denotes the intersection number of two cycles
in $f^{-1}(b)$.

###  4.1 Picard-Lefschetz theory in dimension zero

Let $f(x)\in\mathbb{C}[x]$ be a polynomial of degree $d$ with real roots
$t_{i}$, where $i=1,\dots,d$, and $d-1$ critical values.

######  Theorem 4.2.

For the regular value $b=0$, we have the following:

  * •

$H_{0}(f^{-1}(b),\mathbb{Z})$ is generated by $\delta_{i}=[t_{i}]-[t_{i+1}]$
for $i=1,\dots,d-1$;

  * •

Intersection matrix for $H_{0}(f^{-1}(b),\mathbb{Z})$ with respect to this
basis is

$<\delta_{i},\delta_{j}>=\begin{cases}2&\text{if }i=j\\\ -1&\text{if}\
i=j+1\\\ 0&\text{if}\ |i-j|>1.\end{cases}$

(See e.g. [13] or [6].)

######  Lemma 4.1.

For any two vanishing cycles $\delta_{i},\ \delta_{j}\in
H_{0}(f^{-1}(b),\mathbb{Z})$ there is a monodromy $\lambda$ such that
$\lambda(\delta_{i})=\delta_{j}$.

Consider

(11) $\displaystyle R(x)=\prod_{i=1}^{n}(x-t_{i})$

where $(t_{i}\in\mathbb{R}^{+}\ ,\ t_{i}<t_{i+1})$ such that $R$ has $s-1$
different critical values. Let us define

(12) $\displaystyle g(x)=\prod_{i=1}^{a+1}(x-s_{i})$ $\displaystyle where$
$\displaystyle a>1$

and meet the following conditions

  1. 1.

$s_{i}$’s are positive real numbers and $s_{i}\neq s_{j}$ ,

  2. 2.

The function $g$ has different critical values and also $R(x)=s_{i}$ has $n$
real roots. $s_{i}$s are in an interval $I$ such that $g^{-1}(I)$ is a union
of $n$ intervals.

  3. 3.

The function $g\circ R(x)$ is a Morse function.

######  Notation 4.1.

Let us denote by $C\cup\tilde{C}$ the set of critical values of $g\circ R$
where $C$ is the set of the critical values of $g$, and $\tilde{C}$ is the
image of the set of the critical points of $R$ under $g\circ R$. All the
critical points of $g$ and $R$ are real. Therefore,
$C=\\{c_{i}=g(p_{i})|p_{i}\in V(g_{x})\ ,\ p_{1}<p_{2}<\dots<p_{a}\\}$ when
$n$ is odd, and $C=\\{c_{a+1-i}=g(p_{i})|p_{i}\in V(g_{x})\ ,\
p_{1}<p_{2}<\dots<p_{a}\\}$ when $n$ is even. Also the order of $\tilde{C}$ is
as usual $\\{\tilde{c}_{a+j}=g\circ R(q_{j})\ |\ q_{j}\in V(R_{x})\ ,\
q_{1}<q_{2}<\dots<q_{n-1}\\}$.

Take the distinguished system of paths related to the function $g\circ R$ such
that all the paths are in the upper half plane. Let $\gamma_{c}$ be the
vanishing cycle along the path $\lambda_{c}$ of the fiber $g^{-1}(0)$.
Therefore, $R_{*}^{-1}(\gamma_{c})=\\{\delta_{c}^{i}|i=1,\dots,n\\}$ is the
set of vanishing cycles along the path $\lambda_{c}$ of the fiber $(g\circ
R)^{-1}(0)$.

######  Theorem 4.3.

If $b=0$, then the zero homology group of $(g\circ R)^{-1}(b)$ is generated by

$\delta_{c}^{i},\delta_{\tilde{c}}\ \ where\ \ c\in C\ ,\
\tilde{c}\in\tilde{C}\ ,\ i=1,\dots,n.$

######  Theorem 4.4.

For the regular value point b=0 of $g\circ R$, the intersection matrix of
$H_{0}((g\circ R)^{-1}(b),\mathbb{Z})$ with respect to the basis in Theorem
4.3 is

$<\delta_{i}^{j}\ ,\ \delta_{i^{\prime}}^{j^{\prime}}>=\begin{cases}2&\text{if
}{(i=i^{\prime},j=j^{\prime})}\\\ -1&\text{if
}\begin{cases}(j=j^{\prime},i^{\prime}=i+1,i<a)\\\
\vee({j^{\prime}}=0,i=a,{i^{\prime}}=a+2k-1,j=k+(\frac{(1+(-1)^{n})}{2}))\\\
\vee(j^{\prime}=0,i=1,i^{\prime}=a+2k,j=2k+(\frac{(1+(-1)^{n+1})}{2}))\end{cases}\\\
1&\text{if
}\begin{cases}(j^{\prime}=0,i=a,i^{\prime}=a+2k-1,j=k+(\frac{1+(-1)^{n+1}}{2}))\\\
\vee(j^{\prime}=0,i=1,i^{\prime}=a+2k,j=2k+(\frac{1+(-1)^{n}}{2}))\end{cases}\\\
0&\text{if }\begin{cases}(j=j^{\prime},|i^{\prime}-i|>1)\vee(j=j^{\prime}\ ,\
i,i^{\prime}>a,i\neq i^{\prime})\\\ \vee(i=i^{\prime},j\neq
j^{\prime}).\end{cases}\end{cases}$

by $\delta_{i}^{j}$ and $\delta_{a+k}^{0}$ we mean $\delta_{c_{i}}^{j}$ and
$\delta_{\tilde{c}_{a+k}}$ respectively, and $k\in\mathbb{N}$.

###### Proof.

Without loss of generality we can assume that $n$ is odd. Consider the
vanishing cycle $\delta_{\tilde{c}_{a+2k-1}}:=[r]-[r^{\prime}]$ , where
$r,r^{\prime}$ are two consecutive roots of $g\circ R$. Thus
$R(r)=R(r^{\prime})=s_{a+1}$ is a root of the function $g$. Let
$s_{a},s_{a+1}$ be two consecutive roots of $g$ and also let
$R^{-1}(s_{a})=\\{l_{j}|l_{1}<l_{2}<\dots<l_{n}\\}$ such that
$l_{k}<r^{\prime}<r<l_{k+1}$ be four consecutive roots of $g\circ R$. Two
vanishing cycles on that basis are $\delta_{a}^{2k-1}=[r^{\prime}]-[l_{k}]\ $
and $\ \delta_{a}^{2k}=[r]-[l_{k+1}]$; therefore,

$<\delta_{a+2k-1},\delta_{a}^{2k-1}>=-<\delta_{a+2k-1},\delta_{a}^{2k}>=-1$
$<[r]-[r^{\prime}],[r^{\prime}]-[l_{k}]>=-<[r]-[r^{\prime}],[r]-[l_{k+1}]>=-1.$

For the vanishing cycle $\delta_{a+2k}:=[r_{2}]-[r_{1}]$ where $r_{1},r_{2}$
are two consecutive roots of $g\circ R$ we have $R_{*}(\delta_{a+2k})=0$ and
$R(r_{1})=R(r_{2})=s_{1}$ where $s_{1}$ is a root of $g$. For the root $s_{2}$
of $g$ let $R^{-1}(s_{2})=\\{l_{j}|l_{1}<l_{2}<\dots<l_{n}\\}$, so we have
$l_{2k}<r_{1}<r_{2}<l_{2k+1}$ are consecutive roots of $g\circ R$. Therefore,
$\delta_{1}^{2k}=[r_{1}]-[l_{2k}]$ and $\delta_{1}^{2k+1}=[l_{2k+1}]-[r_{2}]$
are vanishing cycles and they are in the basis so we have

$<\delta_{a+2k},\delta_{1}^{2k}>=-<\delta_{a+2k},\delta_{1}^{2k+1}>=1$
$<[r_{2}]-[r_{1}],[l_{2k}]-[r_{1}]>=-<[r_{2}]-[r_{1}],[l_{2k+1}]-[r_{2}]>=1.$

Also, the above procedure can work for an even number $n$ but only by changing
the order of $C$. The function $R$ induces the surjective morphism

$R_{*}:H_{0}((g\circ R)^{-1}(b),\mathbb{Z})\rightarrow
H_{0}(g^{-1}(b),\mathbb{Z}).$

If $\gamma_{c_{i}},\gamma_{c_{i+1}}\in H_{0}(g^{-1}(b),\mathbb{Z})$ are two
vanishing cycles where $c_{i},c_{i+1}\in C$, then each set
$R_{*}^{-1}(\gamma_{c_{m}})=\\{\delta_{c_{m}}^{j}|j=1,2,\dots,n\\}$ contains
$n$ separated vanishing cycles where $m=i,i+1$. For each cycle
$\delta_{c_{i}}^{j}$ there is exactly one cycle $\delta_{c_{i+1}}^{j}$ such
that $<\delta_{c_{i}}^{j},\delta_{c_{i+1}}^{j}>=-1$. By definition of the
functions $R$ and $g$ and also Theorem 4.2, we can have the other equalities.
∎

######  Definition 4.5.

The Dynkin diagram of a Morse polynomial with different critical values is a
graph defined in the following way: Its vertices are in one-to-one
correspondence with a distinguished basis of vanishing cycles $\delta_{i}$,
$i=1,2,...,\mu=d-1$. The i-th and j-th vertices of the graph are joined with
an edge of multiplicity $<\delta_{i},\delta_{j}>$. The intersection indexes
$(-1)^{n}$ are depicted by dash lines, where $n$ is the dimension of vanishing
cycles.

For an illustration of the Dynkin diagram $g\circ R$ with respect to this
basis ( when $s$ is odd) see the figure 1.

Figure 1: Dynkin diagram of pull-back when $n$ is odd.

######  Lemma 4.2.

The action of monodromy group $\pi_{1}(\mathbb{C}\setminus C\cup\tilde{C},b)$
on a tangency vanishing cycle generates all

$\delta_{\tilde{c}}\ ,\ \tilde{c}\in\tilde{C}\ \ and\ \
\delta_{c}^{i}-\delta_{c}^{j}\ \ where\ \ c\in C\ ,\ i,j=1,\dots,n.$

###### Proof.

Each tangency vanishing cycle $\delta_{\tilde{c}_{a+2i-1}}$ (resp.
$\delta_{\tilde{c}_{a+2i}}$ ) has an intersection with two vanishing cycles
$\delta_{c_{a}}^{2i-1},\delta_{c_{a}}^{2i}$ (resp.
$\delta_{c_{1}}^{2i},\delta_{c_{1}}^{2i+1}$ ) with different signs. By using
Picard-Lefschetz formula, the action of monodromy $\lambda_{c_{a}}$ (resp.
$\lambda_{c_{1}}$) on $\delta_{\tilde{c}_{a+2i-1}}$ (resp.
$\delta_{\tilde{c}_{a+2i}}$ ) generates
$\delta_{c_{a}}^{2i-1}-\delta_{c_{a}}^{2i}$ (resp.
$\delta_{c_{1}}^{2i}-\delta_{c_{1}}^{2i+1}$ ). According to Lemma 4.1, the
action of the monodromy group $\pi:=<\lambda_{c}|\ c\in
C>\subset\pi_{1}(\mathbb{C}\setminus((C\cup\tilde{C}),b)$ on
$R_{*}(\delta_{c}^{j})$ (for all $c\in C$ and $j$) generates zero homology
group $H_{0}(g^{-1}(b),\mathbb{Z})$. Therefore, for a fixed $j$, the action of
$\pi$ on $\delta_{c_{a}}^{j}$ or $\delta_{c_{1}}^{j}$ can generate all
$\delta_{c}^{j}$. In other words, the action $\pi$ on
$\delta_{c_{a}}^{2i}-\delta_{c_{a}}^{2i-1}$ (resp.
$\delta_{c_{1}}^{2i+1}-\delta_{c_{1}}^{2i}$ ) generates all
$\delta_{c}^{2i}-\delta_{c}^{2i-1}$ (resp.
$\delta_{c}^{2i+1}-\delta_{c}^{2i}$). Since Dynkin diagram is connected, the
action of monodromy $\lambda_{\tilde{c}_{a+2i}}$
($\lambda_{\tilde{c}_{a+2i+1}}$) on $\delta_{c_{1}}^{2i}$ can generate
$\delta_{\tilde{c}_{a+2i}}$ (resp. $\delta_{\tilde{c}_{a+2i+1}}$). By
repeating this procedure we can generate all $\delta_{\tilde{c}}$ because the
degree of each vertex of the Dynkin diagram is at most 2. Since the number of
tangency vanishing cycles is $n=deg(R)$ we can generate independent cycles
$\delta_{c}^{i+1}-\delta_{c}^{i}$, where $i=1,\dots,n-1$. Therefore, these
cycles can generate all $\delta_{c}^{j^{\prime}}-\delta_{c}^{j}$. ∎

######  Corollary 4.1.

The morphism $R_{*}:H_{0}((g\circ R)^{-1}(b),\mathbb{Z})\rightarrow
H_{0}(g^{-1}(b),\mathbb{Z})$ is surjective and

$Ker(R_{*})=<\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b).\delta_{\tilde{c}}>,$

where $<\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b).\delta_{\tilde{c}}>$ is
the group generated by the action of the monodromy group on the tangency
vanishing cycle $\delta_{\tilde{c}}$.

######  Proposition 4.1.

The group generated by the action of the monodromy group
$\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b)$ on a vanishing cycle
$\delta_{c}^{j}\in R_{*}^{-1}(\gamma_{c})$, where $\gamma_{c}\in
H_{0}(g^{-1}(b),\mathbb{Z})$, is equal to $H_{0}((g\circ
R)^{-1}(b),\mathbb{Z})$.

###  4.2 Direct Sum of Polynomials

Let $F$ and $f$ be the functions as in 2 and 4. We are going to study the
topology of a regular fiber of $f\circ F$.

######  Notation 4.2.

We denote by $C_{1}$ (resp. $C_{2}$) the set of critical values of $g$ (resp.
$h$), and also denote by $\tilde{C}_{1}$ (resp. $\tilde{C}_{2}$) the set of
the image of the critical points of $R$ (resp. $S$) under $g\circ R$ (resp.
$h\circ S$). Thus $C_{1}\cup\tilde{C}_{1}$ and $C_{2}\cup\tilde{C}_{2}$ are
the set of critical values of $g\circ R$ and $h\circ S$ respectively. Without
loss of generality, we can suppose that
$(C_{1}\cup\tilde{C}_{1})\cap(C_{2}\cup\tilde{C}_{2})=\emptyset$.

We take two systems of distinguished paths $\lambda_{c}$ relative to the
functions $g\circ R$ and $h\circ S$, where
$c\in(C_{1}\cup\tilde{C}_{1})\cup(C_{2}\cup\tilde{C}_{2})$ and $\lambda_{c}$
starts from $b=0$ and ends at $c$: see Figure (3). Note that for the function
$h\circ S$ we choose a distinguished system of paths such that all of the
paths are in the lower half plane, and they preserve the order of
$C_{2}\cup\tilde{C}_{2}$ as in Notation 4.1. Let $\delta\in H_{0}((g\circ
R)^{-1}(0),\mathbb{Z})$ and $\gamma\in H_{0}((h\circ S)^{-1}(0),\mathbb{Z})$
be two vanishing cycles along the paths $\lambda_{c}$ and $\lambda_{a}$
respectively. Let $t_{s}:[0,1]\rightarrow\mathbb{C}$ be a path defined by

$t_{s}:=\begin{cases}\lambda_{c}(1-2s)&0\leq s\leq\frac{1}{2}\ ,\
c\in(C_{1}\cup\tilde{C}_{1})\\\ \lambda_{a}(2s-1)&\frac{1}{2}\leq s\leq 1\ ,\
a\in(C_{2}\cup\tilde{C}_{2}).\end{cases}$

The cycle $\delta$ vanishes along $t_{.}^{-1}$ when $s$ tends to zero and
$\gamma$ vanishes along $t_{.}$ when s tends to 1.

Figure 2: A distinguished system of paths where $c_{i}\in C_{1},a_{i}\in
C_{2}$ and $\tilde{c}_{i}\in\tilde{C}_{1},\tilde{a}_{i}\in\tilde{C}_{2}$

######  Definition 4.6.

The cycle

$\delta*\gamma\cong\delta*_{t_{.}}\gamma:=\cup_{s\in[0,1]}\delta_{t_{s}}\times\gamma_{t_{s}}\in
H_{1}((f\circ F)^{-1}(0),\mathbb{Z})$

is an oriented cycle. Note that its orientation changes when the direction of
path $t_{.}$ is changed. The triple
$(t_{s},\delta,\gamma)=(t_{s},\delta_{t_{.}},\gamma_{t_{.}})$ is called an
admissible triple.

Figure 3: Joining of vanishing cycles

Let

$\delta_{c_{i}}^{j}\ ,\ \delta_{\tilde{c}_{a+k}}\in H_{0}((g\circ
R)^{-1}(b),\mathbb{Z})\ \ where\ \ i=1,\dots,a\ ,\ j=1,\dots,n\ ,\
k=1\dots,n-1,$

and

$\gamma_{c_{i}}^{j}\ ,\ \gamma_{\tilde{c}_{a+k}}\in H_{0}((h\circ
S)^{-1}(b),\mathbb{Z})\ \ where\ \ i=1,\dots,a\ ,\ j=1,\dots,n\ ,\
k=1\dots,n-1$

be the corresponding distinguished basis of vanishing cycles.

######  Theorem 4.5.

The $\mathbb{Z}$-module $H_{1}((f\circ F)^{-1}(0),\mathbb{Z})$ is free and is
generated by

$\alpha:=\delta*\gamma\ \ \ where\ \ \ \delta\in H_{0}((g\circ
R)^{-1}(b),\mathbb{Z})\ \ ,\ \ \gamma\in H_{0}((h\circ
S)^{-1}(b),\mathbb{Z}),$

and where we have taken the admissible triples

$(\lambda_{c}.\lambda_{a}^{-1},\delta,\gamma)\ here\ c\in
C_{1}\cup\tilde{C}_{1}\ \ and\ \ a\in C_{2}\cup\tilde{C}_{2}.$

See e.g. ([13] Chapter 7).
Take $h\circ S=b^{\prime}-(h^{\prime}\circ S^{\prime})$, where $b^{\prime}$ is
a fixed complex number and $h^{\prime}\circ S^{\prime}$ is a perturbation of
$h\circ S$. The set of critical values of $h^{\prime}\circ S^{\prime}$ is
denoted by $(C_{2}^{\prime}\cup\tilde{C}_{2}^{\prime})$ and therefore the set
of critical values of $h\circ S$ is
$C_{2}\cup\tilde{C}_{2}=b^{\prime}-(C_{2}^{\prime}\cup\tilde{C}_{2}^{\prime})$.
We define $(f\circ F)(x,y):=g\circ R(x)+h^{\prime}\circ S^{\prime}(y)$. Assume
that
$(C_{1}\cup\tilde{C}_{1})\cap(C_{2}^{\prime}\cup\tilde{C}_{2}^{\prime})=\emptyset$,
and since the set of critical values of $f\circ F$ is
$(C_{1}\cup\tilde{C}_{1})+(C_{2}^{\prime}\cup\tilde{C}_{2}^{\prime})$ then
$b^{\prime}$ is a regular value of $f\circ F$. Let $(t_{s},\delta,\gamma)$ be
an admissible triple where $t_{s}$ starts from $c$ and ends at
$b^{\prime}-a^{\prime}$ (here $c\in C_{1}\cup\tilde{C}_{1}$ and $a^{\prime}\in
C_{2}^{\prime}\cup\tilde{C}_{2}^{\prime}$). Therefore, the path
$t_{.}+a^{\prime}$ starts from $c+a^{\prime}$ ands end at $b^{\prime}$. For
instance, see Figure 4:

Figure 4: A distinguished system of paths

######  Proposition 4.2.

The topological cycle $\delta*\gamma$ is a vanishing cycle along the path
$t_{.}+a^{\prime}$ with respect to fibration $f\circ F=t$.

See e.g. ([13] Chapter 7).

######  Definition 4.7.

  1. 1.

A vanishing cycle around the critical point $p$ where $p\in F^{-1}(Sing(f))$
is called a pull-back vanishing cycle.

  2. 2.

A vanishing cycle around a tangency critical point is called a tangency
vanishing cycle.

  3. 3.

A vanishing cycle around a critical point $p$ where $p\in V(R_{x})\cap
V(S_{x})$ is called an exceptional vanishing cycle.

For simplicity we denote by $\delta_{i}^{j}$ the cycle $\delta_{c_{i}}^{j}$
where $c_{i}\in C_{1}$, $i=1,\dots,a$ and $j=1,\dots,n$ (resp. by
$\gamma_{i}^{j}$ the cycle $\gamma_{a_{i}}^{j}$ where $a_{i}\in C_{1}$,
$i=1,\dots,a$ and $j=1,\dots,n$). Also, we denote by $\delta_{k}$ the cycle
$\delta_{\tilde{c}_{k}}$ where $k=a+1,\dots,a+(n-1)$ (resp. by $\gamma_{k}$
the cycle $\gamma_{\tilde{c}_{k}}$ where $k=a+1,\dots,a+(n-1)$).

######  Theorem 4.6.

Let $b=0$ be the regular value of the function $f$. Let $\delta_{i}$ where
$i=1,\dots,a$ be the distinguished set of vanishing cycles in
$H_{0}(g^{-1}(b),\mathbb{Z})$ and also, let $\gamma_{j}$ where $j=1,\dots,a$
be the distinguished set of vanishing cycles in $H_{0}(h^{-1}(b),\mathbb{Z})$.
Therefore, the intersection matrix of $H_{1}(f^{-1}(b),\mathbb{Z})$ in the
basis

$\delta_{i}*\gamma_{j}\ \ \ where\ \ \ \ i,j=1,2,\dots,a,$

is of the form

$<\delta_{i}*\gamma_{j}\ ,\ \delta_{l}*\gamma_{k}>=$
$\begin{cases}(-1)^{a}&\text{if}\begin{cases}(i=l,k=j+1,j=odd)\vee\\\
(j=k,l=i+1,i=even)\vee\\\ (i=odd,j=even,l=i+1\ or\ l=i-1,k=j+1)\end{cases}\\\
(-1)^{a+1}&\text{if}\begin{cases}(i=l,k=j+1,j=even)\vee\\\
(j=k,l=i+1,i=odd)\vee\\\ (i=even,j=odd,l=i-1\ or\ i+1,k=l+1)\end{cases}\\\
0&otherwise.\end{cases}$

See e.g. ([13] Chapter 7), and [1].
The Dynkin diagram of $f$, when $a$ is even, is shown in Figure 5.

Figure 5: Dynkin diagram

In Figure 5, according to the distinguished set of paths, the paths such as
$t_{i,j}:=\lambda_{j}.\lambda_{i}^{-1}$ have transversal intersection with
$t_{i,j+1}=\lambda_{j+1}.\lambda_{i}^{-1}$ (resp.
$t_{i+1,j}:=\lambda_{j}.\lambda_{i+1}^{-1}$) at the point $b=0$, and
$d(t_{ij})\wedge d(t_{i,j+1})=-d(t_{i,j+1})\wedge d(t_{i,j+2})$ (resp.
$d(t_{ij})\wedge d(t_{i+1,j})=-d(t_{i+1,j})\wedge d(t_{i+2,j})$).

######  Theorem 4.7.

For the regular value $b=0$ of $f\circ F=g(R)+h(S)$, we choose a distinguished
set of vanishing cycles $\delta_{i}^{j},\ \delta_{k}$ where
$i=1,2,\dots,a,j=1,\dots,n$ and $k=a+1,\dots,a+(n-1)$ (resp. $\gamma_{i}^{j}\
,\ \gamma_{k}$ where $i=1,2,\dots,a,j=1,\dots,n$ and $k=a+1,\dots,a+(n-1)$) in
$H_{0}((g\circ R)^{-1}(b),\mathbb{Z})$, (resp. $H_{0}((h\circ
S)^{-1}(b),\mathbb{Z})$). The intersection matrix in this basis

$\delta_{i}^{j}*\gamma_{i^{\prime}}^{j^{\prime}}\ ,\
\delta_{k}*\gamma_{i}^{j}\ ,\ \delta_{i}^{j}*\gamma_{k}\ ,\
\delta_{k}*\gamma_{k^{\prime}}$

where

$i,i^{\prime}=1,\dots,a\ ,\ j,j^{\prime}=1,\dots,n\ ,\
k,k^{\prime}=a+1,\dots,a+(n-1),$

of $H_{1}((f\circ F)^{-1}(0),\mathbb{Z})$ is given by

$<\delta_{i}^{m}*\gamma_{j}^{s},\delta_{l}^{m^{\prime}}*\gamma_{k}^{s^{\prime}}>=<\delta_{i}^{1}*\gamma_{j}^{1},\delta_{l}^{1}*\gamma_{k}^{1}>,\
\ \ for\ \ m=m^{\prime}\ ,\ s=s^{\prime},1\leq i,j,k,l\leq a,$

where

$<\delta_{i}^{1}*\gamma_{j}^{1},\delta_{l}^{1}*\gamma_{k}^{1}>=(-1)^{n+1}<R(\delta_{i}^{1})*S(\gamma_{j}^{1}),R(\delta_{l}^{1})*S(\gamma_{k}^{1})>.$

Here, the intersection can be explained by using Proposition 4.6 and

${\left\\{\begin{array}[]{lll}<\delta_{a+i}^{0}*\gamma_{j}^{1},\delta_{a+i}^{0}*\gamma_{k}^{s^{\prime}}>=<\delta_{a}^{s}*\gamma_{j}^{s},\delta_{a}^{s}*\gamma_{k}^{s^{\prime}}>\\\
<\delta_{i}^{m}*\gamma_{a+j}^{0},\delta_{l}^{m^{\prime}}*\gamma_{a+j}^{0}>=<\delta_{i}^{m}*\gamma_{a}^{m},\delta_{l}^{m^{\prime}}*\gamma_{a}^{m}>,\end{array}\right.}$

for the others we can use

$<\delta_{i}^{m}*\gamma_{a}^{s},\delta_{i}^{m}*\gamma_{a+s}^{0}>=<\delta_{i}^{m}*\gamma_{a+s}^{0},\delta_{i}^{m}*\gamma_{a}^{s+1}>=$
$\begin{cases}-1&\text{if }\begin{cases}(n=2n^{\prime}+1\ ,\ s=2t+1)\vee\\\
(n=2n^{\prime}\ ,\ a=2a^{\prime}+1,\ s=2t+1)\end{cases}\\\ 1&\text{if
}(n=2n^{\prime}\ ,\ a=2a^{\prime}+1\ ,\ s=2t+1),\end{cases}$
$<\delta_{1}^{m}*\gamma_{j}^{s},\delta_{a+m}^{0}*\gamma_{j}^{s}>=<\delta_{a+m}^{0}*\gamma_{j}^{s},\delta_{1}^{m+1}*\gamma_{j}^{s}>=$
$\begin{cases}1&\text{if }\begin{cases}(n=2n^{\prime}+1\ ,\ m=2t+1)\vee\\\
(n=2n^{\prime}\ ,\ a=2a^{\prime}+1,\ m=2t+1)\end{cases}\\\ -1&\text{if
}(n=2n^{\prime}\ ,\ a=2a^{\prime}+1\ ,\ m=2t+1).\end{cases}$

See e.g. ([13] Chapter 7).
Since the dimension of $(f\circ F)^{-1}(b)$ is one, for the two vanishing
cycles $\alpha,\beta\in H_{1}((f\circ F)^{-1}(b),\mathbb{Z})$, we have
$<\alpha,\beta>=-<\beta,\alpha>$ and $<\alpha,\alpha>=0$, i.e. the
intersection matrix is skew-symmetric.
The Dynkin diagram of $f\circ F$ when $n$ is odd and a=3 is shown in Figure 6.

Figure 6: Dynkin diagram of $f\circ F$, when $n$ is odd and a=3

In Figure 6, black vertices correspond to the tangency vanishing cycles,
squires vertices correspond to the exceptional vanishing cycles and all the
other vertices correspond to pull-back vanishing cycles. The white cycle
vertices correspond to some of the pull-back vanishing cycles with the same
image under $F_{*}$. The direction of the intersections are to be considered
from left to right and top to bottom in this figure.

######  Definition 4.8.

An isomorphism of graphs $G$ and $H$ is a bijection between the vertex sets of
$G$ and $H$

$f:V(G)\rightarrow V(H),$

such that any two vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only
if $f(u)$ and $f(v)$ are adjacent in H.

Let us denote by H (resp. G), the Dynkin diagram of $f\circ F$ (resp. $f$)
with respect to the distinguished set of vanishing cycles related to the
critical points of $f\circ F$ (resp. $f$). We consider the group generated by
the action of the monodromy group $\pi:=<\lambda_{c}|c\in
C_{1}+C_{2}>\subset\pi_{1}(\mathbb{C}\setminus((C_{1}\cup
C_{2})+(\tilde{C}_{1}\cup\tilde{C}_{2})),b)$ on a pull-back vanishing cycle
$\delta_{c}^{i}*\gamma_{a}^{j}$. We know that this group is generated by some
pull-back vanishing cycles, so it introduces a sub-graph of $H$ which is
denoted by $G_{ij}$.

For each $i,j=1,\dots,n$ the graph $G_{ij}$ is isomorphic to the graph $G$ .
Therefore, if we remove the vertices corresponding to the tangency and
exceptional vanishing cycles, then $H$ is divided into $n^{2}$ graph $G_{ij}$.

######  Definition 4.9.

The cycle $\delta$ in a regular fiber $f^{-1}(b)$ is called simple if the
homology group $H_{1}(f^{-1}(b),\mathbb{Z})$ is generated by the action of
monodromy group $\pi_{1}(\mathbb{C}\setminus C,b)$ on $\delta$ (where $C$ is
the set of critical values of $f$).

######  Theorem 4.8.

Each vanishing cycle (respective to the distinguished set of paths related to
the critical values) in a regular fiber of $f$ is simple.

See e.g. [14].

######  Theorem 4.9.

For the regular value $b=0$ of the Morse function $f\circ F=g\circ R+h\circ S$
(a composition of the functions which were defined in (2) and (4)), the action
of the monodromy group on a tangency vanishing cycle generates

$\delta_{\tilde{c}}*\gamma_{a}\ \ ,\ \ \delta_{c}*\gamma_{\tilde{a}}\ \ ,\ \
\delta_{\tilde{c}}*\gamma_{\tilde{a}}\ \ and\ \
\delta_{c}^{i}*\gamma_{a}^{j}-\delta_{c}^{i^{\prime}}*\gamma_{a}^{j^{\prime}},$

where $\tilde{c}\in\tilde{C}_{1}\ ,\ c\in C_{1}\ ,\ a\in C_{2},\
\tilde{a}\in\tilde{C}_{2}$ and $\ i,j,i^{\prime},j^{\prime}=1,2,\dots,n$.

###### Proof.

  1. 1.

Let $L$ be a line of $D$. Each vanishing cycle around a critical point in $D$
is tangency or exceptional. Consider $f\circ F|_{L}-l$, when restriction of
the functions $g\circ R$ or $h\circ S$ under $L$ are the constant value $l$.
If $L=\\{y=cons\\}$ (resp. $L=\\{x=cons\\}$), then the cycles around the
critical point in $L$ correspond to zero vanishing cycle $g\circ R$ (resp.
$h\circ S$). Furthermore, a tangency vanishing cycle corresponds to a pull-
back (zero) vanishing cycle, so by Proposition 4.1, the action of monodromy
group on that cycle generates all the other vanishing cycles. This implies
that, this action on our tangency vanishing cycle will generate the tangency
and exceptional vanishing cycles around the critical points in $L$. The
algebraic set $D$ consists of $n$ lines parallel with $x,y$ axes, so
$V(R_{x})\cap V(S_{y})\neq\emptyset$. Since the critical point in two vertical
lines of $D$ have different values except the point in the intersection, then
the action of monodromy on a tangency vanishing cycle around a point of these
lines can generate all the other vanishing cycles of these lines. In other
words, this action on a tangency vanishing cycle can generate all the tangency
and exceptional vanishing cycles.

  2. 2.

If we remove the vertices of $H$ that correspond to the tangency and
exceptional vanishing cycles, then the new graph contains $n^{2}$ sub-graphs
$G_{i,j}$ where $1\leq i,j\leq n$. The tangency vertex
$\delta_{t}^{i}*\gamma_{a+j}^{0}$ (resp. $\delta_{a+i}^{0}*\gamma_{t}^{j}$)
connects the sub-graphs $G_{i,j}$ to $G_{i,j+1}$(resp.$G_{i,j}$ to
$G_{i+1,j}$). In general $\delta_{t}^{i}*\gamma_{a+j}^{0}$ connects
$\delta_{t}^{i}*\gamma_{a}^{j}$ to $\delta_{t}^{i}*\gamma_{a}^{j+1}$ when $j$
is odd and also, it connects $\delta_{t}^{i}*\gamma_{1}^{j}$ to
$\delta_{t}^{i}*\gamma_{1}^{j+1}$ when $j$ is even. According to Picard-
Lefschetz formula the action of monodromy $\lambda_{t,a}$ (resp.
$\lambda_{t,1}$) around the critical point $t+a$ (resp. $t+1$) when $j$ is odd
(resp. $j$ is even) on the tangency vanishing cycle
$\delta_{t}^{i}*\gamma_{a+j}^{0}$ is as follows:

$\displaystyle\lambda_{t,a}(\delta_{t}^{i}*\gamma_{a+j}^{0})$ $\displaystyle=$
$\displaystyle\delta_{t}^{i}*\gamma_{a+j}^{0}-\sum_{\alpha\in
F_{*}^{-1}(\delta_{t}*\gamma_{a})}<\alpha,\delta_{t}^{i}*\gamma_{a+j}^{0}>\alpha$
$\displaystyle=$
$\displaystyle\delta_{t}^{i}*\gamma_{a+j}^{0}-(\delta_{t}^{i}*\gamma_{a}^{j}-\delta_{t}^{i}*\gamma_{a}^{j+1}),$

when $j$ is even, the action of monodromy $\lambda_{t,1}$ on
$\delta_{t}^{i}*\gamma_{a+j}^{0}$ generates
$\delta_{t}^{i}*\gamma_{1}^{j}-\delta_{t}^{i}*\gamma_{1}^{j+1}$. According to
Theorem 4.8, the action of the monodromy group $\pi$ on
$\delta_{t}^{i}*\gamma_{a}^{j}$ generates corresponding vanishing cycles to
the vertices in $G_{i,j}$. Therefore, the action of the monodromy group $\pi$
on the tangency vanishing cycle $\delta_{t}^{i}*\gamma_{a+j}^{0}$ generates
the whole vanishing cycle
$\delta_{l}^{i}*\gamma_{k}^{j}-\delta_{l}^{i}*\gamma_{k}^{j+1}$ where $1\leq
l,k\leq a$. By the same process the action of the monodromy group $\pi$ on the
tangency vanishing cycle $\delta_{a+i}^{0}*\gamma_{t}^{j}$ generates the whole
vanishing cycle
$\delta_{l}^{i}*\gamma_{k}^{j}-\delta_{l}^{i+1}*\gamma_{k}^{j}$ where $1\leq
l,k\leq a$.

In general we can generate

$\bigtriangledown:=\\{\delta_{c}^{i+1}*\gamma_{a}^{j}-\delta_{c}^{i}*\gamma_{a}^{j},\delta_{c}^{i}*\gamma_{a}^{j+1}-\delta_{c}^{i}*\gamma_{a}^{j}|i,j=1,\dots,n-1,c\in
C_{1},a\in C_{2}\\}$

which is a basis for the group generated by

$\delta_{c}^{i}*\gamma_{a}^{j}-\delta_{c}^{i^{\prime}}*\gamma_{a}^{j^{\prime}}\
\ \forall c\in C_{1},\forall a\in C_{2},\ \
i,j,i^{\prime},j^{\prime}=1,\dots,n.$

∎

######  Corollary 4.2.

The group which is generated by the action of the monodromy group on a
tangency vanishing cycle can also be generated by

$\delta_{\tilde{c}}*\gamma_{a}\ \ ,\ \ \delta_{c}*\gamma_{\tilde{a}}\ \ ,\ \
\delta_{\tilde{c}}*\gamma_{\tilde{a}}\ \ and\ \
\delta_{c}^{i}*\gamma_{a}^{j}-\delta_{c}^{i^{\prime}}*\gamma_{a}^{j^{\prime}}$

where $\tilde{c}\in\tilde{C}_{1}\ ,\ c\in C_{1}\ ,\ a\in C_{2},\
\tilde{a}\in\tilde{C}_{2}$ and $\ i,j,i^{\prime},j^{\prime}=1,2,\dots,n$.

######  Theorem 4.10.

Let $F:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ be defined by
$(x,y)\rightarrow(R,S)$ as in (2) and let $f$ be a polynomial as in (4). The
linear map

$F_{*}:H_{1}((f\circ F)^{-1}(b),\mathbb{Z})\rightarrow
H_{1}((f)^{-1}(b),\mathbb{Z})$

is surjective, and $ker(F_{*})$ is generated by the action of monodromy group
$\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b)$ on a tangency vanishing
$\delta$.

###### Proof.

It is clear that for each $c\in C$ the cycles $\delta_{c}^{i}-\delta_{c}^{j}$
belong to $ker(F_{*})$ where $\delta_{c}^{i},\delta_{c}^{j}$ are the pull-back
vanishing cycles around the singularities with value $c$. Each tangency
vanishing cycle $\delta_{t}$ is divided into two paths with homotopic images
under $F$ by $D$. Thus, $\delta_{t}\in ker(F_{*})$. Each exceptional vanishing
cycle $\delta_{e}$ is divided into 4 paths by $D$. The images of those 4 paths
under $F$ are homotopic, hence $F_{*}(\delta_{e})=0$. By using the Corollary
4.2 we conclude that
$<\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b).\delta_{t}>\subset
ker(F_{*})$. It’s obvious that the morphism $F_{*}$ is surjective and so

(13) $\begin{split}null(F_{*})&=\\#(V((g\circ R)_{x})\cap V((h\circ
S)_{y}))-\\#(V(g_{x})\cap V(h_{y}))\\\ &=(na+n-1)^{2}-a^{2}.\end{split}$

If $F_{*}^{-1}(\gamma_{c})=\\{\delta_{c}^{i}|i=1,\dots,n^{2}\\}$, where
$\gamma_{c}\in H_{1}(f^{-1}(b),\mathbb{Z})$, and since all the elements in the
set $\triangle_{c}=:\\{\delta_{c}^{i+1}-\delta_{c}^{i}|i=1,\dots,n^{2}-1\\}$
are independent, then its elements can generate all
$\delta_{c}^{i}-\delta_{c}^{j}$ for all $i,j=1,\dots,n^{2}$. The number of all
the elements of the set $\cup_{c\in C}\triangle_{c}$ is $a^{2}(n^{2}-1)$. The
tangency and exceptional vanishing cycles, which are in $ker(F_{*})$, are
independent elements and their number is $2na(n-1)+(n-1)^{2}$. Therefore, by
considering the equality (13) we have
$<\pi_{1}(\mathbb{C}\setminus(C\cup\tilde{C}),b).\delta_{t}>=ker(F_{*})$. ∎

Acknowledgment: I would like to express my gratitude to my advisor Hossein
Movasati, whose contributions, stimulating suggestion and encouragement have
helped me to coordinate my Ph.D. thesis especially in regards to writing this
paper.

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Yadollah Zare

Instituto Nacional de Matemática Pura e Aplicada-IMPA

Rio de Janeiro

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>