# Flat structure on the space of isomonodromic deformations

Mitsuo Kato, Toshiyuki Mano and Jiro Sekiguchi

###### Abstract

The WDVV equation was found by physicists in 2D topological field theory. B.
Dubrovin gave a geometric interpretation for the WDVV equation by introducing
the notion of Frobenius manifold. Moreover Dubrovin derived a kind of
isomonodromic deformations of linear differential equations from semisimple
(massive) Frobenius manifolds. In the three dimensional case, the WDVV
equation is equivalent to a one parameter family of the sixth Painlevé
equation. The aim of the present paper is to investigate a generalization of
the WDVV equation so that the generalized equation is equivalent to the
isomonodromic deformation of generic linear differential equations of Okubo
type. As a consequence we show the existence of a “flat coordinate system”
associated to the extended WDVV equation. Application of our formulation to
the case of finite complex reflection groups implies the existence of a “flat
generator system” of a well-generated complex reflection group. This result is
an analogue of that of K. Saito for the case of finite real reflection groups.
AMS 2010 Subject Classification: 34M56, 33E17, 35N10, 32S25.
Key words: flat structure, WDVV equation, complex reflection group, Painlevé
equation.

##  1 Introduction

The aim of this article is to formulate a generalization of the flat structure
and to give its application. We start this introduction with explaining a
historical background of the notions of flat structure and Frobenius manifold
(cf. [10],[14],[30]). At the end of 1970’s, K. Saito introduced the notion of
flat structure in order to study a structure of the parameter spaces of
isolated hypersurface singularities. After his pioneering work, B. Dubrovin
unified both the flat structure formulated by K. Saito and the WDVV equation
(which is an abbreviation of Witten-Dijkgraaf-Verlinde-Verlinde equation and
arises from the 2D topological field theory [9], [40]) as the Frobenius
manifold structure. Dubrovin not only formulated the notion of Frobenius
manifold but also applied it to the study of the Painlevé VI equation. As is
well known, the Painlevé VI equation is a nonlinear ordinary differential
equation having four parameters. Dubrovin constructed a one-parameter family
of the Painlevé VI equation from the three dimensional Frobenius manifolds.

We now explain an idea how a solution of the Painlevé VI equation arises from
the WDVV equation based on Dubrovin [10] and Sabbah [30]. For this aim, we
briefly explain the definition of the WDVV equation. Let
$F=F(t_{1},t_{2},\ldots,t_{n})$ be a function of the following form

$F=\left\\{\begin{array}[]{ll}\frac{1}{2}t_{1}t_{n}^{2}+\sum_{j=2}^{m}t_{j}t_{2m+2-j}t_{n}+\frac{1}{2}t_{m+1}^{2}t_{n}+F_{0}(t_{1},\ldots,t_{n-1})&(n=2m+1)\\\
\frac{1}{2}t_{1}t_{n}^{2}+\sum_{j=2}^{m}t_{j}t_{2m+1-j}t_{n}+F_{0}(t_{1},\ldots,t_{n-1})&(n=2m)\\\
\end{array}\right.$

Assume that $F$ is weighted homogeneous, which means that there are non-zero
numbers $w_{1},w_{2},\ldots,w_{n},d$ such that
$\sum_{j=1}^{n}w_{j}t_{j}\partial_{t_{j}}F=dF$. We put
$E=\sum_{j=1}^{n}w_{j}t_{j}\partial_{t_{j}}$. $E$ is called an Euler vector
field. Using $F$, we define $g_{j}=\partial_{t_{n-j+1}}F\>(j=1,2,\ldots,n)$
and put $\vec{g}=(g_{1},g_{2},\ldots,g_{n})$. By differentiating $\vec{g}$
with respect to $t_{1},\ldots,t_{n}$, we obtain $n$ vectors
$\partial_{t_{j}}\vec{g}$ and construct a matrix $C=(C_{ij})$ by

$C=\left(\begin{array}[]{c}\partial_{t_{1}}\vec{g}\\\
\partial_{t_{2}}\vec{g}\\\ \vdots\\\ \partial_{t_{n}}\vec{g}\\\
\end{array}\right).$

In particular it holds that $C_{ij}=\partial_{t_{i}}g_{j}$ and
$\partial_{t_{n}}\vec{g}=(t_{1},t_{2},\ldots,t_{n})$. Moreover we put
$\tilde{B}^{(k)}=\partial_{t_{k}}C$. It follows from the definition that
$(\tilde{B}^{(k)})_{ij}=\partial_{t_{k}}\partial_{t_{i}}g_{j}$. As a
consequence, we have $(\tilde{B}^{(k)})_{ij}=(\tilde{B}^{(i)})_{kj}$. It is
clear from the definition that $C-t_{n}I_{n}$ is independent of $t_{n}$. This
implies that $\tilde{B}^{(n)}=I_{n}$. We need the matrix $T=-EC$ for later
consideration.

If all the matrices $\tilde{B}^{(p)}\>(p=1,2,\ldots,n)$ commute with each
other, the function $F$ is called a prepotential or a free energy (cf. [10]).
Writing down the matrix entries of
$\tilde{B}^{(p)}\tilde{B}^{(q)}=\tilde{B}^{(q)}\tilde{B}^{(p)}$, we obtain
nonlinear differential equations for $F$. The collection of such differential
equations is called the WDVV equation. As a consequence, a prepotential is a
solution of the WDVV equation. Moreover in this case,
$(t_{1},t_{2},\ldots,t_{n})$ is called a flat coordinate.

We define a diagonal matrix $B_{\infty}$ by $B_{\infty}={\rm
diag}(r+w_{1},r+w_{2},\ldots,r+w_{n})$ for some constant $r\in\mathbb{C}$ and
also define $n\times n$ matrices $B^{(p)}\>(p=1,2,\ldots,n)$ by

$B^{(p)}=T^{-1}{\tilde{B}}^{(p)}B_{\infty}.$ (1)

Using $B^{(p)}$, we introduce a system of differential equations

$\partial_{t_{p}}Y=B^{(p)}Y\quad(p=1,2,\ldots,n).$ (2)

We see that the system (2) is integrable. We put
$\displaystyle{T_{0}=t_{n}I_{n}+\frac{1}{w_{n}}T}$. Since $T+w_{n}t_{n}I_{n}$
does not depend on $t_{n}$ and since $B^{(n)}=T^{-1}B_{\infty}$, the
differential equation

$\partial_{t_{n}}Y=B^{(n)}Y$ (3)

turns out to be

$(t_{n}I_{n}-T_{0})\partial_{t_{n}}Y=-\frac{1}{w_{n}}B_{\infty}Y.$ (4)

Regarding (4) as an ordinary differential equation with respect to the
variable $t_{n}$, (4) is called an ordinary differential equation of Okubo
type. In this sense, the system (2) is one of generalizations of Okubo type
ordinary differential equation to several variables case.

We are in a position to describe the relationship between systems of
differential equations (2) and solutions of the Painlevé VI equation. For our
purpose we assume that $n=3$ and $w_{3}=1$. Since $h=\det(-T)$ is a cubic
polynomial in $t_{3}$, let $z_{j}(t^{\prime})\>(j=1,2,3)$ be defined by
$h(t_{1},t_{2},t_{3})=\prod_{i=1}^{3}(t_{3}-z_{i}(t^{\prime})),$ where
$t^{\prime}=(t_{1},t_{2})$. We consider
$B^{(3)}=T^{-1}\tilde{B}^{(3)}B_{\infty}=T^{-1}B_{\infty}$ as before. It
follows from the definition that, if $i\not=j$, the $(i,j)$-entry of
$hB^{(3)}$ is a linear function of $t_{3}$. Noting this, we define
$z_{ij}(t^{\prime})\>(i\not=j)$ by the condition that
$t_{3}=z_{ij}(t^{\prime})$ is the zero of the $(i,j)$-entry of $hB^{(3)}$.
Then
$w_{ij}=\frac{z_{ij}(t^{\prime})-z_{1}(t^{\prime})}{z_{2}(t^{\prime})-z_{1}(t^{\prime})}$
is a solution of the Painlevé VI equation as a function of
$t=\frac{z_{3}(t^{\prime})-z_{1}(t^{\prime})}{z_{2}(t^{\prime})-z_{1}(t^{\prime})}.$
Especially, when we start from an algebraic prepotential $F$ (i.e. an
algebraic solution of the WDVV equation), we obtain an algebraic solution of
the Painlevé VI equation. There are many studies on the construction of
algebraic solutions of the Painlevé VI equation; N. J. Hitchin [15], [16], B.
Dubrovin [10], B. Dubrovin -M. Mazzocco [11], P. Boalch [2], [3], [4], [5], A.
V. Kitaev [23], [24], A. V. Kitaev- R. Vidunas [25], [39], K. Iwasaki [17] and
others, and the classification of algebraic solutions of Painlevé VI was
accomplished by O. Lisovyy and Y. Tykhyy [27].

It is underlined here that parts of the arguments so far go well without the
existence of a solution of the WDVV equation. This leads us to the following
consideration. We start with introducing an $n$-tuple of functions
$\vec{g}=(g_{1}(t_{1},\dots,t_{n}),\ldots,g_{n}(t_{1},\dots,t_{n}))$ such that
$Eg_{j}=(w_{j}+w_{n})g_{j}\quad(j=1,2,\ldots,n)$ and that

$g_{j}=\left\\{\begin{array}[]{ll}t_{j}t_{n}+g_{j}^{(0)}(t_{1},\ldots,t_{n-1})&(j=1,2,\ldots,n-1),\\\
\displaystyle{\frac{1}{2}t_{n}^{2}+g_{n}^{(0)}(t_{1},\ldots,t_{n-1})}&(j=n)\\\
\end{array}\right.$ (5)

with functions $g_{j}^{(0)}(t_{1},\ldots,t_{n-1})$ of
$t^{\prime}=(t_{1},\ldots,t_{n-1})$. If there is a solution
$F=F(t_{1},\dots,t_{n})$ of the WDVV equation, we find that
$g_{j}=\partial_{t_{n-j+1}}F\>(j=1,2,\ldots,n)$ satisfy (5). We return to our
situation. Using $g_{j}\>(j=1,2,\ldots,n)$, we define an $n\times n$ matrix
$C$ such that $C_{ij}=\partial_{t_{i}}g_{j}$. It is easy to see that
$C_{nj}=t_{j}\>(j=1,2,\ldots,n)$. We define matrices
${\tilde{B}}^{(p)}=\partial_{t_{p}}C\>(p=1,2,\ldots,n)$ and
$T=-\sum_{j=1}^{n}w_{j}t_{j}\partial_{t_{j}}C=-\sum_{j=1}^{n}w_{j}t_{j}\tilde{B}^{(j)}.$
If
${\tilde{B}}^{(p)}{\tilde{B}}^{(q)}={\tilde{B}}^{(q)}{\tilde{B}}^{(p)}\>(\forall
p,\>q=1,2,\ldots,n)$, then $\vec{g}=(g_{1},g_{2},\ldots,g_{n})$ is called a
potential vector field and $(t_{1},t_{2},\ldots,t_{n})$ is a flat coordinate.
Similar to the case of prepotentials, the potential vector field is a solution
of a certain system of nonlinear differential equations arising from the
commutativity of matrices ${\tilde{B}}^{(p)}\>(p=1,2,\cdots,n)$. In this
sense, the collection of the relations
${\tilde{B}}^{(p)}{\tilde{B}}^{(q)}={\tilde{B}}^{(q)}{\tilde{B}}^{(p)}\>(\forall
p,\>q)$ is called the extended WDVV equation for $\vec{g}$. One of the reasons
why we are led to the study on flat structures not necessary having
prepotentials is to understand the relationship between a special kind of
holonomic systems of three variables and solutions of the Painlevé VI
equation. In this direction systems of differential equations of Okubo type in
several variables play an important role in attaching a flat structure to a
solution to the Painlevé VI equation.

Our main results in this article are summarized as follows:

(I) To introduce a flat structure on the space of isomonodromic deformations
of differential equations of Okubo type;

(II) To show a correspondence between the extended WDVV equation in three
dimensional case and the full parameter family of the Painlevé VI equation;

(III) To construct a flat structure on the quotient space of the affine space
by the action of a well-generated complex reflection group.

The result (II) is a generalization of that by Dubrovin [10], and (III) is an
analogue to the result by K. Saito on real reflection groups (cf. [33], [32]).

We are going to explain the contents of this article briefly. In Section 2,
after a preliminary study on matrices with function entries, we introduce the
notion of systems of differential equations of Okubo type in several variables
which is a generalization of ordinary differential equation of Okubo type
(Definition 2.1). In Section 3, we first treat a monic polynomial of degree
$n$

$h(x)=h(x^{\prime},x_{n})=x_{n}^{n}-s_{1}(x^{\prime})x_{n}^{n-1}+\cdots+(-1)^{n}x_{n}(x^{\prime})$

where $x^{\prime}=(x_{1},\cdots,x_{n-1})$ and each $s_{i}(x^{\prime})$ is
holomorphic with respect to $x^{\prime}$. We consider properties of a vector
field $V$ on $(x^{\prime},x_{n})$-space with holomorphically (or
meromorphically) dependent on $x^{\prime}$ and polynomially dependent on
$x_{n}$ such that $Vh/h$ is a polynomial of $x_{n}$ with coefficients
dependent holomorphically (or meromorphically) on $x^{\prime}$. Such a vector
field is called a (meromorphic) logarithmic vector field along the divisor
defined by $h=0$. Then we give a definition of free divisor in the sense of K.
Saito. Moreover we prove some properties of them for later use. In particular,
we construct a standard form of generators of (meromorphic) logarithmic vector
fields along the divisor (Lemma 3.3). In Section 4, we treat the notion of
Saito structure (without a metric) which is introduced in Sabbah [30]. For our
purpose, we need to treat a generalization of the WDVV equation which is not
mentioned in [30] explicitly (Proposition 4.3, Definition 4.3). Section 5 is
the main part of the present article. We show that an Okubo type system
introduced in Section 2 has a flat structure (without a potential) if its flat
coordinate is non-degenerate in some sense (Theorem 5.1). Saito structure
(without a metric) [30] plays a central role in our formulation. Combining
this result and the discussion in Appendix A, we can conclude the equivalence
between the extended WDVV equation in the three dimensional case and the
(full-parameter) Painlevé VI equation (Corollary 5.1). In the last part of
this section we treat the case where the flat structure has a potential. This
case corresponds to Dubrovin’s result i.e., the equivalence between the three
dimensional WDVV equation and a one-parameter family of the Painlevé VI
equation (Corollary 5.2). In Section 6, we treat a problem on the existence of
a flat coordinate system for an irreducible complex reflection group. In the
case of a real reflection group, K. Saito [32] proved the existence of a flat
coordinate system (see also [33]). Actually in the real case, the existence of
a potential is also shown by Dubrovin [10]. In the complex case, we show that
if a complex reflection group $G$ is well-generated, then there is a flat
structure (without a metric) for the affine space whose coordinate system
consists of basic $G$-invariant polynomials (Theorem 6.1). It is underlined
here that the existence of a potential is not assured for the complex case.
Instead we show that the potential vector field corresponding to a complex
reflection group has polynomial entries (Corollary 6.1). In other words, we
can obtain a polynomial solution of the extended WDVV equation from a well-
generated complex reflection group. In Section 7, we treat algebraic solutions
of the Painlevé VI equation. As is shown in Section 5, the extended WDVV
equation in three variables case is reduced to the Painlevé VI equation. This
enables us to attach a flat structure corresponding to a solution to the
Painlevé VI equation. In particular, to each algebraic solution to the
Painlevé VI equation, there associates a flat coordinate (without a
potential). We show such flat coordinates associated to some examples of
algebraic solutions. In Appendix A, we discuss the relationship between Okubo
type systems in several variables and solutions of the Painlevé VI equation
with the help of the result of Jimbo and Miwa [19]. In Appendix B, we explain
a method of constructing an Okubo type system in $(n+1)$ variables of rank $n$
from a completely integrable Pfaffian system of $(n+1)$ variables of rank
$(n-1)$ by using a modification of a technique on middle convolution developed
by Haraoka [12].

Finally, we close this introduction with addressing some problems. Firstly, as
stated above, the extended WDVV equation in the three dimensional case is
equivalent to the Painlevé VI equation. Also in the higher dimensional case,
the extended WDVV equation is equivalent to the isomonodromic deformations of
differential equations of Okubo type of higher rank. Recently, higher order
analogues of the Painlevé equations are obtained by several authors ([22],
[37]). It is hopeful to construct algebraic solutions of those higher order
Painlevé equations by using the result in Section 6. To execute the concrete
calculations seems to be an interesting problem. Secondly, K. Saito
constructed his theory of flat structures starting from the theory of
unfolding of singularities. Can we relate the theory of singularities not only
to real reflection groups but also to complex reflection groups? The authors
have obtained some observations in this direction (cf. [20]).

Acknowledgments.

Professor Yoshishige Haraoka taught the first author (M.K.) that integrable
systems in three variables are useful to derive the Painlevé VI solutions.
This is the starting point of our work. The authors would like to thank
Professor Haraoka for his advice. The second author (T.M.) is partially
supported by JSPS Grant-in-Aid for Young Scientists (B) (No.25800082). The
third author (J.S.) is partially supported by JSPS Grant-in-Aid for Scientific
Research (C) (No.26400111).

##  2 A generalization of differential equations of Okubo type to several
variables case

In this section, we first treat some preliminaries on matrices with function
entries and then we introduce a generalization of differential equations of
Okubo type to several variables case as completely integrable Pfaffian
systems. They are equivalent to the isomonodromic deformations of ordinary
differential equations of Okubo type as we will show in Appendix A (cf. Remark
2.3).

###  2.1 Preliminaries on matrices with function entries

Before entering into the argument, we recall the definition of an ordinary
differential equation of Okubo type. Let $T$ and $B$ be $n\times n$-matrices.
When $T$ is a diagonal matrix, the ordinary differential equation

$(zI_{n}-T)\frac{dY}{dz}=-BY$ (6)

is called a differential equation of Okubo type ([28]). In this paper, we call
(6) a differential equation of Okubo type also when $T$ is semisimple (but not
necessarily diagonal). We put the following assumptions on (6):

(A1) The eigenvalues $\\{z_{1},\dots,z_{n}\\}$ of $T$ are mutually distinct;

(A2) The eigenvalues $\\{\lambda_{1}\dots,\lambda_{n}\\}$ of $B$ satisfy the
condition $\lambda_{i}-\lambda_{j}\notin\mathbb{Z}$ for $i\neq j$.

Our aim in this section is to extend (6) to a completely integrable Pfaffian
system of several variables and study its properties. As usual,
$x=(x_{1},\ldots,x_{n})$ is a coordinate of $\mathbb{C}^{n}$. We start with
introducing $n\times n$ matrices $T=T(x)$ and $B_{\infty}$ satisfy the
conditions:

(B1) the entries of $T$ are holomorphic functions on a simply-connected domain
$U$ in $\mathbb{C}^{n}$,

(B2) $B_{\infty}$ is a diagonal matrix whose diagonal entries
$\lambda_{1},\ldots,\lambda_{n}$ are constants,

(B3) $\lambda_{i}-\lambda_{j}\not\in\mathbb{Z}$ for $i\not=j$.
Let $H(x,z)=\det(zI_{n}-T(x))$ be the characteristic polynomial of $T(x)$,
which is a monic polynomial in $z$ of degree $n$ and analytic in $x$:
$H(x,z)=z^{n}-S_{1}(x)z^{n-1}+\cdots+(-1)^{n}S_{n}(x)$. We assume the
following conditions on $H(x,z)$:

(B4) $\det\Bigl{(}\displaystyle\frac{\partial S_{j}(x)}{\partial
x_{i}}\Bigr{)}_{i,j=1,\dots,n}\neq 0$ at generic points of $U$.
It follows from (B4) that the discriminant
$\delta_{H}(x)=\prod_{i<j}(z_{i}(x)-z_{j}(x))^{2}$ of
$H(x,z)=\prod_{i=1}^{n}(z-z_{i}(x))$ is not identically zero and we define a
divisor $\Delta_{H}=\\{\delta_{H}(x)=0\\}\cup\\{\det\bigl{(}\frac{\partial
S_{j}(x)}{\partial x_{i}}\bigr{)}=0\\}$ on $U$. We note that the eigenvalues
$z_{1}(x),\dots,z_{n}(x)$ of $T$ can be considered single-valued holomorphic
functions in $x$ on any simply-connected domain $W\subset
U\setminus\Delta_{H}$ by fixing their branches. If $W\subset
U\setminus\Delta_{H}$ is a domain of holomorphy and $1,2$-connected (i.e., the
first and second homotopy groups vanish), then we can take an invertible
matrix $P=P(x)$ whose entries are single-valued holomorphic functions on $W$
such that

$P^{-1}TP=\mbox{diag}\,[z_{1}(x),\dots,z_{n}(x)].$ (7)

In the following, we promise that

(W) the letter $W$ always denotes a $1,2$-connected domain of holomorphy in
$U\setminus\Delta_{H}$.

We put $B^{(z)}=-(zI_{n}-T(x))^{-1}B_{\infty}$ and decompose $B^{(z)}$ into
partial fractions on $W$

$B^{(z)}=\sum_{i=1}^{n}\frac{B^{(z)}_{i}}{z-z_{i}(x)},$ (8)

where $B_{i}^{(z)}$ $(i=1,\dots,n)$ are independent of $z$. In addition to
(B1)-(B4), we assume the condition:

(B5) for each of $B^{(z)}_{i}$ $(i=1,\dots,n)$,
$r_{i}:=\mbox{tr}\,B^{(z)}_{i}\neq\pm 1$ on $U\setminus\Delta_{H}$.

######  Lemma 2.1.

Assume that $B_{\infty}$ is invertible. Let $B^{(i)}\>(i=1,\ldots,n)$ be
$n\times n$ matrices satisfying

$H(x,z)B^{(i)}\in(\mathcal{O}_{U}\otimes_{\mathbb{C}}\mathbb{C}[z])^{n\times
n},$ (9)

where $\mathcal{O}_{U}$ denotes the ring of holomorphic functions on $U$ and

$\displaystyle\frac{\partial B^{(i)}}{\partial z}-\frac{\partial
B^{(z)}}{\partial x_{i}}+[B^{(i)},B^{(z)}]=O,\ \ \ i=1,\dots,n,$ (10)
$\displaystyle\frac{\partial B^{(i)}}{\partial x_{j}}-\frac{\partial
B^{(j)}}{\partial x_{i}}+[B^{(i)},B^{(j)}]=O,\ \ \ i,j=1,\dots,n.$ (11)

Then there are $n\times n$ matrices
$\tilde{B}^{(i)},B^{(i)}_{\epsilon}\in\mathcal{O}_{U}^{n\times n}$ such that

$B^{(i)}=-(zI_{n}-T(x))^{-1}\tilde{B}^{(i)}B_{\infty}+B^{(i)}_{\epsilon},\ \ \
\ i=1,\dots,n,$ (12)

and that
$B^{(i)}_{\epsilon}=\mathcal{E}^{-1}\displaystyle\frac{\partial\mathcal{E}}{\partial
x_{i}}$ in terms of a diagonal matrix $\mathcal{E}=\mbox{{\rm
diag}}(\epsilon_{1},\dots,\epsilon_{n})$ whose entries are holomorphic
functions on $U$. Moreover, $T,\tilde{B}^{(i)}$ $(i=1,\dots,n)$ are mutually
commutative:

$[T,{\tilde{B}}^{(i)}]=O,\quad[{\tilde{B}}^{(i)},{\tilde{B}}^{(j)}]=O\quad(\forall
i,j).$ (13)

In particular, it holds that

$P^{-1}TP=\mbox{diag}\,[z_{1}(x),\dots,z_{n}(x)],$ (14)
$P^{-1}\tilde{B}^{(i)}P=\mbox{diag}\,\Bigl{[}-\displaystyle\frac{\partial
z_{1}(x)}{\partial x_{i}},\dots,-\displaystyle\frac{\partial
z_{n}(x)}{\partial x_{i}}\Bigr{]},\ \ \ \ i=1,\dots,n,$ (15)

on $W$.

###### Proof.

In virtue of the definition of $B^{(z)}$, we can write

$H(x,z)B^{(z)}=\sum_{i=0}^{n-1}(HB^{(z)})_{i}z^{i},$ (16)

where

$(HB^{(z)})_{i}\in\mathcal{O}_{U}^{n\times n},\ \ \
(HB^{(z)})_{n-1}=-B_{\infty}.$

In virtue of the assumption (9), we can write

$H(x,z)B^{(i)}=\sum_{j=0}^{m_{i}}(HB^{(i)})_{j}z^{j},\ \ \ \ i=1,\dots,n.$
(17)

As the first step of the proof of the lemma, we are going to show $m_{i}\leq
n$ and its consequence. By rewriting the equation (10) for the matrices
$B^{(i)},\>B^{(z)}$ to that for the matrices $HB^{(i)},\>HB^{(z)}$, we have

$H\frac{\partial(HB^{(i)})}{\partial z}-\frac{\partial H}{\partial
z}HB^{(i)}+[HB^{(i)},HB^{(z)}]=H\frac{\partial(HB^{(z)})}{\partial
x_{i}}-\frac{\partial H}{\partial x_{i}}HB^{(z)},\ \ \ i=1,\dots,n.$ (18)

The equation (18) combined with (16) and (17) implies that the left hand side
of the resulting equation is a polynomial in $z$ of degree $m_{i}+n-1$ and its
right hand side is of degree $2n-2$. Besides, the coefficient of the term
$z^{m_{i}+n-1}$ in the left hand side reads

$(m_{i}-n)(HB^{(i)})_{m_{i}}+[(HB^{(i)})_{m_{i}},(HB^{(z)})_{n-1}].$

As a consequence we find that $m_{i}\leq n$, because
$(HB^{(z)})_{n-1}=-B_{\infty}$ is diagonal and
$\lambda_{i}-\lambda_{j}\not\in\mathbb{Z}$ for $i\neq j$. Now we treat the
case $m_{i}=n$. Since $[(HB^{(i)})_{n},(HB^{(z)})_{n-1}]=O$ and since
$(HB^{(z)})_{n-1}=-B_{\infty}$ is diagonal, we conclude that $(HB^{(i)})_{n}$
is also diagonal. Hence $B^{(i)}$ $(i=1,\dots,n)$ is decomposed into partial
fractions as follows on $W$:

$B^{(i)}=\sum_{j=1}^{n}\frac{B^{(i)}_{j}}{z-z_{j}}+B^{(i)}_{\epsilon},$ (19)

where $B^{(i)}_{\epsilon}=(HB^{(i)})_{n}$ is a diagonal matrix which is
defined on $U$. Substituting (19) for (11) and then comparing the values at
$z=\infty$ of the both sides, we have

$\frac{\partial B^{(i)}_{\epsilon}}{\partial x_{j}}-\frac{\partial
B^{(j)}_{\epsilon}}{\partial x_{i}}=O\quad(i,j=1,\ldots,n).$

It means that $B^{(i)}_{\epsilon}$ is integrable, namely there is a diagonal
matrix $\mathcal{E}$ such that
$B^{(i)}_{\epsilon}=\mathcal{E}^{-1}\displaystyle\frac{\partial\mathcal{E}}{\partial
x_{i}}$.

As the second step of the proof, we shall show the equalities

$B^{(i)}_{j}=-\frac{\partial z_{j}}{\partial x_{i}}B^{(z)}_{j},\ \ \ \
i,j=1,\dots,n,$ (20)

for the residue matrices $B_{i}^{(z)},\>B_{i}^{(j)}$ in (8) and (19). We have

$\displaystyle-B^{(i)}_{j}-B^{(z)}_{j}\frac{\partial z_{j}}{\partial
x_{i}}+[B^{(i)}_{j},B^{(z)}_{j}]=O,\ \ \ \ i,j=1,\dots,n,$ (21)
$\displaystyle\frac{\partial B^{(z)}_{j}}{\partial x_{i}}=\sum_{k\neq
j}\frac{[B^{(i)}_{j},B^{(z)}_{k}]+[B^{(i)}_{k},B^{(z)}_{j}]}{z_{j}-z_{k}}+[B^{(i)}_{\epsilon},B^{(z)}_{j}],\
\ \ \ i,j=1,\dots,n$ (22)

by substituting (8) and (19) for the integrability condition (10). Now fix
$j\in\\{1,\ldots,n\\}$. We separately consider the cases where $B^{(z)}_{j}$
is semisimple or not.

Case 1 We assume that $B_{j}^{(z)}$ is semisimple, ${\rm
rank}(B_{j}^{(z)})\leq 1$ and ${\rm tr}(B_{j}^{(z)})=r_{j}\>(\not=\pm 1)$.
Then it follows that

$(I_{n}-B^{(z)}_{j})B^{(z)}_{j}=(1-r_{j})B^{(z)}_{j}$ (23)

from the assumption. Moreover there are linearly independent vectors
$v^{(j)}_{1},\dots,v^{(j)}_{n}$ of $\mathbb{C}^{n}$ such that
$v_{k}^{(j)}\in{\rm ker}B_{j}^{(z)}\>(k=1,\ldots,n-1)$ and $v_{n}^{(n)}$ is an
eigenvector of $B_{j}^{(z)}$ belonging to the eigenvalue $r_{j}$. Then from
(21) we have

$-(I_{n}+B^{(z)}_{j})B^{(i)}_{j}v^{(j)}_{k}=0,\ \ \ \ k=1,\dots,n-1,$

from which $v^{(j)}_{k}\in\mbox{ker}\,B^{(i)}_{j}$ follows. In fact, $(-1)$ is
not an eigenvalue of $B^{(z)}_{j}$. Hence we have
$\mbox{rank}\,B^{(i)}_{j}\leq 1$. Similarly from (21), we have

$\bigl{(}(r_{j}-1)I_{n}-B^{(z)}_{j}\bigr{)}B^{(i)}_{j}v^{(j)}_{n}=\frac{\partial
z_{j}}{\partial x_{i}}B^{(z)}_{j}v^{(j)}_{n}.$ (24)

In turn, from (23) and $r_{j}\neq 1$, we have

$\bigl{(}(r_{j}-1)I_{n}-B^{(z)}_{j}\bigr{)}^{-1}B^{(z)}_{j}=-B^{(z)}_{j}.$
(25)

Combining (24) with (25), we obtain

$B^{(i)}_{j}v^{(j)}_{n}=-r_{j}\frac{\partial z_{j}}{\partial
x_{i}}v^{(j)}_{n},$

which implies that $v^{(j)}_{n}$ is an eigenvector of $B^{(i)}_{j}$ belonging
to the eigenvalue $(-r_{j}\frac{\partial z_{j}}{\partial x_{i}})$. Hence (20)
is proved.

Case 2 We assume $\mbox{tr}(B^{(z)}_{j})=0$, $\mbox{rank}(B^{(z)}_{j})=1$ and
that $B^{(z)}_{j}$ is not semisimple. In virtue of the assumption, we have
$(B^{(z)}_{j})^{2}=O$. In the similar manner to Case 1, we can prove that
$\mbox{ker}B^{(z)}_{j}$ is included in $\mbox{ker}B^{(i)}_{j}$. Then it
follows that $B^{(i)}_{j}B^{(z)}_{j}=O$. From (21), we have

$-B^{(z)}_{j}\Bigl{(}B^{(i)}_{j}+B^{(z)}_{j}\frac{\partial z_{j}}{\partial
x_{i}}-[B^{(i)}_{j},B^{(z)}_{j}]\Bigr{)}=-B^{(z)}_{j}B^{(i)}_{j}=O.$

Then, from (21) again, (20) holds.

As the last step of the proof, we take a matrix $P$ that satisfies (14). Then

$P^{-1}B^{(z)}P=-\begin{pmatrix}z-z_{1}(x)&{}&{}\\\ {}&\ddots&{}\\\
{}&{}&z-z_{n}(x)\end{pmatrix}^{-1}P^{-1}B_{\infty}P,$

and from (20), we have

$P^{-1}B^{(i)}P=-\begin{pmatrix}z-z_{1}(x)&{}&{}\\\ {}&\ddots&{}\\\
{}&{}&z-z_{n}(x)\end{pmatrix}^{-1}\begin{pmatrix}-\frac{\partial
z_{1}(x)}{\partial x_{i}}&{}&{}\\\ {}&\ddots&{}\\\ {}&{}&-\frac{\partial
z_{n}(x)}{\partial
x_{i}}\end{pmatrix}P^{-1}B_{\infty}P+P^{-1}B^{(i)}_{\epsilon}P.$

Hence we obtain

$B^{(i)}=-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty}+B^{(i)}_{\epsilon}$

for

$\tilde{B}^{(i)}=P\mbox{diag}\,\Bigl{[}-\frac{\partial z_{1}(x)}{\partial
x_{i}},\dots,-\frac{\partial z_{n}(x)}{\partial x_{i}}\Bigr{]}P^{-1}.$

This proves (14), (15) and thus (13) on $W$. Since $B_{\infty}$ is invertible,
we find that $\tilde{B}^{(i)}\in\mathcal{O}_{U}^{n\times n}$. In virtue of the
identity theorem, (13) holds on $U$. We have thus proved the lemma completely.
∎

Since we mainly treat the case $B_{\epsilon}^{(i)}=O\>(i=1,\ldots,n)$ in Lemma
2.1, we show the lemma below which gives a necessary and sufficient condition
for a Pfaffian system introduced later and defined by using
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ to be completely integrable.

######  Lemma 2.2.

Let $T,{\tilde{B}}^{(i)}\>(i=1,\ldots,n)$ be $n\times n$ matrices contained in
${\cal O}_{U}^{n\times n}$, $B_{\infty}$ be a constant diagonal matrix, and
define $B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ by the relation

$B^{(z)}=-(zI_{n}-T)^{-1}B_{\infty},\ \ \
B^{(i)}=-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty}.$ (26)

If $T,\>{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$ satisfy the equations

$\displaystyle[T,{\tilde{B}}^{(i)}]=O,\quad[{\tilde{B}}^{(i)},{\tilde{B}}^{(j)}]=O\quad(\forall
i,j),$ (27) $\displaystyle\frac{\partial T}{\partial
x_{i}}+\tilde{B}^{(i)}+[\tilde{B}^{(i)},B_{\infty}]=O,\ \ \ \ i=1,\dots,n,$
(28) $\displaystyle\frac{\partial\tilde{B}^{(i)}}{\partial
x_{j}}-\frac{\partial\tilde{B}^{(j)}}{\partial x_{i}}=O,\ \ \ i,j=1,\dots,n,$
(29)

then $B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ satisfy the equations
$(\ref{eq:icokubopfaffz}),(\ref{eq:icokubopfaff})$ in Lemma 2.1. Moreover, in
case where $\lambda_{i}\neq 0$ $(i=1,\ldots,n)$, the equations
$(\ref{eq:commute}),(\ref{eq:ci1}),(\ref{eq:ci3})$ hold for
$T,\>{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$ if and only if
$(\ref{eq:icokubopfaffz}),(\ref{eq:icokubopfaff})$ hold for
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$.

###### Proof.

Assume that $T,{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$ and
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ satisfy the relation (26) and that
$T,\>{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$ satisfy the equations
(27),(28),(29). In virtue of (27), it follows from the definition that

$\frac{\partial B^{(i)}}{\partial z}-\frac{\partial B^{(z)}}{\partial
x_{i}}+[B^{(i)},B^{(z)}]=(zI_{n}-T)^{-1}\left(\frac{\partial T}{\partial
x_{i}}+{\tilde{B}}^{(i)}+[{\tilde{B}}^{(i)},B_{\infty}]\right)(zI_{n}-T)^{-1}B_{\infty}.$

Similarly, it follows that

$\displaystyle(zI_{n}-T)\Bigl{(}\frac{\partial B^{(i)}}{\partial
x_{j}}-\frac{\partial B^{(j)}}{\partial x_{i}}+[B^{(i)},B^{(j)}]\Bigr{)}$
$\displaystyle=$
$\displaystyle\Biggl{(}\Bigl{(}\frac{\partial\tilde{B}^{(j)}}{\partial
x_{i}}-\frac{\partial\tilde{B}^{(i)}}{\partial
x_{j}}\Bigr{)}(zI_{n}-T)+\Bigl{(}\frac{\partial T}{\partial
x_{i}}+\tilde{B}^{(i)}+[\tilde{B}^{(i)},B_{\infty}]\Bigr{)}\tilde{B}^{(j)}$
$\displaystyle\ \ -\Bigl{(}\frac{\partial T}{\partial
x_{j}}+\tilde{B}^{(j)}+[\tilde{B}^{(j)},B_{\infty}]\Bigr{)}\tilde{B}^{(i)}\Biggr{)}(zI_{n}-T)^{-1}B_{\infty}.$

As a consequence, it follows that if (27), (28) and (29) hold for
$T,\>{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$, then
$(\ref{eq:icokubopfaffz}),(\ref{eq:icokubopfaff})$ hold for
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$. In virtue of Lemma 2.1, (27) follows from
(10) and (11). Hence, it is easy to see that if $B_{\infty}$ is invertible,
then (27), (28) and (29) are equivalent to
$(\ref{eq:icokubopfaffz}),(\ref{eq:icokubopfaff})$. ∎

######  Remark 2.1.

The assumption (B3) shows that the condition (29) is a consequence of the
condition (28). Indeed, differentiate (28) with respect to $x_{j}$ and take
the subtraction of it from the one with changing the places of $i$ and $j$.
Then we have

$\displaystyle\frac{\partial}{\partial
x_{j}}\bigl{(}\tilde{B}^{(i)}+[\tilde{B}^{(i)},B_{\infty}]\bigr{)}-\frac{\partial}{\partial
x_{i}}\bigl{(}\tilde{B}^{(j)}+[\tilde{B}^{(j)},B_{\infty}]\bigr{)}=O.$

It is rewritten to

$\displaystyle\frac{\partial\tilde{B}^{(i)}}{\partial
x_{j}}-\frac{\partial\tilde{B}^{(j)}}{\partial
x_{i}}+\Bigl{[}\frac{\partial\tilde{B}^{(i)}}{\partial
x_{j}}-\frac{\partial\tilde{B}^{(j)}}{\partial x_{i}},B_{\infty}\Bigr{]}=O,$

which combined with (B3) implies (29).

######  Remark 2.2.

Let $T$, $B_{\infty}$ and $\tilde{B}^{(i)}\>(i=1,\ldots,n)$ be matrices as
before. Assume that $T$, $B_{\infty}$, $\tilde{B}^{(i)}\>(i=1,\ldots,n)$
satisfy the conditions $(\ref{eq:commute})$, $(\ref{eq:ci1})$ and
$(\ref{eq:ci3})$. Then replacing $B_{\infty}$ by $B_{\infty}-\lambda I_{n}$
for any $\lambda\in\mathbb{C}$, we see that $T$, $B_{\infty}-\lambda I_{n}$,
${\tilde{B}}^{(i)}\>(i=1,\ldots,n)$ also satisfy $(\ref{eq:commute})$,
$(\ref{eq:ci1})$ and $(\ref{eq:ci3})$.

###  2.2 Systems of differential equations of Okubo type in several variables

The reason why we studied matrices with function entries in the previous
subsection is to introduce the Pfaffian system

$dY=\Bigl{(}B^{(z)}dz+\sum_{i=1}^{n}B^{(i)}dx_{i}\Bigr{)}Y,$ (30)

where $B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ are $n\times n$ matrices. The
integrability condition of (30) is given by (10), (11). In virtue of Lemmas
2.1, 2.2, we focus our attention to the case where
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ satisfy the conditions in Lemma 2.2. To
avoid the confusion, we explain the condition of the matrices
$B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ precisely.

######  Definition 2.1.

Let $B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ be $n\times n$ matrices. Then the
Pfaffian system (30) is called a system of differential equations of Okubo
type in several variables or shortly an Okubo type system in several variables
if $B^{(z)},\>B^{(i)}\>(i=1,\ldots,n)$ are defined by (26), where
$T,\tilde{B}^{(i)}\>(i=1,\ldots,n),\>B_{\infty}$ are $n\times n$ matrices with
the conditions (B1)-(B5) and (27), (28), (29).

The following proposition is a direct consequence of Lemmas 2.1, 2.2.

######  Proposition 2.1.

Any Okubo type system in several variables (30) is completely integrable.

######  Remark 2.3.

The completely integrable Pfaffian system (30) can be regarded as a Lax
formalism of the isomonodromic deformation of an ordinary differential
equation of Okubo type (6). In fact (30) is equivalent to

$dY=\sum_{i=1}^{n}B^{(z)}_{i}d\log(z-z_{i})Y.$ (31)

Then the system of nonlinear differential equations (28) is equivalent to the
Schlesinger system

$dB^{(z)}_{i}=\sum_{j\neq i}[B^{(z)}_{j},B^{(z)}_{i}]d\log(z_{i}-z_{j}),\ \ \
\ \ i=1,\dots,n.$ (32)

(See Appendix A for details.)

######  Remark 2.4.

Given an Okubo type system in several variables (30) defined by $T$,
$B_{\infty}$, $\tilde{B}^{(i)}\>(i=1,\ldots,n)$, then $T+c(x)I_{n}$,
$B_{\infty}$, $\tilde{B}^{(i)}-\frac{\partial c(x)}{\partial
x_{i}}I_{n}\>(i=1,\dots,n)$ also define an Okubo type system in several
variables for any $c(x)\in\mathcal{O}_{U}$, that is $T+c(x)I_{n}$,
$B_{\infty}$, $\tilde{B}^{(i)}-\frac{\partial c(x)}{\partial
x_{i}}I_{n}\>(i=1,\dots,n)$ satisfy (27), (28) and (29).

######  Lemma 2.3.

It is possible to change the variables $x_{1},\dots,x_{n}$ so that $T=T(x)$
satisfies the following condition:

Condition (T): $T(x)=-x_{n}+T_{0}(x^{\prime})$, where $T_{0}(x^{\prime})$
depends only on $x^{\prime}=(x_{1},\dots,x_{n-1})$.

###### Proof.

Since $B_{\infty}$ is diagonalized and $B_{\epsilon}^{(i)}=O$, if there are
two matrices $T(x)$ and $T^{\prime}(x)$ such that their characteristic
polynomials coincide (i.e., $\det(zI_{n}-T(x))=\det(zI_{n}-T^{\prime}(x))$),
then they are connected by a constant diagonal matrix $K$:
$T^{\prime}(x)=K^{-1}T(x)K$. Take $c(x)$ in Remark 2.4 as
$c(x)=-\frac{1}{n}\mbox{tr}\,T(x)$ and put $T_{0}(x)=T(x)+c(x)I_{n}$. Then
$\mbox{tr}\,T_{0}(x)=0$ and its characteristic polynomial
$H_{0}(x,z)=\det(zI_{n}-T_{0}(x))$ is written as

$H_{0}(x,z)=z^{n}+S^{0}_{2}(x)z^{n-2}+\cdots(-1)^{n}S^{0}_{n}(x).$

In virtue of this expression, we see that $T_{0}(x)$ depends only on $n-1$
functions $S^{0}_{2}(x),\dots,S^{0}_{n}(x)$. Then we can choose $n-1$
variables $x_{1}^{\prime},\dots,x_{n-1}^{\prime}$ such that
$T_{0}(x)=T_{0}(x^{\prime})$. Thus $T(x)$ is written as
$T(x)=-x_{n}^{\prime}I_{n}+T_{0}(x^{\prime})$ with respect to new independent
variables $x_{1}^{\prime},\dots,x_{n}^{\prime}$. ∎

In the following, we always assume Condition (T) and the simply connected
domain $U$ has a form
$U=U^{\prime}\times\mathbb{C}\subset\mathbb{C}^{n-1}\times\mathbb{C}$.

######  Remark 2.5.

Under Condition (T), $\tilde{B}^{(n)}=I_{n}$,
$\tilde{B}^{(i)}=\tilde{B}^{(i)}(x^{\prime})\in\mathcal{O}_{U^{\prime}}^{n\times
n}$ $(i=1,\dots,n-1)$. and (30) is equivalent to the Pfaffian system for
$Y_{0}=Y_{0}(x)$

$dY_{0}=\Bigl{(}\sum_{i=1}^{n}B_{0}^{(i)}dx_{i}\Bigr{)}Y_{0},$ (33)

where $B^{(i)}_{0}=T^{-1}\tilde{B}^{(i)}B_{\infty}$. We call (33) the reduced
form of (30).

Here we give a lemma needed later (Lemma 6.5 and Appendix B) although it is
known to experts.

######  Lemma 2.4.

We consider a linear ordinary differential equation

$\frac{dY}{dz}=\sum_{i=1}^{n}\frac{B^{\prime}_{i}}{z-z_{i}}Y.$ (34)

We put $B^{\prime}_{\infty}=-\sum_{i=1}^{n}B^{\prime}_{i}$. Assume
$B^{\prime}_{\infty}=\mbox{diag}\,[\lambda_{1},\dots,\lambda_{n}]$ with
$\lambda_{i}\neq 0$ $(i=1,\dots,n)$ and $\mbox{rank}\,B^{\prime}_{i}\leq 1$
$(i=1,\dots,n)$. Then (34) can be rewritten to one of Okubo type
$(\ref{eq:okubo})$.

###### Proof.

By the assumptions $\mbox{rank}\,B^{\prime}_{i}\leq 1$ and that
$B^{\prime}_{\infty}$ is invertible, we can write

$B^{\prime}_{i}=-\begin{pmatrix}b_{1i}\\\ \vdots\\\
b_{ni}\end{pmatrix}\begin{pmatrix}a_{i1}&\cdots&a_{in}\end{pmatrix}B^{\prime}_{\infty},\
\ \ \ i=1,\dots,n.$

Put

$P=\begin{pmatrix}b_{11}&\cdots&b_{1n}\\\ \vdots&\ddots&\vdots\\\
b_{n1}&\cdots&b_{nn}\end{pmatrix},$

then, by $B^{\prime}_{\infty}=-\displaystyle\sum_{i=1}^{n}B^{\prime}_{i}$, we
have

$P^{-1}=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\\ \vdots&\ddots&\vdots\\\
a_{n1}&\cdots&a_{nn}\end{pmatrix}.$

Replacing the unknown functions $Y$ by $\bar{Y}=P^{-1}Y$, the equation (34)
turns out to be

$\frac{d\bar{Y}}{dz}=\sum_{i=1}^{n}\frac{P^{-1}B^{\prime}_{i}P}{z-z_{i}}\bar{Y}.$
(35)

Since

$\displaystyle P^{-1}B^{\prime}_{i}P$
$\displaystyle=-P^{-1}\begin{pmatrix}b_{1i}\\\ \vdots\\\
b_{ni}\end{pmatrix}\begin{pmatrix}a_{i1}&\cdots&a_{in}\end{pmatrix}B^{\prime}_{\infty}P$
$\displaystyle=-\begin{pmatrix}{}&O&{}\\\
\displaystyle\sum_{k=1}^{n}\lambda_{k}a_{ik}b_{k1}&\cdots&\displaystyle\sum_{k=1}^{n}\lambda_{k}a_{ik}b_{kn}\\\
{}&O&{}\end{pmatrix},$

we obtain the equation

$\sum_{i=1}^{n}\frac{P^{-1}B^{\prime}_{i}P}{z-z_{i}}=-\begin{pmatrix}z-z_{1}&{}&{}\\\
{}&\ddots&{}\\\ {}&{}&z-z_{n}\end{pmatrix}^{-1}P^{-1}B^{\prime}_{\infty}P,$

which means that (35) is a differential equation of Okubo type (6). ∎

##  3 Logarithmic vector fields

In this section, we introduce (meromorphic) logarithmic vector fields along a
divisor and the notion of free divisor in the sense of K. Saito. Moreover we
prove some basic properties of them for later use. In particular, we construct
a standard form of generators of (meromorphic) logarithmic vector fields along
a divisor (Lemma 3.3, cf. Remark 3.3).

###  3.1 Studies of polynomial $h(x)$

We use the notations $x^{\prime}=(x_{1},\dots,x_{n-1})$ and
$x=(x^{\prime},x_{n})=(x_{1},\dots,x_{n})$. Let
$h(x)=H(x,0)=\det(-T(x))\in\mathcal{O}_{U}$ or equivalently
$H(x,z)=h(x^{\prime},x_{n}+z)$ under Condition (T). Let $D$ be a divisor in
$U$ defined by $h(x)$ i.e., $D=\\{(x)\in U;h(x)=0\\}$. We assume that the
domain $U\subset\mathbb{C}^{n}$ (and $W\subset U$ resp.) has the form of
$U=U^{\prime}\times\mathbb{C}\subset\mathbb{C}^{n-1}\times\mathbb{C}$ (and
$W=W^{\prime}\times\mathbb{C}\subset U^{\prime}\times\mathbb{C}$ resp.). In
virtue of Condition (T), $h(x)$ is a monic polynomial in $x_{n}$ of degree
$n$:

$h(x)=h(x^{\prime},x_{n})=x_{n}^{n}-s_{1}(x^{\prime})x_{n}^{n-1}+\dots+(-1)^{n}s_{n}(x^{\prime}),$
(36)

where $s_{i}(x^{\prime})\in\mathcal{O}_{U^{\prime}}$.

Let $z_{1}^{0}(x^{\prime}),\dots,z_{n}^{0}(x^{\prime})$ be the roots of
$h(x^{\prime},x_{n})$ as a polynomial of $x_{n}$, that is

$\displaystyle
h(x^{\prime},x_{n})=\prod_{m=1}^{n}(x_{n}-z_{m}^{0}(x^{\prime})),$

and let

$\delta_{h}=\delta_{h}(x^{\prime})=\prod_{k<l}(z_{k}^{0}(x^{\prime})-z_{l}^{0}(x^{\prime}))^{2}$

be the discriminant of $h(x)$.

######  Remark 3.1.

The roots $z_{m}^{0}(x^{\prime})$ of $h(x^{\prime},x_{n})$ and $z_{m}(x)$ of
$H(x,z)$ are related to each other by $z_{m}(x)=-x_{n}+z_{m}^{0}(x^{\prime})$.

In virtue of the assumption (B4), it holds that

$\det\begin{pmatrix}\frac{\partial S_{1}(x)}{\partial
x_{1}}&\dots&\frac{\partial S_{n}(x)}{\partial x_{1}}\\\ \dots\\\
\frac{\partial S_{1}(x)}{\partial x_{n-1}}&\dots&\frac{\partial
S_{n}(x)}{\partial x_{n-1}}\\\ \frac{\partial S_{1}(x)}{\partial
x_{n}}&\dots&\frac{\partial S_{n}(x)}{\partial
x_{n}}\end{pmatrix}=\det\begin{pmatrix}\frac{\partial
s_{1}(x^{\prime})}{\partial x_{1}}&\frac{\partial s_{2}(x^{\prime})}{\partial
x_{1}}&\dots&\frac{\partial s_{n}(x^{\prime})}{\partial x_{1}}\\\ \dots\\\
\frac{\partial s_{1}(x^{\prime})}{\partial x_{n-1}}&\frac{\partial
s_{2}(x^{\prime})}{\partial x_{n-1}}&\dots&\frac{\partial
s_{n}(x^{\prime})}{\partial x_{n-1}}\\\
-n&-(n-1)s_{1}(x^{\prime})&\dots&-s_{n-1}(x^{\prime})\end{pmatrix}\neq 0,$
(37)

which is equivalent to

$\det\begin{pmatrix}\frac{\partial z_{1}^{0}(x^{\prime})}{\partial
x_{1}}&\frac{\partial z_{2}^{0}(x^{\prime})}{\partial
x_{1}}&\dots&\frac{\partial z_{n}^{0}(x^{\prime})}{\partial x_{1}}\\\
\frac{\partial z_{1}^{0}(x^{\prime})}{\partial x_{2}}&\frac{\partial
z_{2}^{0}(x^{\prime})}{\partial x_{2}}&\dots&\frac{\partial
z_{n}^{0}(x^{\prime})}{\partial x_{2}}\\\ \dots\\\ \frac{\partial
z_{1}^{0}(x^{\prime})}{\partial x_{n-1}}&\frac{\partial
z_{2}^{0}(x^{\prime})}{\partial x_{n-1}}&\dots&\frac{\partial
z_{n}^{0}(x^{\prime})}{\partial x_{n-1}}\\\
-1&-1&\dots&-1\end{pmatrix}=\det\begin{pmatrix}\frac{\partial
z_{1}(x)}{\partial x_{1}}&\frac{\partial z_{2}(x)}{\partial
x_{1}}&\dots&\frac{\partial z_{n}(x)}{\partial x_{1}}\\\ \frac{\partial
z_{1}(x)}{\partial x_{2}}&\frac{\partial z_{2}(x)}{\partial
x_{2}}&\dots&\frac{\partial z_{n}(x)}{\partial x_{2}}\\\
\vdots&\vdots&\dots&\vdots\\\ \frac{\partial z_{1}(x)}{\partial
x_{n}}&\frac{\partial z_{2}(x)}{\partial x_{n}}&\dots&\frac{\partial
z_{n}(x)}{\partial x_{n}}\end{pmatrix}\neq 0$ (38)

on $W^{\prime}$. On each $W^{\prime}$, we define matrices
$P_{h}(x^{\prime}),M_{h}(x^{\prime}),M_{V^{h}}(x)$ by

$P_{h}(x^{\prime})=\begin{pmatrix}\frac{\partial
z_{1}^{0}(x^{\prime})}{\partial x_{1}}&\frac{\partial
z_{2}^{0}(x^{\prime})}{\partial x_{1}}&\dots&\frac{\partial
z_{n}^{0}(x^{\prime})}{\partial x_{1}}\\\ \frac{\partial
z_{1}^{0}(x^{\prime})}{\partial x_{2}}&\frac{\partial
z_{2}^{0}(x^{\prime})}{\partial x_{2}}&\dots&\frac{\partial
z_{n}^{0}(x^{\prime})}{\partial x_{2}}\\\ \dots\\\ \frac{\partial
z_{1}^{0}(x^{\prime})}{\partial x_{n-1}}&\frac{\partial
z_{2}^{0}(x^{\prime})}{\partial x_{n-1}}&\dots&\frac{\partial
z_{n}^{0}(x^{\prime})}{\partial x_{n-1}}\\\
-1&-1&\dots&-1\end{pmatrix}=\begin{pmatrix}\frac{\partial z_{1}(x)}{\partial
x_{1}}&\frac{\partial z_{2}(x)}{\partial x_{1}}&\dots&\frac{\partial
z_{n}(x)}{\partial x_{1}}\\\ \frac{\partial z_{1}(x)}{\partial
x_{2}}&\frac{\partial z_{2}(x)}{\partial x_{2}}&\dots&\frac{\partial
z_{n}(x)}{\partial x_{2}}\\\ \dots&\vdots&\dots&\vdots\\\ \frac{\partial
z_{1}(x)}{\partial x_{n}}&\frac{\partial z_{2}(x)}{\partial
x_{n}}&\dots&\frac{\partial z_{n}(x)}{\partial x_{n}}\end{pmatrix},$ (39)

and

$M_{h}(x^{\prime})=P_{h}\,\mbox{diag}[z_{1}^{0},z_{2}^{0},\dots,z_{n}^{0}]\,P_{h}^{-1},\quad
M_{V^{(h)}}(x)=-P_{h}\,\mbox{diag}[z_{1},z_{2},\dots,z_{n}]\,P_{h}^{-1}.$ (40)

By definition, we have

$M_{V^{(h)}}(x)=x_{n}I_{n}-M_{h}(x^{\prime}).$ (41)

######  Lemma 3.1.

The matrix $M_{h}(x^{\prime})$ has the following properties:

$({\rm i})$ The entries of $M_{h}(x^{\prime})$ are meromorphic functions on
$U^{\prime}$.

$({\rm ii})$ On each $W^{\prime}$, we have

$\left(z_{m}^{0}(x^{\prime})I_{n}-M_{h}(x^{\prime})\right)\begin{pmatrix}\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{1}}\\\ \frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{2}}\\\ \dots\\\ \frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{n-1}}\\\ -1\end{pmatrix}=0,\quad 1\leq
m\leq n.$ (42)

###### Proof.

Proof of (i). Differentiate the both side of

$\prod_{m=1}^{n}(z-z_{m}(x))=z^{n}+\sum_{k=1}^{n}(-1)^{k}S_{k}(x)z^{n-k},$

by $x_{i}$, and let $z=z_{m}(x)$. Then we have

$-{\partial z_{m}\over\partial x_{i}}\prod_{l\neq
m}(z_{m}(x)-z_{l}(x))=\sum_{k=1}^{n}(-1)^{k}{\partial S_{k}\over\partial
x_{i}}z_{m}(x)^{n-k}.$ (43)

Since $M_{h}(x^{\prime})$ is invariant under the permutations of
$\\{z_{1},z_{2},\dots,z_{n}\\}$, all entries of $M_{h}(x^{\prime})$ are
holomorphic on $U^{\prime}\setminus\\{\delta_{h}(x^{\prime})\cdot\det
P_{h}(x^{\prime})=0\\}$. From (39), (40) and (43), we find that all entries of
$M_{h}(x^{\prime})$ have at most poles along
$\\{\delta_{h}(x^{\prime})\cdot\det P_{h}(x^{\prime})=0\\}$. This proves (i).

(ii) is apparent from the definition of $M_{h}(x^{\prime})$. ∎

###  3.2 Logarithmic vector fields

We give the definitions of logarithmic vector field along $D$ and free divisor
following K. Saito [31]:

######  Definition 3.1.

Let $\mathcal{M}_{U^{\prime}}$ be the field of meromorphic functions on
$U^{\prime}$. A vector field
$V=\sum_{k=1}^{n}v_{k}(x^{\prime},x_{n})\partial_{x_{k}}$ with
$v_{k}(x^{\prime},x_{n})\in\mathcal{M}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$
is called a meromorphic logarithmic vector field along $D$ if
$(Vh)/h\in\mathcal{M}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$, or
equivalently if $(Vh)|_{D}=0$. If moreover,
$v_{k}(x^{\prime},x_{n})\in\mathcal{O}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$,
$V$ is called a logarithmic vector field along $D$.

Let $\mbox{Der}(-\log D)$ be the set of logarithmic vector fields along $D$,
which is naturally an
$\mathcal{O}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$-module. The
divisor $D$ is said to be free if $\mbox{Der}(-\log D)$ is a free
$\mathcal{O}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$-module.

######  Remark 3.2.

It is known that $D$ is free if there are logarithmic vector fields
$V_{i}=\sum_{j=1}^{n}v_{i,j}(x)\partial_{x_{j}}$, $i=1,\dots,n$, along $D$
such that $\det(v_{i,j}(x))=h(x)$ (Saito’s criterion [31]). We call the matrix
$(v_{i,j}(x))$ a Saito matrix.

######  Lemma 3.2.

A vector field $V=\sum_{k=1}^{n}v_{k}(x^{\prime},x_{n})\partial_{x_{k}}$ with
$v_{k}(x^{\prime},x_{n})\in\mathcal{M}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$
is a meromorphic logarithmic vector field along $D$ if and only if

$\sum_{k=1}^{n-1}v_{k}(x^{\prime},z_{m}^{0}(x^{\prime}))\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial
x_{k}}-v_{n}(x^{\prime},z_{m}^{0}(x^{\prime}))=0,\quad 1\leq m\leq n.$ (44)

###### Proof.

Since $Vh(x^{\prime},z_{m}^{0}(x^{\prime}))=V\cdot 0=0$, we have

$\sum_{k=1}^{n-1}v_{k}(x^{\prime},z_{m}^{0}(x^{\prime}))\left(\frac{\partial
h(x^{\prime},x_{n})}{\partial x_{k}}+\frac{\partial
h(x^{\prime},x_{n})}{\partial x_{n}}\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial
x_{k}}\right)\Big{|}_{x_{n}=z_{m}^{0}(x^{\prime})}=0.$ (45)

It also follows that

$(Vh(x^{\prime},x_{n}))|_{x_{n}=z_{m}^{0}(x^{\prime})}=\sum_{k=1}^{n}v_{k}(x^{\prime},x_{n})\left(\frac{\partial
h(x^{\prime},x_{n})}{\partial
x_{k}}\right)\Big{|}_{x_{n}=z_{m}^{0}(x^{\prime})}.$ (46)

From (45) and (46), we have

$\displaystyle(Vh(x^{\prime},x_{n}))|_{x_{n}=z_{m}^{0}(x^{\prime})}$
$\displaystyle=$
$\displaystyle-\left(\sum_{k=1}^{n-1}v_{k}(x^{\prime},z_{m}^{0}(x^{\prime}))\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial
x_{k}}-v_{n}(x^{\prime},z_{m}^{0}(x^{\prime}))\right)\frac{\partial
h(x^{\prime},x_{n})}{\partial x_{n}}\Big{|}_{x_{n}=z_{m}^{0}(x^{\prime})}.$
(47)

Since $\delta_{h}(x^{\prime})\neq 0$ for generic $x^{\prime}$, it holds that
$\frac{\partial h(x^{\prime},x_{n})}{\partial
x_{n}}|_{x_{n}=z_{m}^{0}(x^{\prime})}\neq 0$ for generic $x^{\prime}$. This
implies that (44) holds if and only if
$(Vh(x^{\prime},x_{n}))|_{x_{n}=z_{m}^{0}(x^{\prime})}=0$, $1\leq m\leq n$, or
equivalently if and only if $(Vh)|_{D}=0$. This proves the lemma. ∎

We introduce vector fields $V_{i}^{(h)}$ $(i=1,\dots,n)$ by

$\begin{pmatrix}V^{(h)}_{n}\\\ V^{(h)}_{n-1}\\\ \vdots\\\
V^{(h)}_{1}\end{pmatrix}=M_{V^{(h)}}\begin{pmatrix}\partial_{x_{1}}\\\
\partial_{x_{2}}\\\ \vdots\\\ \partial_{x_{n}}\end{pmatrix}.$ (48)

######  Lemma 3.3.

The vector fields $V^{(h)}_{i}$, $1\leq i\leq n$ are meromorphic logarithmic
vector fields along $D$ satisfying

$(V^{(h)}_{i}h)/h=\begin{cases}\quad n&\mbox{if}\quad i=1,\\\ -{\partial
s_{1}(x^{\prime})\over\partial x_{n-i+1}}&\mbox{if}\quad i>1.\end{cases}$ (49)

###### Proof.

From the equality (42) and Lemma 3.2, $V^{(h)}_{k}$, $1\leq k\leq n$ are
meromorphic logarithmic vector fields. Let
$\alpha_{i}(x)=(V^{(h)}_{i}h)/h\,(\in\mathcal{M}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}])$.
From the definition (48) of $V^{(h)}_{i}$, we find that the degree of
$V^{(h)}_{i}h$ with respect to $x_{n}$ are $n$. Hence
$\alpha_{i}\in\mathcal{M}_{U^{\prime}}$. From the definition of $V^{(h)}_{i}$,
we also find that the coefficient of $V^{(h)}_{1}h$ to $x_{n}^{n}$ is equal to
$n$, and that of $V^{(h)}_{i}h$ to $x_{n}^{n}$ is equal to $-{\partial
s_{1}(x^{\prime})\over\partial x_{n-i+1}}$ for $i>1$. This proves the lemma. ∎

The function $\det M_{V^{(h)}}(x^{\prime},x_{n})$ is a monic polynomial in
$x_{n}$ of degree $n$, and from the equality (42), $\det
M_{V^{(h)}}(x^{\prime},z_{m}^{0}(x^{\prime}))=0$ for all solutions
$z_{m}^{0}(x^{\prime})$ of $h(x^{\prime},x_{n})=0$. This implies

$\det M_{V^{(h)}}(x^{\prime},x_{n})=h(x^{\prime},x_{n}).$ (50)

######  Lemma 3.4.

Assume $V=\sum_{k=1}^{n}v_{k}(x^{\prime})\partial_{x_{k}}$, with
$v_{k}(x^{\prime})\in\mathcal{M}_{U^{\prime}}$ is a meromorphic logarithmic
vector field along $D$. Then $V=0$.

###### Proof.

From Lemma 3.2, it holds that $\sum_{k=1}^{n-1}v_{k}(x^{\prime})\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{k}}-v_{n}(x^{\prime})=0,\ 1\leq m\leq n,$
which is equivalent to

$\begin{pmatrix}v_{1}(x^{\prime})&v_{2}(x^{\prime})&\dots&v_{n}(x^{\prime})\end{pmatrix}P_{h}(x^{\prime})=O,$

on each $W^{\prime}$. Since $\det P_{h}(x^{\prime})\neq 0$, the lemma holds. ∎

######  Lemma 3.5.

Let $V$ be a meromorphic logarithmic vector field along $D$. Then there are
unique meromorphic functions
$a_{i}(x)=a_{i}(x^{\prime},x_{n})\in\mathcal{M}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$,
$1\leq i\leq n$ such that $V=\sum_{i=1}^{n}a_{i}(x)V^{(h)}_{i}$. If, moreover,
$V$ and $V_{i}^{(h)}$, $1\leq i\leq n$ are logarithmic vector fields along $D$
then
$a_{i}(x)\in\mathcal{O}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$.

###### Proof.

The uniqueness of the “coefficients” $a_{i}(x)$ follows from
$\det(x_{n}I_{n}-M_{h}(x^{\prime}))\neq 0$.

Let $V=\sum_{i=1}^{n}v_{i}(x)\partial_{x_{i}}$ be a meromorphic logarithmic
vector field along $D$, and let
$v_{i}(x)=v_{i}(x^{\prime},x_{n})=\sum_{k=0}^{K_{i}}v_{i,k}(x^{\prime})x_{n}^{k}$,
for some $v_{i,k}(x^{\prime})\in\mathcal{M}_{U^{\prime}}$. We assume
$v_{i,K_{i}}(x^{\prime})\neq 0$ if $K_{i}>0$. Put
$\deg_{x_{n}}V=\max\\{K_{1},K_{2},\dots K_{n}\\}$. We prove the lemma by
induction on $\deg_{x_{n}}V$.

If $\deg_{x_{n}}V=0$, then $V=0$ from Lemma 3.4. Hence the claim is true.
Assume $\deg_{x_{n}}V>0$. Then
$V^{\prime}=V-\sum_{i=1}^{n}\sum_{k=1}^{K_{i}}v_{i,k}(x^{\prime})x_{n}^{k-1}V^{(h)}_{n-i+1}$
is a meromorphic logarithmic vector field along $D$ with
$\deg_{x_{n}}V^{\prime}<\deg_{x_{n}}V$. Consequently the lemma is true for
$V^{\prime}$, whence true for $V$.

If $V$ is a logarithmic vector field along $D$, that is
$v_{i,k}(x^{\prime})\in\mathcal{O}_{U^{\prime}}$, then
$a_{i}(x)\in\mathcal{O}_{U^{\prime}}\otimes_{\mathbb{C}}\mathbb{C}[x_{n}]$ by
the construction of them stated above. ∎

######  Lemma 3.6.

Assume that there are logarithmic vector fields along $D$,
$V_{i}=\sum_{j=1}^{n}v_{i,j}(x)\partial_{x_{j}}$, $i=1,\dots,n$ such that
$v_{i,j}(x)-x_{n}\delta_{n+1-i,j}\in\mathcal{O}_{U^{\prime}}$, where
$\delta_{ij}$ denotes Kronecker’s delta. Then it holds that
$V_{i}=V_{i}^{(h)}$, $i=1,\dots,n$. In particular, $V_{i}^{(h)}$,
$i=1,\dots,n,$ are logarithmic vector fields along $D$, and $D$ is free.

###### Proof.

Let $V_{i}^{(h)}=\sum_{j=1}^{n}v_{i,j}^{(h)}(x)\partial_{x_{j}}$. Then we see
$v_{i,j}^{(h)}(x)-x_{n}\delta_{n+1-i,j}\in\mathcal{M}_{U^{\prime}}$ from (48).
Thus we have $v_{i,j}(x)-v_{i,j}^{(h)}(x)\in\mathcal{M}_{U^{\prime}}$,
$j=1,\dots,n$, which implies $V_{i}-V_{i}^{(h)}=0$ in virtue of Lemma 3.4.
Then the equality (50) implies that $M_{V^{(h)}}$ is a Saito matrix and $D$ is
free. ∎

The following lemma is obvious from (14) and (15).

######  Lemma 3.7.

For any polynomial
$F(x,b_{1},b_{2},\dots,b_{n},t)\in\mathcal{M}_{U}\otimes_{\mathbb{C}}\mathbb{C}[b_{1},b_{2},\dots,b_{n},t]$,
the equality $F(x,\tilde{B}^{(1)},\tilde{B}^{(2)},\dots,\tilde{B}^{(n)},T)=0$
holds if and only if
$F(x,-{\partial z_{m}(x)\over\partial x_{1}},-{\partial z_{m}(x)\over\partial
x_{2}},\dots,-{\partial z_{m}(x)\over\partial x_{n}},z_{m}(x))=0$ hold for all
$m\in\\{1,2,\dots n\\}$.

Put $B_{0}^{(k)}=T^{-1}\tilde{B}^{(k)}B_{\infty}$, and let
$A^{(k)}=A^{(k)}(x)$, $1\leq k\leq n$ be the $n\times n$ matrices defined by

$\begin{pmatrix}A^{(n)}\\\ A^{(n-1)}\\\ \vdots\\\
A^{(1)}\end{pmatrix}=M_{V^{(h)}}\begin{pmatrix}B^{(1)}_{0}\\\ B^{(2)}_{0}\\\
\vdots\\\ B^{(n)}_{0}\end{pmatrix},$ (51)

that is, $A^{(k)}=\sum_{j=1}^{n}(M_{V^{(h)}})_{n-k+1,\,j}\,B^{(j)}_{0}$.

######  Lemma 3.8.

The differential equation $(\ref{eq:okubopfaff0})$ is equivalent to

$V_{k}^{(h)}Y_{0}=A^{(k)}Y_{0},\ 1\leq k\leq n,$ (52)

and it holds that

$A^{(k)}=-\tilde{B}^{(n-k+1)}B_{\infty}.$ (53)

###### Proof.

Since

$\displaystyle
V^{(h)}_{k}Y_{0}=\sum_{j=1}^{n}(M_{V^{(h)}})_{n-k+1,\,j}\frac{\partial
Y_{0}}{\partial
x_{j}}=\sum_{j=1}^{n}(M_{V^{(h)}})_{n-k+1,\,j}B^{(j)}_{0}Y_{0}=A^{(k)}Y_{0},$

the equality (52) holds.

We prove (53). We have

$\displaystyle\begin{pmatrix}A^{(n)}\\\ A^{(n-1)}\\\ \vdots\\\
A^{(1)}\end{pmatrix}$ $\displaystyle=M_{V^{(h)}}\begin{pmatrix}B^{(1)}_{0}\\\
B^{(2)}_{0}\\\ \vdots\\\
B^{(n)}_{0}\end{pmatrix}=M_{V^{(h)}}\begin{pmatrix}T^{-1}\tilde{B}^{(1)}B_{\infty}\\\
T^{-1}\tilde{B}^{(2)}B_{\infty}\\\ \vdots\\\
T^{-1}\tilde{B}^{(n)}B_{\infty}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}T^{-1}\tilde{A}^{({n})}B_{\infty}\\\
T^{-1}\tilde{A}^{(n-1)}B_{\infty}\\\ \vdots\\\
T^{-1}\tilde{A}^{(1)}B_{\infty}\end{pmatrix},$

where we put

$\begin{pmatrix}\tilde{A}^{(n)}\\\ \tilde{A}^{(n-1)}\\\ \vdots\\\
\tilde{A}^{(1)}\end{pmatrix}=M_{V^{(h)}}\begin{pmatrix}\tilde{B}^{(1)}\\\
\tilde{B}^{(2)}\\\ \vdots\\\
\tilde{B}^{(n)}\end{pmatrix}=(x_{n}I_{n}-M_{h})\begin{pmatrix}\tilde{B}^{(1)}\\\
\tilde{B}^{(2)}\\\ \vdots\\\ \tilde{B}^{(n)}\end{pmatrix}.$

We now use the ring isomorphism in Lemma 3.7. From (42), we have

$\displaystyle M_{h}\begin{pmatrix}\frac{\partial z_{m}(x)}{\partial
x_{1}}&\frac{\partial z_{m}(x)}{\partial x_{2}}&\dots&\frac{\partial
z_{m}(x)}{\partial
x_{n}}\end{pmatrix}^{t}=z_{m}^{0}(x^{\prime})\begin{pmatrix}\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{1}}&\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{2}}&\dots&\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{n-1}}&-1\end{pmatrix}^{t}$
$\displaystyle\qquad=\begin{pmatrix}(x_{n}+z_{m})\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{1}}&(x_{n}+z_{m})\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{2}}&\dots&(x_{n}+z_{m})\frac{\partial
z_{m}^{0}(x^{\prime})}{\partial x_{n}}\end{pmatrix}^{t}.$

Then from the ring isomorphism in Lemma 3.7, we have

$M_{h}\begin{pmatrix}\tilde{B}^{(1)}&\dots&\tilde{B}^{(n)}\end{pmatrix}^{t}=\begin{pmatrix}(x_{n}+T)\tilde{B}^{(1)}&\dots&(x_{n}+T)\tilde{B}^{(n)}\end{pmatrix}^{t},$

which implies $\tilde{A}^{(k)}=-T\tilde{B}^{(n-k+1)}$. Consequently we have
$A^{(k)}=-\tilde{B}^{(n-k+1)}B_{\infty}$. ∎

###  3.3 Global case

In this subsection we assume that the function $h(x^{\prime},x_{n})$ is a
polynomial in $(x^{\prime},x_{n})$ and weighted homogeneous with respect to a
weight $w(\cdot)$ with

$0<w(x_{1})\leq w(x_{2})\leq\dots\leq w(x_{n-1})<w(x_{n})=1.$

In this case, we replace $\mathcal{O}_{U^{\prime}}$ and
$\mathcal{M}_{U^{\prime}}$ in the previous subsection by
$\mathbb{C}[x^{\prime}]$ and $\mathbb{C}(x^{\prime})$ respectively, and assume
that the integrable system (33) is weighted homogeneous, that is, all the
entries of $B^{(k)}_{0}$, $Y_{0}$ are weighted homogeneous. In this situation,
we obtain more detailed descriptions for (meromorphic) logarithmic vector
fields along $D$.

######  Lemma 3.9.

It holds that $V^{(h)}_{1}=\sum_{k=1}^{n}w(x_{k})x_{k}\partial_{x_{k}}$,
namely, $V^{(h)}_{1}$ is the Euler operator.

###### Proof.

Let $h(x^{\prime},x_{n})=\prod_{m=1}^{n}(x_{n}-z_{m}^{0}(x^{\prime}))$, as
before. For each $m$, we have

$\displaystyle 0$ $\displaystyle=\sum_{k=1}^{n-1}w(x_{k})x_{k}{\partial
h(x^{\prime},z_{m}^{0}(x^{\prime}))\over\partial x_{k}}$
$\displaystyle=\sum_{k=1}^{n-1}w(x_{k})x_{k}\left({\partial
h(x^{\prime},x_{n})\over\partial x_{k}}+{\partial
z_{m}^{0}(x^{\prime})\over\partial x_{k}}{\partial
h(x^{\prime},x_{n})\over\partial
x_{n}}\right)\Big{|}_{x_{n}=z_{m}^{0}(x^{\prime})},$ $\displaystyle 0$
$\displaystyle=w(h)h(x^{\prime},x_{n})|_{x_{n}=z_{m}^{0}(x^{\prime})}=\left(\sum_{k=1}^{n}w(x_{k})x_{k}{\partial
h(x^{\prime},x_{n})\over\partial
x_{k}}\right)\Big{|}_{x_{n}=z_{m}^{0}(x^{\prime})},$

which implies

$z_{m}^{0}(x^{\prime})=\sum_{k=1}^{n-1}w(x_{k})x_{k}{\partial
z_{m}^{0}(x^{\prime})\over\partial x_{k}},\quad 1\leq m\leq n.$

These equalities combined with (42) imply

$(M_{h})_{n,k}=\left\\{\begin{array}[]{ccc}-w(x_{k})x_{k}&\mbox{if}&1\leq
k\leq n-1\\\ 0&\mbox{if}&k=n.\end{array}\right.$ (54)

Then (48) proves the lemma. ∎

Note $w(z_{m}^{0}(x^{\prime}))=w(x_{n})=1$. Hence the equality (42) implies
$w((M_{h})_{i,j})+w({\partial z_{m}^{0}\over\partial
x_{j}})=w(z_{m}^{0})+w({\partial z_{m}^{0}\over\partial x_{i}})$ for $1\leq
i,j\leq n$, that is,

$w\left((M_{h})_{i,j}\right)=1-w(x_{i})+w(x_{j}),\quad 1\leq i,j\leq n.$ (55)

Since $w\left((M_{h})_{i,j}\right)=w\left((M_{V^{(h)}})_{i,j}\right)$, we have
a kind of duality

$w(x_{i})+w\left(V^{(h)}_{n-i+1}\right)=1,\quad 1\leq i\leq n,$ (56)

between $\\{w(x_{i})\\}$ and $\\{w(V_{i}^{(h)})\\}$. Consequently
$w(V^{(h)}_{1})=0$, and $w(V^{(h)}_{i})>0$ for $i\geq 2$.

Let $Y_{0}=(y_{1},y_{2},\dots,y_{n})^{t}$,
$B_{\infty}=$diag$[\lambda_{1},\lambda_{2},\dots,\lambda_{n}]$. Then, from
Lemma 3.8, it follows that $V^{(h)}_{1}Y_{0}=-B_{\infty}Y_{0}$, which implies

$w(y_{i})=-\lambda_{i},\quad 1\leq i\leq n.$ (57)

From the equality ${\partial\over\partial x_{n}}Y_{0}=B^{(n)}_{0}Y_{0}$, we
obtain $w(y_{i})-1=w((B^{(n)}_{0})_{i,j})+w(y_{j})$, whence

$w((B^{(n)}_{0})_{i,j})=-\lambda_{i}+\lambda_{j}-1,\quad 1\leq i,j\leq n.$
(58)

Since
$B^{(n)}_{0}=T^{-1}B_{\infty}=-(x_{n}I_{n}-T_{0}(x^{\prime}))^{-1}B_{\infty}$
as in Remark 2.5, we have $w((T_{0})_{i,j})=-w((B^{(n)}_{0})_{j,i})$, that is,

$w((T_{0})_{i,j})=1-\lambda_{i}+\lambda_{j},\quad 1\leq i,j\leq n.$ (59)

######  Lemma 3.10.

Let $V$ be a non-zero weighted homogeneous logarithmic vector field along $D$.
Then $w(V)\geq 0$. Moreover $w(V)=0$ holds if and only if $V=cV^{(h)}_{1}$ for
a constant number $c$.

###### Proof.

Let $V=\sum_{j=1}^{n}v_{j}(x)\partial_{x_{j}}$,
$v_{j}=v_{j}(x^{\prime},x_{n})\in\mathbb{C}[x]$. If $w(V)<0$, then
$v_{j}(x)\in\mathbb{C}[x^{\prime}]$. This proves $V=0$ from Lemma 3.4. If
$w(V)=0$, then $v_{j}(x)\in\mathbb{C}[x^{\prime}]$ for $j\leq n-1$, and
$v_{n}(x)-cx_{n}\in\mathbb{C}[x^{\prime}]$ for some $c\in\mathbb{C}$. Then
$V-cV^{(h)}_{1}=0$ from Lemma 3.4. ∎

For weighted homogeneous rational logarithmic vector fields
$V_{i}=\sum_{j=1}^{n}v_{i,j}(x)\partial_{x_{j}}=\sum_{j=1}^{n}v_{i,j}(x^{\prime},x_{n})\partial_{x_{j}}$
along $D$ satisfying $w(V_{1})\leq w(V_{2})\leq\dots\leq w(V_{n})$, we define

$M_{V}=(v_{n-i+1,\,j})_{1\leq i,j\leq n},$

that is, $M_{V}$ is the $n\times n$ matrix whose $(i,j)$-entry $(M_{V})_{i,j}$
of $M_{V}$ is equal to $v_{n-i+1,\,j}$.

######  Lemma 3.11.

Let $V_{i}$ be weighted homogeneous logarithmic vector fields with
$w(V_{1})\leq w(V_{2})\leq\dots\leq w(V_{n})$. Assume $\det
M_{V}(x)|_{x^{\prime}=0}=cx_{n}^{n}$ for a non-zero constant number $c$. Then
the following two facts hold.

$({\rm i})$ $w(V_{i})=w(V^{(h)}_{i})\,(=1-w(x_{n-i+1}))$, $1\leq i\leq n$.

$({\rm ii})$ There is a matrix $G(x^{\prime})\in GL(n,\mathbb{C}[x^{\prime}])$
such that $w(G(x^{\prime})_{i,j})=w(x_{j})-w(x_{i})$, and that

$\begin{pmatrix}V^{(h)}_{n}&V^{(h)}_{n-1}&\dots&V^{(h)}_{1}\end{pmatrix}^{t}=G(x^{\prime})\begin{pmatrix}V_{n}&V_{n-1}&\dots&V_{1}\end{pmatrix}^{t}$

holds. In particular, $V_{i}^{(h)}(x)$, $1\leq i\leq n$ are also logarithmic
vector fields, and hence $D$ is free.

###### Proof.

Proof of (i). Since $w(V_{i})\geq 0$ from Lemma 3.10 it holds that
$(M_{V})|_{x=0}=0$. Since $\det(M_{V})|_{x^{\prime}=0}=cx_{n}^{n}$, there are
$\nu_{i}$, $1\leq i\leq n$ such that
$\\{\nu_{1},\nu_{2},\dots,\nu_{n}\\}=\\{1,2,\dots n\\}$ and that
$v_{i,\nu_{i}}|_{x^{\prime}=0}=c_{i}x_{n}$ for some non-zero constant numbers
$c_{i}$, $1\leq i\leq n$. In this case, it holds that

$w(V_{i})=1-w(x_{\nu_{i}}),\ 1\leq i\leq n.$

In particular, we have

$w(V_{i})>0\Longleftrightarrow\nu_{i}<n,\quad\mbox{and}\quad
w(V_{i})<w(V_{j})\Longrightarrow\nu_{i}>\nu_{j}.$

These proves $w(V_{i})=1-w(x_{n-i+1})$.

Proof of (ii). From (i), we have

$w(v_{n-i+1,j})=1-w(x_{i})+w(x_{j}),$

and since $\det M_{V}(x)|_{x^{\prime}=0}=cx_{n}^{n}$, it holds that

$M_{V}(x)|_{x^{\prime}=0}=x_{n}R,$ (60)

where

$R=\begin{pmatrix}R_{1}&O&\dots&O\\\ O&R_{2}&\dots&O\\\
\cdot\cdot&\cdot\cdot&\dots&\cdot\cdot\\\ O&O&\dots&R_{k}\end{pmatrix}$

for some $R_{i}\in GL(n_{i},\mathbb{C})$ with $\sum_{i=1}^{k}n_{i}=n$. From
$w(x_{n-1})<w(x_{n})=1$, we have $n_{k}=1$. Then it holds
$R^{-1}M_{V}(x)|_{x^{\prime}=0}=x_{n}I_{n}.$

Since all the entries of $R^{-1}M_{V}(x)$ are linear function of $x_{n}$ with
the coefficients being weighted homogeneous polynomials in $x^{\prime}$, there
is an upper triangular matrix $\tilde{R}(x^{\prime})\in
GL(n,\mathbb{C}[x^{\prime}])$ such that

$\displaystyle
V_{1}^{(h)}\tilde{R}_{i,j}(x^{\prime})=(-w(x_{i})+w(x_{j}))\tilde{R}_{i,j}(x^{\prime}),$
(61) $\displaystyle\tilde{R}(x^{\prime})R^{-1}M_{V}(x)-x_{n}I_{n}\in GL(n,{\bf
C}[x^{\prime}]).$ (62)

The equality (62) and Lemma 3.6 implies

$\tilde{R}(x^{\prime})R^{-1}\begin{pmatrix}V_{n}&V_{n-1}&\dots&V_{1}\end{pmatrix}^{t}=\begin{pmatrix}V^{(h)}_{n}&V^{(h)}_{n-1}&\dots&V^{(h)}_{1}\end{pmatrix}^{t},$

which proves (ii). ∎

Lemma 3.11 and Lemma 3.5 imply the following.

######  Lemma 3.12.

Assume that $D$ is free. Then $V^{(h)}_{i}$, $1\leq i\leq n$ are logarithmic
vector fields along $D$. And, for any logarithmic vector field $V$ along $D$,
there are polynomials $a_{k}(x)$, $1\leq k\leq n$ in $x$ such that
$V=\sum_{k=1}^{n}a_{k}V^{(h)}_{k}$. In particular,
$\\{V_{i}^{(h)}\\}_{i=1,\dots,n}$ forms a unique system of generators of
logarithmic vector fields along $D$ such that its Saito matrix $M_{V^{(h)}}$
satisfies $M_{V^{(h)}}-x_{n}I_{n}\in\mathbb{C}[x^{\prime}]^{n\times n}$.

######  Remark 3.3.

In the case where $h(x)$ is the discriminant of a well-generated complex
reflection group, D. Bessis [1] showed the existence of a system of generators
of logarithmic vector fields along $D$ whose Saito matrix $M_{V}$ has the form
$M_{V}-x_{n}I_{n}\in\mathbb{C}[x^{\prime}]^{n\times n}$. Such a system of
generators is called flat in [1] (cf. Section 6).

##  4 Saito structure (without a metric)

B. Dubrovin [10] proved that there is a correspondence between semisimple
Frobenius manifolds and the spaces of isomonodromic deformations of linear
differential equations with special conditions. We shall give a framework
which captures isomonodromic deformations of generic differential equations of
Okubo type by relaxing some conditions for the definition of Frobenius
structure.

Our key ingredient is the following notion presented in C. Sabbah’s book [30]:

######  Definition 4.1 (C. Sabbah [30]).

Let $X$ be a complex analytic manifold of dimension $n$ (in this paper, we
mainly suppose that $X$ is a simply-connected domain $U$ in $\mathbb{C}^{n}$),
$TX$ be its tangent bundle, and $\Theta_{X}$ be the sheaf of holomorphic
sections of $TX$. A Saito structure (without a metric) on $X$ is a data that
consists of the following objects (i),(ii),(iii) that are subject to the
conditions (a), (b) below:

(i)

a flat torsion free connection  $\triangle$ on $TX$,

(ii)

a symmetric Higgs field $\Phi$ on $TX$,

(iii)

two global sections (vector fields) $e$ and $E$ of $TX$, respectively called
unit field and Euler field of the structure.

(a)

A meromorphic connection $\nabla$ on the bundle $\pi^{*}TX$ on
$\mathbb{P}^{1}\times X$ defined by

$\nabla=\pi^{*}\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}-(z\cdot{\rm
Id}+\Phi(E))^{-1}(\pi^{*}\Phi)\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}E-(z\cdot{\rm
Id}+\Phi(E))^{-1}\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}E\,dz$
(63)

is integrable, where $\pi$ is the projection $\pi:\mathbb{P}^{1}\times
X\rightarrow X$ and $z$ is a non-homogeneous coordinate of $\mathbb{P}^{1}$,

(b)

the field $e$ is  $\triangle$ -horizontal (i.e.,
$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}(e)=0$) and satisfies
$\Phi_{e}={\rm Id}$, where we regard $\Phi$ as an ${\it
End}_{\mathcal{O}_{X}}(\Theta_{X})$-valued $1$-form and $\Phi_{e}\in{\it
End}_{\mathcal{O}_{X}}(\Theta_{X})$ denotes the contraction of the vector
field $e$ and the $1$-form $\Phi$.

######  Remark 4.1.

In [30], $\nabla$ is defined as a meromorphic connection with an irregular
singularity, which is equivalent to our definition (63) via the Fourier-
Laplace transform of meromorphic differential operators.

######  Remark 4.2.

To the Higgs field $\Phi$ there associates a product $\star$ on $\Theta_{X}$
defined by $\xi\star\eta=\Phi_{\xi}(\eta)$ for $\xi,\eta\in\Theta_{X}$. The
Higgs field $\Phi$ is said to be symmetric if the product $\star$ is
commutative. The condition $\Phi_{e}={\rm Id}$ in Definition 4.1 (b) implies
that the field $e$ is the unit of the product $\star$. The integrability of
the connection $\nabla$ implies that of the Higgs field $\Phi$, which is
equivalent to the associativity of $\star$. So the product $\star$ associated
to a Saito structure is commutative and associative.

Since the connection  $\triangle$ is flat torsion free, we can take a flat
coordinate system $(t_{1},\dots,t_{n})$ such that
$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}(\partial_{t_{i}})=0$
$(i=1,\dots,n)$ at least on a simply-connected open set of $X$. We assume the
existence of a flat coordinate system $(t_{1},\dots,t_{n})$ on $X$ replacing
$X$ by such an open set if necessary. In addition, we assume the following
conditions:

(C1) $e=\partial_{t_{n}}$,

(C2) $E=w_{1}t_{1}\partial_{t_{1}}+\cdots+w_{n}t_{n}\partial_{t_{n}}$ for
$w_{i}\in\mathbb{C}\>(i=1,\dots,n)$,

(C3) $w_{n}=1$ and $w_{i}-w_{j}\not\in\mathbb{Z}$ for $i\neq j$.
In this paper, a function $f\in\mathcal{O}_{X}$ is said to be weighted
homogeneous with a weight $w(f)\in\mathbb{C}$ if $f$ is an eigenfunction of
the Euler operator: $Ef=w(f)f$. In particular, the flat coordinates
$t_{i}\>(i=1,\dots,n)$ are weighted homogeneous with $w(t_{i})=w_{i}$.

The following lemma is easy but useful.

######  Lemma 4.1.

For $w\in\mathbb{C}\setminus\\{0\\}$, let
$f_{i}\in\mathcal{O}_{X}\>(i=1,\dots,n)$ be weighted homogeneous function with
$w(f_{i})=w-w_{i}$. If the $1$-form
$\omega=\displaystyle\sum_{i=1}^{n}f_{i}dt_{i}$ is closed (i.e., $d\omega=0$),
then

$f=w^{-1}\sum_{i=1}^{n}w_{i}t_{i}f_{i}$

is a unique weighted homogeneous function with $w(f)=w$ such that $df=\omega$.

###### Proof.

It can be proved straightforwardly. ∎

We write $\Phi\in\mbox{{\it
End}}_{\mathcal{O}_{X}}(\Theta_{X})\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X}$ as
$\Phi=\displaystyle\sum_{k=1}^{n}\Phi^{(k)}dt_{k}$, where
$\Phi^{(k)}\in\mbox{{\it End}}_{\mathcal{O}_{X}}(\Theta_{X})\>(k=1,\ldots,n)$.
We fix the basis $\\{\partial_{t_{1}},\dots,\partial_{t_{n}}\\}$ of
$\Theta_{X}$ and introduce the following matrices:

(i) $\tilde{\mathcal{B}}^{(k)}\>(k=1,\dots,n)$ is the representation matrix of
$\Phi^{(k)}$, namely the $(i,j)$-entry $\tilde{\mathcal{B}}^{(k)}_{ij}$ is
defined by

$\Phi^{(k)}(\partial_{t_{i}})=\sum_{j=1}^{n}\tilde{\mathcal{B}}^{(k)}_{ij}\partial_{t_{j}}\quad(i=1,\ldots,n),$
(64)

(ii) $\mathcal{T}$ and $\mathcal{B}_{\infty}$ are the representation matrices
of $-\Phi(E)$ and $\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}E$
respectively, namely

$-\Phi_{\partial_{t_{i}}}(E)=\sum_{j=1}^{n}\mathcal{T}_{ij}\partial_{t_{j}},\
\ \ \
\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}_{\partial_{t_{i}}}(E)=\sum_{j=1}^{n}(\mathcal{B}_{\infty})_{ij}\partial_{t_{j}}.$
(65)

We assume that $-\Phi(E)$ is generically regular semisimple on $X$, that is
the discriminant of $\det(z-\mathcal{T})$ does not identically vanish on $X$.

######  Lemma 4.2.

$\mathcal{B}_{\infty}=\mbox{diag}\,[w_{1},\dots,w_{n}]$.

###### Proof.

Since $E=\sum_{i=1}^{n}w_{i}t_{i}\partial_{t_{i}}$ and
$(\partial_{t_{1}},\dots,\partial_{t_{n}})$ is flat, we have

$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}(E)=\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\Bigl{(}\sum_{i=1}^{n}w_{i}t_{i}\partial_{t_{i}}\Bigr{)}=\sum_{i=1}^{n}w_{i}dt_{i}\partial_{t_{i}}.$

∎

######  Remark 4.3.

The weighted homogeneous flat coordinate system is unique up to multiplication
by non-zero constants in virtue of (C3), that is, for another weighted
homogeneous flat coordinate system $(t_{1}^{\prime},\dots,t_{n}^{\prime})$,
there are constants $c_{i}\neq 0,$ $i=1,\dots,n-1,$ such that
$(t_{1}^{\prime},\dots,t_{n-1}^{\prime},t_{n}^{\prime})=(c_{1}t_{1},\dots,c_{n-1}t_{n-1},t_{n})$
(notice that $\partial_{t_{n}^{\prime}}=\partial_{t_{n}}=e$). Correspondingly,
the representation matrix $\tilde{B}^{(k)^{\prime}}$ of $\Phi$ with respect to
$(\partial_{t_{1}}^{\prime},\dots,\partial_{t_{n}}^{\prime})$ is written as
$\tilde{B}^{(k)^{\prime}}_{ij}=c_{i}^{-1}c_{j}c_{k}^{-1}\tilde{B}^{(k)}_{ij}$.

######  Proposition 4.1.

The meromorphic connection $\nabla$ is integrable if and only if
$\mathcal{T}$, $\mathcal{B}_{\infty}$ and
$\tilde{\mathcal{B}}^{(i)}\>(i=1,\dots,n)$ are subject to the following
relations

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial\tilde{\mathcal{B}}^{(i)}}{\partial
t_{j}}=\frac{\partial\tilde{\mathcal{B}}^{(j)}}{\partial t_{i}},\ \
i,j=1,\dots,n,\\\
\bigl{[}\tilde{\mathcal{B}}^{(i)},\tilde{\mathcal{B}}^{(j)}\bigr{]}=O,\ \
i,j=1,\dots,n,\\\ \bigl{[}\mathcal{T},\tilde{\mathcal{B}}^{(i)}\bigr{]}=O,\ \
i=1,\dots,n,\\\ \displaystyle\frac{\partial\mathcal{T}}{\partial
t_{i}}+\tilde{\mathcal{B}}^{(i)}+[\tilde{\mathcal{B}}^{(i)},\mathcal{B}_{\infty}]=O,\
\ i=1,\dots,n.\end{array}\right.$ (66)

###### Proof.

See (2.6) on p.203, [30]. ∎

######  Remark 4.4.

The relation (66) is nothing but the conditions (27), (28), (29) for the
matrices $T,\>{\tilde{B}}^{(i)}\>(i=1,\ldots,n),B_{\infty}$ in the definition
of an Okubo type system in several variables in Section 2, which means that
the existence of a Saito structure gives rise to an Okubo type system in
several variables. In the next section, we will find a criterion for
determining whether a given Okubo type system in several variables arises from
a Saito structure in this way.

######  Lemma 4.3.

A Higgs field $\Phi$ is symmetric if and only if
$\tilde{\mathcal{B}}^{(k)}_{ij}=\tilde{\mathcal{B}}^{(i)}_{kj}$.

###### Proof.

It is obvious from
$\partial_{t_{k}}\star\partial_{t_{i}}=\Phi^{(k)}(\partial_{t_{i}})=\sum_{j=1}^{n}\tilde{\mathcal{B}}^{(k)}_{ij}\partial_{t_{j}}.$
∎

######  Lemma 4.4.

Define vector fields $V_{i}\>(i=1,\ldots,n)$ by

$\begin{pmatrix}V_{n}\\\ \vdots\\\
V_{1}\end{pmatrix}=-\mathcal{T}\begin{pmatrix}\partial_{t_{1}}\\\ \vdots\\\
\partial_{t_{n}}\end{pmatrix}.$ (67)

Then

$V_{k}h=(-1)^{n+1}\mbox{tr}\,\tilde{\mathcal{B}}^{(k)}h,\ \ \ \ k=1,\dots,n$

with $h=h(t)=\det(-\mathcal{T})$. In particular, $V_{i}$, $i=1,\dots,n,$ are
logarithmic vector fields along $D=\\{t\in X;h(t)=0\\}$, and $D$ is a free
divisor.

###### Proof.

By $h=\det(-\mathcal{T})$, it holds that

$(-1)^{n}V_{k}h=\begin{vmatrix}V_{k}\mathcal{T}[1]\\\ \mathcal{T}[2]\\\
\vdots\\\ \mathcal{T}[n]\end{vmatrix}+\begin{vmatrix}\mathcal{T}[1]\\\
V_{k}\mathcal{T}[2]\\\ \vdots\\\
\mathcal{T}[n]\end{vmatrix}+\dots+\begin{vmatrix}\mathcal{T}[1]\\\
\mathcal{T}[2]\\\ \vdots\\\ V_{k}\mathcal{T}[n]\end{vmatrix},\ \ \
k=1,\dots,n,$ (68)

where $\mathcal{T}[i]=(\mathcal{T}_{i1},\dots,\mathcal{T}_{in})$ denotes the
$i$-th raw vector of $\mathcal{T}$. We can write

$V_{k}\mathcal{T}[i]=f^{k}_{i1}\mathcal{T}[1]+\dots+f^{k}_{in}\mathcal{T}[n],\
\ \ \ hf^{k}_{ij}\in\mathcal{O}_{X}$ (69)

Let $F^{(k)}=(f^{k}_{ij})$. Then (69) is rewritten as
$V_{k}\mathcal{T}=F^{(k)}\mathcal{T}$, namely we have
$F^{(k)}=V_{k}\mathcal{T}\cdot\mathcal{T}^{-1}$. In turn, we have

$(-1)^{n}V_{k}h=(f^{k}_{11}+f^{k}_{22}+\dots+f^{k}_{nn})h=\mbox{tr}\,(F^{(k)})\,h.$

Therefore we have only to show that
$\mbox{tr}\,(V_{k}\mathcal{T}\cdot\mathcal{T}^{-1})=-\mbox{tr}\,\tilde{\mathcal{B}}^{(k)}$.

From the integrability condition

$\frac{\partial\mathcal{T}}{\partial
t_{i}}=-\tilde{\mathcal{B}}^{(i)}-[\tilde{\mathcal{B}}^{(i)},\mathcal{B}_{\infty}],$

we have

$V_{k}\mathcal{T}=\sum_{l}\mathcal{T}_{kl}\frac{\partial\mathcal{T}}{\partial
t_{l}}=-\sum_{l}\mathcal{T}_{kl}\tilde{\mathcal{B}}^{(l)}-\Bigl{[}\sum_{l}\mathcal{T}_{kl}\tilde{\mathcal{B}}^{(l)},\mathcal{B}_{\infty}\Bigr{]}.$

Noting the symmetry
$\tilde{\mathcal{B}}^{(l)}_{ij}=\tilde{\mathcal{B}}^{(i)}_{lj}$ and the
commutativity
$\mathcal{T}\tilde{\mathcal{B}}^{(i)}=\tilde{\mathcal{B}}^{(i)}\mathcal{T}$,
we have

$\displaystyle(V_{k}\mathcal{T})_{ij}$
$\displaystyle=-(1+w_{j}-w_{i})\sum_{l}\mathcal{T}_{kl}\tilde{\mathcal{B}}^{(l)}_{ij}$
$\displaystyle=-(1+w_{j}-w_{i})\sum_{l}\mathcal{T}_{kl}\tilde{\mathcal{B}}^{(i)}_{lj}$
$\displaystyle=-(1+w_{j}-w_{i})\sum_{l}\tilde{\mathcal{B}}^{(i)}_{kl}\mathcal{T}_{lj}$
$\displaystyle=-(1+w_{j}-w_{i})\sum_{l}\tilde{\mathcal{B}}^{(k)}_{il}\mathcal{T}_{lj},$

and thus

$\displaystyle V_{k}\mathcal{T}$
$\displaystyle=-\tilde{\mathcal{B}}^{(k)}\mathcal{T}-\tilde{\mathcal{B}}^{(k)}\mathcal{T}\mathcal{B}_{\infty}+\mathcal{B}_{\infty}\tilde{\mathcal{B}}^{(k)}\mathcal{T}$
$\displaystyle=-\tilde{\mathcal{B}}^{(k)}\mathcal{T}-\mathcal{T}\tilde{\mathcal{B}}^{(k)}\mathcal{B}_{\infty}+\mathcal{B}_{\infty}\tilde{\mathcal{B}}^{(k)}\mathcal{T}.$

Hence we obtain

$\displaystyle\mbox{tr}\,(V_{k}\mathcal{T}\cdot\mathcal{T}^{-1})$
$\displaystyle=-\mbox{tr}\,\tilde{\mathcal{B}}^{(k)}-\mbox{tr}\,(\tilde{\mathcal{B}}^{(k)}\mathcal{B}_{\infty})+\mbox{tr}\,(\mathcal{B}_{\infty}\tilde{\mathcal{B}}^{(k)})$
$\displaystyle=-\mbox{tr}\,\tilde{\mathcal{B}}^{(k)}.$

∎

######  Remark 4.5.

It is shown in [30] that $D$ is a free divisor. However our proof employed
here is more informative.

######  Lemma 4.5.

There is a unique matrix $\mathcal{C}$ such that

$\mathcal{T}=-E\mathcal{C},\ \ \
\tilde{\mathcal{B}}^{(i)}=\frac{\partial\mathcal{C}}{\partial t_{i}},\ \
i=1,\dots,n$

and that each matrix entry $\mathcal{C}_{ij}$ of $\mathcal{C}$ is weighted
homogeneous with $w(\mathcal{C}_{ij})=1-w_{i}+w_{j}$.

###### Proof.

Since $\mathcal{T}$ is the representation matrix of $-\Phi(E)$, we have

$\sum_{j=1}^{n}\mathcal{T}_{ij}\partial_{t_{j}}=-\Phi_{\partial_{t_{i}}}(E)=-\sum_{k=1}^{n}w_{k}t_{k}\Phi_{\partial_{t_{i}}}(\partial_{t_{k}})=-\sum_{j,k=1}^{n}w_{k}t_{k}\tilde{\mathcal{B}}^{(i)}_{kj}\partial_{t_{j}}.$
(70)

From the last of the equations (66), we have

$\frac{\partial\mathcal{T}_{ij}}{\partial
t_{k}}=-(1+w_{j}-w_{i})\tilde{\mathcal{B}}^{(k)}_{ij}.$

Therefore it holds that

$E\mathcal{T}_{ij}=-\sum_{k=1}^{n}w_{k}(1+w_{j}-w_{i})t_{k}\tilde{\mathcal{B}}^{(k)}_{ij}=(1+w_{j}-w_{i})\mathcal{T}_{ij},$

which implies that $\mathcal{T}_{ij}$ is weighted homogeneous with
$w(\mathcal{T}_{ij})=1+w_{j}-w_{i}$ and $\tilde{\mathcal{B}}^{(k)}_{ij}$ is
weighted homogeneous with
$w(\tilde{\mathcal{B}}^{(k)}_{ij})=1+w_{j}-w_{i}-w_{k}$. In virtue of Lemma
4.1 and the first of the equations (66), there is a unique matrix
$\mathcal{C}$ such that $\frac{\partial\mathcal{C}}{\partial
t_{i}}=\tilde{\mathcal{B}}^{(i)}$ and that each matrix entry
$\mathcal{C}_{ij}$ of $\mathcal{C}$ is weighted homogeneous with
$w(\mathcal{C}_{ij})=1+w_{j}-w_{i}$. It follows that
$\mathcal{T}=-E\mathcal{C}$ from (70). ∎

######  Lemma 4.6.

Let $(t_{1},\dots,t_{n})$ be a flat coordinate system of the Saito structure.
Then it holds that $\mathcal{T}_{nj}=-w_{j}t_{j}$ (or equivalently
$\mathcal{C}_{nj}=t_{j}$), $j=1,\dots,n.$

###### Proof.

As in Lemma 4.4, define vector fields $V_{i}\>(i=1,\ldots,n)$ by
$\begin{pmatrix}V_{n}&\dots&V_{1}\end{pmatrix}^{t}=-\mathcal{T}\begin{pmatrix}\partial_{t_{1}}&\dots&\partial_{t_{n}}\end{pmatrix}^{t}.$
Then Lemma 4.4 shows that each $V_{i}$ is logarithmic along $D$. Since
$h(t)=\det(-\mathcal{T})$ is weighted homogeneous with $w(h)=n$, it follows
that $Eh=nh$, which implies that $E$ is a logarithmic vector field along $D$.
Then Lemma 3.4 shows $V_{1}=E$, from which we have
$-\mathcal{T}_{nj}=w_{j}t_{j}$. On the other hand, it holds that
$-\mathcal{T}_{nj}=E\mathcal{C}_{nj}=w_{j}\mathcal{C}_{nj}$ since
$\mathcal{C}_{nj}$ is weighted homogeneous with $w(\mathcal{C}_{nj})=w_{j}$.
Hence we obtain $\mathcal{C}_{nj}=t_{j}$. ∎

######  Proposition 4.2.

There is a unique $n$-tuple of analytic functions
$\vec{g}=(g_{1},\dots,g_{n})\in\mathcal{O}_{X}^{n}$ such that

$\mathcal{C}_{ij}=\frac{\partial g_{j}}{\partial t_{i}}$

and that $g_{j}$ is weighted homogeneous with $w(g_{j})=1+w_{j}$.

###### Proof.

By the symmetry of the Higgs field $\Phi$, it holds that

$\frac{\partial\mathcal{C}_{ij}}{\partial
t_{k}}=\frac{\partial\mathcal{C}_{kj}}{\partial t_{i}},\ \ \ i,j,k=1,\dots,n.$

Hence, in virtue of Lemma 4.1, there is $g_{j}$ such that

$\frac{\partial g_{j}}{\partial t_{i}}=\mathcal{C}_{ij},\ \ \ i,j=1,\dots,n,$

and that $g_{j}$ is weighted homogeneous with $w(g_{j})=1+w_{j}$. ∎

######  Definition 4.2 (Konishi-Minabe [26]).

The vector $\vec{g}$ in Proposition 4.2 (or precisely the vector field
$\mathcal{G}=\sum_{i=1}^{n}g_{i}\partial_{t_{i}}$) is called a potential
vector field of the Saito structure.

######  Proposition 4.3.

The potential vector field $\vec{g}=(g_{1},\dots,g_{n})$ is a solution to the
following system of nonlinear equations:

$\displaystyle\sum_{m=1}^{n}\frac{\partial^{2}g_{m}}{\partial t_{k}\partial
t_{i}}\frac{\partial^{2}g_{j}}{\partial t_{l}\partial
t_{m}}=\sum_{m=1}^{n}\frac{\partial^{2}g_{m}}{\partial t_{l}\partial
t_{i}}\frac{\partial^{2}g_{j}}{\partial t_{k}\partial t_{m}},\ \
i,j,k,l=1,\dots,n,$ (71) $\displaystyle\frac{\partial^{2}g_{j}}{\partial
t_{n}\partial t_{i}}=\delta_{ij},\ \ \ \ i,j=1,\dots,n,$ (72) $\displaystyle
Eg_{j}=\sum_{k=1}^{n}w_{k}t_{k}\frac{\partial g_{j}}{\partial
t_{k}}=(1+w_{j})g_{j},\ \ \ j=1,\dots,n.$ (73)

###### Proof.

The associativity of $\star$ (i.e., the integrability of $\Phi$) is equivalent
to
$\tilde{\mathcal{B}}^{(k)}\tilde{\mathcal{B}}^{(l)}=\tilde{\mathcal{B}}^{(l)}\tilde{\mathcal{B}}^{(k)}$.
We obtain (71) if we rewrite this condition in terms of
$\tilde{\mathcal{B}}^{(k)}_{ij}=\displaystyle\frac{\partial^{2}g_{j}}{\partial
t_{k}\partial t_{i}}$. The equation (72) follows from
$\tilde{\mathcal{B}}^{(n)}=I_{n}$ (i.e., $\Phi_{e}=\mbox{Id}$). ∎

######  Definition 4.3.

The system of non-linear differential equations
$(\ref{eq:genWDVV}),(\ref{eq:vpunit}),(\ref{eq:vphomo})$ for the vector
$\vec{g}=(g_{1},\ldots,g_{n})$ is called the extended WDVV equation.

Conversely, starting with a solution of (71), (72) and (73), it is possible to
reconstruct a Saito structure.

######  Proposition 4.4.

Take constants $w_{j}\in\mathbb{C},$ $j=1,\dots,n$ satisfying
$w_{i}-w_{j}\not\in\mathbb{Z}$ and $w_{n}=1$ and assume that
$\vec{g}=(g_{1},\dots,g_{n})$ is a holomorphic solution of (71), (72) and (73)
on a simply connected domain $U$ in $\mathbb{C}^{n}$. Then there is a Saito
structure on $U$ which has $(t_{1},\dots,t_{n})$ as a flat coordinate system.

###### Proof.

Define $E=\sum_{i=1}^{n}w_{i}t_{i}\partial_{t_{i}},$ $e=\partial_{t_{n}},$
$\tilde{\mathcal{B}}^{(k)}_{ij}=\frac{\partial^{2}g_{j}}{\partial
t_{k}\partial t_{i}},$ $\Phi=\sum_{k=1}^{n}\tilde{\mathcal{B}}^{(k)}dt_{k}$
and
$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}(\partial_{t_{i}})=0,$
$i=1,\dots,n$. Then the conditions (a), (b) of Definition 4.1 hold and
$\vec{g}$ is the potential vector field associated to $\mathcal{C}$. ∎

######  Remark 4.6.

As stated in Remark 4.3, any two weighted-homogeneous flat coordinate systems
$(t^{\prime}_{1},\dots,t^{\prime}_{n})$ and $(t_{1},\dots,t_{n})$ are
connected by
$(t_{1}^{\prime},\dots,t_{n-1}^{\prime},t_{n}^{\prime})=(c_{1}t_{1},\dots,c_{n-1}t_{n-1},t_{n})$.
Let $(g_{j}^{\prime}(t^{\prime}_{1},\dots,t^{\prime}_{n}))_{1\leq j\leq n}$
and $(g_{j}(t_{1},\dots,t_{n}))_{1\leq j\leq n}$ be the potential vector
fields with respect to $(t^{\prime}_{1},\dots,t^{\prime}_{n})$ and
$(t_{1},\dots,t_{n})$ respectively. Then it holds that
$g_{j}^{\prime}(t^{\prime}_{1},\dots,t^{\prime}_{n})=c_{j}g_{j}(t_{1},\dots,t_{n}),$
$j=1,\dots,n.$ In other words, given a solution
$(g_{j}(t_{1},\dots,t_{n}))_{1\leq j\leq n}$ to the extended WDVV equation
(71),(72),(73), then
$(c_{j}g_{j}(c_{1}^{-1}t_{1},\dots,c_{n-1}^{-1}t_{n-1},t_{n}))_{1\leq j\leq
n}$ is also a solution to it.

Let $J$ be an $n\times n$-matrix with $J_{ij}=\delta_{i+j,n+1}$,
$i,j=1,\dots,n$, where $\delta_{ij}$ denotes Kronecker’s delta, and, for an
$n\times n$ matrix $A$, define $A^{*}$ by $A^{*}=JA^{t}J$.

######  Proposition 4.5.

Given a Saito structure on $X$, the following conditions are mutually
equivalent:

(i)

For appropriate normalization of the flat coordinate system, it holds that
$\mathcal{C}^{*}=\mathcal{C}$.

(ii)

For appropriate normalization of the flat coordinate system, there is a
holomorphic function $F\in\mathcal{O}_{X}$ such that

$\frac{\partial F}{\partial t_{i}}=g_{n+1-i}=(\vec{g}J)_{i},\ \ \
i=1,\dots,n.$ (74)

(iii)

There is $r\in\mathbb{C}$ such that

$w_{n+1-i}+w_{i}=-2r,\ \ \ \ i=1,\dots,n,$ (75)

and there is a metric $\eta$ (in this paper, “metric” means non-degenerate
symmetric $\mathbb{C}$-bilinear form on $TX$) such that

$\displaystyle\eta(\sigma\star\xi,\zeta)=\eta(\sigma,\xi\star\zeta),\ \ \ \
(\mbox{compatibility to the product})$ (76)
$\displaystyle(\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\eta)(\xi,\zeta):=d(\eta(\xi,\zeta))-\eta(\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\xi,\zeta)-\eta(\xi,\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\zeta)=0,\
\ \ \ (\mbox{horizontality})$ (77)
$\displaystyle(E\eta)(\xi,\zeta):=E(\eta(\xi,\zeta))-\eta(E\xi,\zeta)-\eta(\xi,E\zeta)=-2r\eta(\xi,\zeta),\
\ (\mbox{homogeneity})$ (78)

for any $\sigma,\xi,\zeta\in\Theta_{X}$.

Here, as stated in Remark 4.3 and Remark 4.6, a flat coordinate system admits
indetermination of multiplication by constants. “Normalization” in the above
conditions means to fix this indetermination.

###### Proof.

$({\rm i})\Leftrightarrow({\rm ii})$ By definition,
$\mathcal{C}^{*}=\mathcal{C}$ is equivalent to that $\mathcal{C}J$ is a
symmetric matrix. From

$\frac{\partial g_{n+1-j}}{\partial t_{i}}=(\mathcal{C}J)_{ij},$

we find that the symmetry of $\mathcal{C}J$ is equivalent to the existence of
$F\in\mathcal{O}_{X}$ such that $\frac{\partial F}{\partial t_{i}}=g_{n+1-i}$.

$({\rm i})\Rightarrow({\rm iii})$ First, we show $w_{n+1-i}+w_{i}=-2r,$
$i=1,\dots,n$ for some $r\in\mathbb{C}$. Note that, if
$\mathcal{C}^{*}=\mathcal{C}$ holds, then it also holds that
$\mathcal{T}^{*}=\mathcal{T}$ and
$\tilde{\mathcal{B}}^{(i)*}=\tilde{\mathcal{B}}^{(i)},$ $i=1,\dots,n$. By the
integrability condition, we have

$\frac{\partial\mathcal{T}}{\partial
x_{i}}+\tilde{\mathcal{B}}^{(i)}+[\tilde{\mathcal{B}}^{(i)},\mathcal{B}_{\infty}]=O,\
\ \ \ \frac{\partial\mathcal{T}^{*}}{\partial
x_{i}}+\tilde{\mathcal{B}}^{(i)*}-[\tilde{\mathcal{B}}^{(i)*},\mathcal{B}_{\infty}^{*}]=O.$

Taking the difference of these two equalities, we have

$[\tilde{\mathcal{B}}^{(i)},\mathcal{B}_{\infty}+\mathcal{B}_{\infty}^{*}]=O,\
\ \ \ i=1,\dots,n.$

Since $\mathcal{T}$ is written as

$\mathcal{T}=-\sum_{i=1}^{n}w_{i}t_{i}\tilde{\mathcal{B}}^{(i)},$

we have $[\mathcal{T},\mathcal{B}_{\infty}+\mathcal{B}_{\infty}^{*}]=O$.
Besides, we note that $\mathcal{T}_{nj}=-w_{j}t_{j}\neq 0,$ $j=1,\dots,n$.
Then it follows that
$\mathcal{B}_{\infty}+\mathcal{B}_{\infty}^{*}=\mbox{diag}[w_{1}+w_{n},w_{2}+w_{n-1},\dots,w_{n}+w_{1}]$
is a scalar matrix.

Next, we show the existence of a metric with the desired conditions. Under the
assumption that $\mathcal{C}J$ is a symmetric matrix, define a metric $\eta$
by $\eta(\partial_{t_{i}},\partial_{t_{j}})=J_{ij},$ $i,j=1,\dots,n$ for the
flat coordinate system $(t_{1},\dots,t_{n})$. Then, on one hand, we have

$\eta(\partial_{t_{i}}\star\partial_{t_{k}},\partial_{t_{j}})=\eta\Bigl{(}\sum_{l=1}^{n}\tilde{\mathcal{B}}^{k}_{il}\partial_{t_{l}},\partial_{t_{j}}\Bigr{)}=\sum_{l=1}^{n}\frac{\partial\mathcal{C}_{il}}{\partial{t_{k}}}J_{lj}=\frac{\partial(\mathcal{C}J)_{ij}}{\partial
t_{k}},$

on the other hand

$\eta(\partial_{t_{i}},\partial_{t_{k}}\star\partial_{t_{j}})=\eta\Bigl{(}\partial_{t_{i}},\sum_{l=1}^{n}\tilde{\mathcal{B}}^{k}_{jl}\partial_{t_{l}}\Bigr{)}=\frac{\partial(\mathcal{C}J)_{ji}}{\partial
t_{k}},$

which concludes the compatibility of $\eta$ to the product. The horizontality
and the homogeneity hold obviously.

$({\rm iii})\Rightarrow({\rm i})$ By the horizontality, it follows that
$\eta(\partial_{t_{i}},\partial_{t_{j}})$ $(i,j=1,\dots,n)$ are constants.
From the homogeneity, it holds that

$(w_{i}+w_{j})\eta(\partial_{t_{i}},\partial_{t_{j}})=-2r\,\eta(\partial_{t_{i}},\partial_{t_{j}}),$

which implies $\eta(\partial_{t_{i}},\partial_{t_{j}})=c_{i}J_{ij}$ for non-
zero constants $c_{i}$, $i=1,\dots,n$. By normalizing the flat coordinates, we
may suppose $\eta(\partial_{t_{i}},\partial_{t_{j}})=J_{ij}$. Then it holds
that

$\displaystyle\eta(\partial_{t_{i}}\star\partial_{t_{k}},\partial_{t_{j}})=\tilde{\mathcal{B}}^{(k)}_{i,n+1-j}$
$\displaystyle=$
$\displaystyle\eta(\partial_{t_{i}},\partial_{t_{k}}\star\partial_{t_{j}})=\tilde{\mathcal{B}}^{(k)}_{j,n+1-i},$

which implies $\tilde{\mathcal{B}}^{(k)*}=\tilde{\mathcal{B}}^{(k)}$,
$k=1,\dots,n$. Hence $\mathcal{T}^{*}=\mathcal{T}$ and
$\mathcal{C}^{*}=\mathcal{C}$ also hold. ∎

The function $F$ in Proposition 4.5 is called a prepotential in [10] and a
potential in [30]. Similarly to the proof of Proposition 4.3, we can show that
the prepotential $F$ satisfies the following equations.

######  Proposition 4.6.

The prepotential $F$ is a solution of the system of nonlinear equations

$\displaystyle\sum_{m=1}^{n}\frac{\partial^{3}F}{\partial t_{k}\partial
t_{i}\partial t_{m}}\frac{\partial^{3}F}{\partial t_{l}\partial t_{j}\partial
t_{n+1-m}}=\sum_{m=1}^{n}\frac{\partial^{3}F}{\partial t_{l}\partial
t_{i}\partial t_{m}}\frac{\partial^{3}F}{\partial t_{k}\partial t_{j}\partial
t_{n+1-m}},\ \ i,j,k,l=1,\dots,n,$ (79)
$\displaystyle\frac{\partial^{3}F}{\partial t_{n}\partial t_{i}\partial
t_{j}}=J_{ij},\ \ \ \ i,j=1,\dots,n,$ (80) $\displaystyle
EF=\sum_{k=1}^{n}w_{k}t_{k}\frac{\partial F}{\partial t_{k}}=(1-2r)F,$ (81)

where $-2r=w_{i}+w_{n+1-i},$ $i=1,\dots,n$. The equation (79) is called the
WDVV equation [10].

A Saito structure with a metric $\eta$ in Proposition 4.5 is equivalent to a
Frobenius structure ([30]).

######  Remark 4.7.

The prepotential $F$ retains a part of the indetermination of $\vec{g}$ in
Remark 4.6. Indeed, if $F(t_{1},\dots,t_{n})$ is a solution to the WDVV
equation, then $cF(c_{1}^{-1}t_{1},\dots,c_{n-1}^{-1}t_{n-1},t_{n})$ with
$c_{i}c_{n+1-i}=c$ $(i=1,\dots,n,c\in\mathbb{C}\setminus\\{0\\})$ is also a
solution to the same equation.

##  5 Flat structure on the space of isomonodromic deformations

In this section, we find the condition for that an Okubo type system in
several variables has a flat structure (without a potential). That condition
turns out to be a generic one, from which we obtain the equivalence between
the extended WDVV equation and the (full-parameter) Painlevé VI equation. In
the last part of this section, we treat the case of flat structure with a
potential. This case corresponds to the Frobenius structure by Dubrovin. And
his result on the equivalence between the WDVV equation and a one-parameter
family of the Painlevé VI equation is shown in an alternative way.

###  5.1 Flat structure (without a potential)

We start with an Okubo type system in several variables:

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial Y}{\partial
z}=-(zI_{n}-T)^{-1}B_{\infty}Y,\\\ \displaystyle\frac{\partial Y}{\partial
x_{i}}=-(zI_{n}-T)^{-1}\tilde{B}^{(i)}B_{\infty}Y,\ \ \ \
i=1,\dots,n.\end{array}\right.$ (82)

Let the assumptions on (82) be same as in Section 2. The purpose of this
subsection is to find a necessary and sufficient condition for that the Okubo
type system (82) has a flat structure. Here, to avoid the confusion, we state
the precise definition of that an Okubo type system (82) has a flat structure.

######  Definition 5.1.

We say that an Okubo type system (82) has a flat structure on $U$ if there is
a Saito structure on $U$ such that there is a change of independent variables
$(t_{1},\dots,t_{n})=(t_{1}(x),\dots,t_{n}(x))$, where $(t_{1},\dots,t_{n})$
is a flat coordinate system, and the matrices $\mathcal{T}$,
$\mathcal{B}_{\infty}$, $\tilde{\mathcal{B}}^{(i)}\>(i=1,\dots,n)$ defined
from the Saito structure satisfy $\mathcal{T}=T$,
$\mathcal{B}_{\infty}=B_{\infty}-(\lambda_{n}-1)I_{n}$,
$\tilde{\mathcal{B}}^{(i)}=\sum_{j=1}^{n}\frac{\partial x_{j}}{\partial
t_{i}}\tilde{B}^{(j)}$ $(i=1,\dots,n)$.

We consider (82) restricted on $W\subset U\setminus\Delta_{H}$ for a while.
(Recall that $W$ is subject to (W) in Section 2.) Take an invertible matrix
$P$ such that

$P^{-1}TP=\mbox{diag}\,[z_{1}(x),\dots,z_{n}(x)],\ \ \ \
P^{-1}\tilde{B}^{(i)}P=\mbox{diag}\,\Bigl{[}-\frac{\partial z_{1}(x)}{\partial
x_{i}},\dots,-\frac{\partial z_{n}(x)}{\partial x_{i}}\Bigr{]}.$

We put $Q=P^{-1}$.

######  Lemma 5.1.

An Okubo type system (82) has a flat structure on $W$ if and only if the
following conditions hold:

$({\rm i})$ It holds that

$\displaystyle\frac{\partial Q_{kj}}{\partial z_{l}}=\frac{\partial
Q_{lj}}{\partial z_{k}},\ \ \ j,k,l=1,\dots,n,$ (83)
$\displaystyle\sum_{k=1}^{n}z_{k}\frac{\partial Q_{ij}}{\partial
z_{k}}=(\lambda_{j}-\lambda_{n})Q_{ij},\ \ \ i,j=1,\dots,n.$ (84)

$({\rm ii})$ $P_{nj}\neq 0$ $(j=1,\dots,n)$ at any point on $W$.
Note that the condition $({\rm ii})$ is equivalent to that we can choose $P$
such that $P_{nj}=-1$ $(j=1,\dots,n)$ identically on $W$.

###### Proof.

Firstly, we assume that (82) has a flat structure on $W$. Let
$(t_{1},\dots,t_{n})$ be a flat coordinate system on $W$. Then, by the change
of the independent variables from $\\{x_{1},\dots,x_{n}\\}$ to
$\\{t_{1},\dots,t_{n}\\}$, (82) is changed to

$\frac{\partial Y}{\partial
t_{i}}=-(zI_{n}-T)^{-1}\Bigl{(}\sum_{j=1}^{n}\frac{\partial x_{j}}{\partial
t_{i}}\tilde{B}^{(j)}\Bigr{)}B_{\infty}Y\ \ \ \ (i=1,\dots,n).$ (85)

It holds that $\mathcal{T}=T$,
$\tilde{\mathcal{B}}^{(i)}=\sum_{j=1}^{n}\frac{\partial x_{j}}{\partial
t_{i}}\tilde{B}^{(j)}$ and
$\mathcal{B}_{\infty}=B_{\infty}-(\lambda_{n}-1)I_{n}$ by Definition 5.1. In
virtue of Lemma 4.4,

$\begin{pmatrix}V_{n}\\\ \vdots\\\
V_{1}\end{pmatrix}=-T\begin{pmatrix}\partial_{t_{1}}\\\ \vdots\\\
\partial_{t_{n}}\end{pmatrix}=(t_{n}-T_{0}(t^{\prime}))\begin{pmatrix}\partial_{t_{1}}\\\
\vdots\\\ \partial_{t_{n}}\end{pmatrix}$ (86)

defines logarithmic vector fields along $D$, and then it holds that
$V_{i}=V_{i}^{(h)}$ or equivalently $-T=M_{V^{(h)}}$ from Lemma 3.6.
Therefore, by (40), $P$ can be taken as

$P=P_{h}=\begin{pmatrix}\frac{\partial z_{1}}{\partial
t_{1}}&\cdots&\frac{\partial z_{n}}{\partial t_{1}}\\\ \vdots&\ddots&\vdots\\\
\frac{\partial z_{1}}{\partial t_{n}}&\cdots&\frac{\partial z_{n}}{\partial
t_{n}}\end{pmatrix}$ (87)

and thus

$Q=\begin{pmatrix}\frac{\partial t_{1}}{\partial z_{1}}&\cdots&\frac{\partial
t_{n}}{\partial z_{1}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial
t_{1}}{\partial z_{n}}&\cdots&\frac{\partial t_{n}}{\partial
z_{n}}\end{pmatrix}$ (88)

since $Q$ is the inverse matrix of $P$. In particular, by
$e=\partial_{t_{n}}$, we find

$\frac{\partial z_{1}}{\partial t_{n}}=\cdots=\frac{\partial z_{n}}{\partial
t_{n}}=-1,$

which implies $P_{nj}=-1$. (83) follows from (88). We have

$\sum_{k=1}^{n}z_{k}\frac{\partial^{2}t_{j}}{\partial z_{k}\partial
z_{i}}=E\frac{\partial t_{j}}{\partial z_{i}}=(w_{j}-1)\frac{\partial
t_{j}}{\partial z_{i}}=(w_{j}-w_{n})\frac{\partial t_{j}}{\partial
z_{i}}=(\lambda_{j}-\lambda_{n})\frac{\partial t_{j}}{\partial z_{i}}$

since the weight of $t_{i}$ is equal to $w_{i}$ and it holds that
$w_{i}-w_{n}=\lambda_{i}-\lambda_{n}$ and $w_{n}=1$. This implies (84).

Next, we shall prove the converse. We define $E$ by
$E=\sum_{i=1}^{n}z_{i}\partial_{z_{i}}$. (84) implies that $Q_{ij}$ is
weighted homogeneous with $w(Q_{ij})=\lambda_{j}-\lambda_{n}$ for any $i,j$.
In virtue of the condition (83) and Lemma 4.1, there is a set of variables
$\\{t_{1},\dots,t_{n}\\}$ such that $Q_{ij}=\displaystyle\frac{\partial
t_{j}}{\partial z_{i}}$ and that $t_{j}$ is weighted homogeneous with
$w(t_{j})=1+\lambda_{j}-\lambda_{n}$. We have

$\frac{\partial z_{1}}{\partial t_{n}}=\cdots=\frac{\partial z_{n}}{\partial
t_{n}}=-1$

since we can take $P$ such that $P_{nj}=(Q^{-1})_{nj}=-1$ $(j=1,\dots,n)$ in
virtue of the condition (ii). We put

$e=-\partial_{z_{1}}-\cdots-\partial_{z_{n}}=\partial_{t_{n}}.$ (89)

We define a connection  $\triangle$ by

$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}_{\partial_{z_{j}}}(\partial_{z_{i}})=\sum_{k,l=1}^{n}\frac{\partial^{2}t_{k}}{\partial
z_{i}\partial z_{j}}\frac{\partial z_{l}}{\partial t_{k}}\partial_{z_{l}},$

namely

$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\begin{pmatrix}\partial_{z_{1}}\\\
\vdots\\\
\partial_{z_{n}}\end{pmatrix}=dQ\,Q^{-1}\begin{pmatrix}\partial_{z_{1}}\\\
\vdots\\\ \partial_{z_{n}}\end{pmatrix}.$

Then, we have

$\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\begin{pmatrix}\partial_{t_{1}}\\\
\vdots\\\
\partial_{t_{n}}\end{pmatrix}=\raisebox{6.94444pt}{\rotatebox{180.0}{$\triangle$}}\left(P\begin{pmatrix}\partial_{z_{1}}\\\
\vdots\\\
\partial_{z_{n}}\end{pmatrix}\right)=(dP+PdQ\,Q^{-1})\begin{pmatrix}\partial_{z_{1}}\\\
\vdots\\\ \partial_{z_{n}}\end{pmatrix}=0.$

The product $\star$ (or equivalently the Higgs field $\Phi$) is defined by
$\partial_{z_{i}}\star\partial_{z_{j}}=-\delta_{ij}\partial_{z_{i}},$ which is
obviously commutative and associative. Then we have
$E\star\partial_{z_{j}}=-z_{j}\partial_{z_{j}},$ which implies that
$Z=\mbox{diag}[z_{1},\dots,z_{n}]$ is the representation matrix of $-\Phi(E)$
with respect to the basis $\\{\partial_{z_{1}},\dots,\partial_{z_{n}}\\}$. The
base change from $\\{\partial_{z_{1}},\dots,\partial_{z_{n}}\\}$ to
$\\{\partial_{t_{1}},\dots,\partial_{t_{n}}\\}$ is given by the matrix $P$:

$\begin{pmatrix}\partial_{t_{1}}\\\ \vdots\\\
\partial_{t_{n}}\end{pmatrix}=P\begin{pmatrix}\partial_{z_{1}}\\\ \vdots\\\
\partial_{z_{n}}\end{pmatrix}$

Thus we have obtained a Saito structure on $W$ that has $(t_{1},\dots,t_{n})$
as a flat coordinate system. ∎

######  Remark 5.1.

The coordinate system $(-z_{1},\dots,-z_{n})$ is called a canonical coordinate
in [10] and [30].

######  Proposition 5.1.

Let the notations and the assumptions be same as Lemma 5.1. An Okubo type
system (82) has a flat structure on $W$ if and only if the condition $({\rm
ii})$ in Lemma 5.1 holds.

###### Proof.

In virtue of Lemma 5.1, it is sufficient to show that the condition (i)
follows from the condition (ii). We put

$\tilde{E}^{(i)}=\sum_{k=1}^{n}\frac{\partial x_{k}}{\partial
z_{i}}\tilde{B}^{(k)}=-PE_{i}Q,$ (90)

where $E_{i}=\mbox{diag}[\delta_{i1},\dots,\delta_{in}]$. Then, for
$(n,j)$-entry of $\tilde{E}^{(i)}$, we have $\tilde{E}^{(i)}_{nj}=Q_{ij}$,
$j=1,\dots,n$, noting $P_{nj}=-1$. It holds that

$\displaystyle\frac{\partial\tilde{E}^{(k)}}{\partial
z_{l}}-\frac{\partial\tilde{E}^{(l)}}{\partial z_{k}}=O,\ \ \ k,l=1\dots,n,$
(91) $\displaystyle\frac{\partial T}{\partial
z_{i}}+\tilde{E}^{(i)}+[\tilde{E}^{(i)},B_{\infty}]=O,\ \ \ i=1,\dots,n,$ (92)

from the integrability condition of (82). The equality (91) implies
$\frac{\partial Q_{kj}}{\partial z_{l}}=\frac{\partial Q_{lj}}{\partial
z_{k}}$. It holds that

$T=P\mbox{diag}[z_{1},\dots,z_{n}]Q=-\sum_{k=1}^{n}z_{k}\tilde{E}^{(k)},$

from which we have

$\frac{\partial T}{\partial
z_{i}}=-\tilde{E}^{(i)}-\sum_{k=1}^{n}z_{k}\frac{\partial\tilde{E}^{(k)}}{\partial
z_{i}}=-\tilde{E}^{(i)}-\sum_{k=1}^{n}z_{k}\frac{\partial\tilde{E}^{(i)}}{\partial
z_{k}}.$

Substituting this equality into (92), we have

$\sum_{k=1}^{n}z_{k}\frac{\partial\tilde{E}^{(i)}}{\partial
z_{k}}=[\tilde{E}^{(i)},B_{\infty}].$

Hence we obtain $\sum_{k=1}^{n}z_{k}\frac{\partial Q_{ij}}{\partial
z_{k}}=(\lambda_{j}-\lambda_{n})Q_{ij}$. ∎

######  Remark 5.2.

Actually, Proposition 5.1 is essentially proved in [30] although it is not
explicitly stated (cf. 4.a on p.242, [30]).

######  Lemma 5.2.

We consider an Okubo type system (82). The following two conditions are
equivalent to each other:

$({\rm i})$ $\begin{vmatrix}\frac{\partial T_{n1}}{\partial
x_{1}}&\cdots&\frac{\partial T_{nn}}{\partial x_{1}}\\\ {}&\vdots&{}\\\
\frac{\partial T_{n1}}{\partial x_{n}}&\cdots&\frac{\partial T_{nn}}{\partial
x_{n}}\end{vmatrix}\neq 0$ at any point on $W$,

$({\rm ii})$ $P_{nj}\neq 0$ $(j=1,\dots,n)$ at any point on $W$.

###### Proof.

We shall prove the contrapositions. First, we assume

$\begin{vmatrix}\frac{\partial T_{n1}}{\partial x_{1}}&\cdots&\frac{\partial
T_{nn}}{\partial x_{1}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial
T_{n1}}{\partial x_{n}}&\cdots&\frac{\partial T_{nn}}{\partial
x_{n}}\end{vmatrix}=0$

at some point on $W$. Let $x^{0}\in W$ be that point. Then there are
$f_{i}\in\mathcal{O}_{W}$, $i=1,\dots,n$ such that
$(f_{1}(x^{0}),\dots,f_{n}(x^{0}))\neq 0$ and

$\sum_{i=1}^{n}f_{i}(x^{0})\Bigl{(}\frac{\partial T_{n1}}{\partial
x_{i}}(x^{0}),\dots,\frac{\partial T_{nn}}{\partial x_{i}}(x^{0})\Bigr{)}=0.$
(93)

Since

$\frac{\partial T_{nj}}{\partial
x_{i}}=(\lambda_{n}-\lambda_{j}-1)\tilde{B}^{(i)}_{nj},$

(93) implies that

$\sum_{i=1}^{n}f_{i}(x^{0})\tilde{B}^{(i)}(x^{0})=\begin{pmatrix}{}&*&{}\\\
0&\cdots&0\end{pmatrix}.$ (94)

Notice that $\displaystyle\sum_{i=1}^{n}f_{i}(x^{0})\tilde{B}^{(i)}(x^{0})\neq
O$ from $P^{-1}\tilde{B}^{(i)}P=\mbox{diag}\,[-\frac{\partial z_{1}}{\partial
x_{i}},\dots,-\frac{\partial z_{n}}{\partial x_{i}}]$ and the assumption

$\begin{vmatrix}\frac{\partial z_{1}}{\partial x_{1}}&\cdots&\frac{\partial
z_{n}}{\partial x_{1}}\\\ {}&\vdots&{}\\\ \frac{\partial z_{1}}{\partial
x_{n}}&\cdots&\frac{\partial z_{n}}{\partial x_{n}}\end{vmatrix}\neq 0.$

Since each column of $P(x^{0})$ is an eigenvector of
$\displaystyle\sum_{i=1}^{n}f_{i}(x^{0})\tilde{B}^{(i)}(x^{0})$, we take
$k\in\\{1,\dots,n\\}$ such that
$\begin{pmatrix}P_{1k}(x^{0})&\dots&P_{nk}(x^{0})\end{pmatrix}^{t}$ is an
eigenvector belonging to a non-zero eigenvalue $c$ of
$\displaystyle\sum_{i=1}^{n}f_{i}(x^{0})\tilde{B}^{(i)}(x^{0})$. In virtue of
(94), we have

$\sum_{i=1}^{n}f_{i}(x^{0})\tilde{B}^{(i)}(x^{0})\begin{pmatrix}P_{1k}(x^{0})\\\
\vdots\\\ P_{nk}(x^{0})\end{pmatrix}=\begin{pmatrix}*\\\ *\\\
0\end{pmatrix}=c\begin{pmatrix}P_{1k}(x^{0})\\\ \vdots\\\
P_{nk}(x^{0})\end{pmatrix},\ \ \ \ c\neq 0,$

which implies $P_{nk}(x^{0})=0$.

Next we shall prove the converse. We assume that there are $x^{0}\in W$ and
$k\in\\{1,2,\dots,n\\}$ such that $P_{nk}(x^{0})=0$. From

$\displaystyle\tilde{B}^{(i)}(x^{0})\begin{pmatrix}P_{1k}(x^{0})\\\ \vdots\\\
P_{nk}(x^{0})\end{pmatrix}$ $\displaystyle=\begin{pmatrix}{}&*&{}\\\
\frac{\partial T_{n1}}{\partial x_{i}}(x^{0})&\cdots&\frac{\partial
T_{nn}}{\partial
x_{i}}(x^{0})\end{pmatrix}\begin{pmatrix}\lambda_{n}-\lambda_{1}-1&{}&{}\\\
{}&\ddots&{}\\\ {}&{}&-1\end{pmatrix}^{-1}\begin{pmatrix}P_{1k}(x^{0})\\\
\vdots\\\ P_{nk}(x^{0})\end{pmatrix}$ $\displaystyle=-\frac{\partial
z_{k}}{\partial x_{i}}(x^{0})\begin{pmatrix}P_{1k}(x^{0})\\\ \vdots\\\
P_{nk}(x^{0})\end{pmatrix}=\begin{pmatrix}*\\\ *\\\ 0\end{pmatrix},$

we obtain

$\begin{pmatrix}\frac{\partial T_{n1}}{\partial
x_{1}}(x^{0})&\cdots&\frac{\partial T_{nn}}{\partial x_{1}}(x^{0})\\\
\vdots&\ddots&\vdots\\\ \frac{\partial T_{n1}}{\partial
x_{n}}(x^{0})&\cdots&\frac{\partial T_{nn}}{\partial
x_{n}}(x^{0})\end{pmatrix}\begin{pmatrix}\lambda_{n}-\lambda_{1}-1&{}&{}\\\
{}&\ddots&{}\\\ {}&{}&-1\end{pmatrix}^{-1}\begin{pmatrix}P_{1k}(x^{0})\\\
\vdots\\\ P_{nk}(x^{0})\end{pmatrix}=0,$

which implies

$\begin{vmatrix}\frac{\partial T_{n1}}{\partial
x_{1}}(x^{0})&\cdots&\frac{\partial T_{nn}}{\partial x_{1}}(x^{0})\\\
\vdots&\ddots&\vdots\\\ \frac{\partial T_{n1}}{\partial
x_{n}}(x^{0})&\cdots&\frac{\partial T_{nn}}{\partial
x_{n}}(x^{0})\end{vmatrix}=0.$

∎

From now on, we consider (82) on $U$ not on $W\subset U\setminus\Delta_{H}$.

######  Theorem 5.1.

An Okubo type system (82) has a flat structure on $U$ if and only if

$\begin{vmatrix}\frac{\partial T_{n1}}{\partial x_{1}}&\cdots&\frac{\partial
T_{nn}}{\partial x_{1}}\\\ {}&\vdots&{}\\\ \frac{\partial T_{n1}}{\partial
x_{n}}&\cdots&\frac{\partial T_{nn}}{\partial x_{n}}\end{vmatrix}\neq 0$ (95)

at any point on $U$. When (82) has a flat structure on $U$, the set of
variables $t_{j}=-(\lambda_{j}-\lambda_{n}+1)^{-1}T_{nj}$, $j=1,\dots,n$ is a
flat coordinate system.

###### Proof.

First, we assume that (82) has a flat structure. Change the independent
variables of (82) from $(x_{1},\dots,x_{n})$ to a flat coordinate system
$(t_{1},\dots,t_{n})$. Then it holds that $T=\mathcal{T}$ and
$\mathcal{T}_{nj}=-w_{j}t_{j}$, from which we have

$\begin{vmatrix}\frac{\partial T_{n1}}{\partial t_{1}}&\dots&\frac{\partial
T_{nn}}{\partial t_{1}}\\\ \dots\\\ \frac{\partial T_{n1}}{\partial
t_{n}}&\dots&\frac{\partial T_{nn}}{\partial
t_{n}}\end{vmatrix}=(-1)^{n}w_{1}\cdots w_{n}\neq 0.$

Conversely, we assume (95). In virtue of Lemma 5.2 and Proposition 5.1, (82)
has a flat structure on any $W\subset U\setminus\Delta_{H}$. We also see that
$\\{t_{j}=-(\lambda_{j}-\lambda_{n}+1)^{-1}T_{nj}\\}$ is a flat coordinate
system. Then
$E:=\sum_{k=1}^{n}(\lambda_{k}-\lambda_{n}+1)t_{k}\partial_{t_{k}}$,
$e:=\partial_{t_{n}}$
$\Phi:=\sum_{j=1}^{n}\tilde{B}^{(j)}dx_{j}=\sum_{j,k}\frac{\partial
x_{j}}{\partial t_{k}}\tilde{B}^{(j)}dt_{k}$ satisfy the conditions of Saito
structure on $W$. Due to the identity theorem, they satisfy the conditions
also on $U$. Hence (82) has a flat structure on $U$. ∎

######  Corollary 5.1.

We consider the case of $n=3$. The extended WDVV equation

$\displaystyle\sum_{m=1}^{3}\frac{\partial^{2}g_{m}}{\partial t_{k}\partial
t_{i}}\frac{\partial^{2}g_{j}}{\partial t_{l}\partial
t_{m}}=\sum_{m=1}^{3}\frac{\partial^{2}g_{m}}{\partial t_{l}\partial
t_{i}}\frac{\partial^{2}g_{j}}{\partial t_{k}\partial t_{m}},\ \
i,j,k,l=1,2,3,$ $\displaystyle\frac{\partial^{2}g_{j}}{\partial t_{3}\partial
t_{i}}=\delta_{ij},\ \ \ \ i,j=1,2,3,$ $\displaystyle
Eg_{j}=\sum_{k=1}^{3}w_{k}t_{k}\frac{\partial g_{j}}{\partial
t_{k}}=(1+w_{j})g_{j},\ \ \ j=1,2,3.$

is generically equivalent to the Painlevé VI equation

$\displaystyle\frac{d^{2}y}{dt^{2}}=$
$\displaystyle\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)\left(\frac{dy}{dt}\right)^{2}-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)\frac{dy}{dt}$
$\displaystyle+\frac{y(y-1)(y-t)}{t^{2}(t-1)^{2}}\left(\alpha+\beta\frac{t}{y^{2}}+\gamma\frac{t-1}{(y-1)^{2}}+\delta\frac{t(t-1)}{(y-t)^{2}}\right).$

Let $T$ be an $3\times 3$ matrix whose entries are given by

$T_{ij}=-(1-w_{i}+w_{j})\frac{\partial g_{i}}{\partial t_{j}},$

and take $P$ such that $P^{-1}TP$ is a diagonal matrix. Let

$r_{i}=-\left(P\begin{pmatrix}w_{1}-w_{3}&{}&{}\\\ {}&w_{2}-w_{3}&{}\\\
{}&{}&0\end{pmatrix}P^{-1}\right)_{ii},\ \ \ i=1,2,3$

and $\theta_{\infty}=w_{1}-w_{2}$, where we remark that $r_{i}$ is an explicit
form of that in $(B5)$ in Section 2. Then the correspondence between the
parameters is given by

$\alpha=\frac{1}{2}(\theta_{\infty}-1)^{2},\ \ \beta=-\frac{1}{2}r_{1}^{2},\ \
\gamma=\frac{1}{2}r_{2}^{2},\ \ \delta=\frac{1}{2}(1-r_{3}^{2}).$

###### Proof.

Since the condition (95) in Theorem 5.1 is generic one, we have the
correspondence between generic Okubo type systems in several variables and
solutions to the extended WDVV equation. The equivalence between the Okubo
type systems of rank $3$ and the Painlevé VI equation is shown in Proposition
A.1. ∎

Theorem 5.1 on the existence of a flat structure associated to a given Okubo
type system can be rephrased as follows.

######  Theorem 5.2.

We consider the reduced form (33) in Remark 2.5 of an Okubo type system (82)
with $\det B_{\infty}\neq 0$:

$dY_{0}=B_{0}Y_{0}=\Bigl{(}\sum_{i=1}^{n}T^{-1}\tilde{B}^{(i)}B_{\infty}dx_{i}\Bigr{)}Y_{0}.$
(96)

Then the following two conditions are equivalent:

(i)

It holds that

$\begin{vmatrix}\frac{\partial T_{n1}}{\partial x_{1}}&\cdots&\frac{\partial
T_{nn}}{\partial x_{1}}\\\ {}&\vdots&{}\\\ \frac{\partial T_{n1}}{\partial
x_{n}}&\cdots&\frac{\partial T_{nn}}{\partial x_{n}}\end{vmatrix}\neq 0$

at any point on $U$.

(ii)

For the divisor $D=\\{\det(-T)=0\\}$, there are logarithmic vector fields
along $D$ $V_{1},\dots,V_{n}$ such that $(\det(-T))^{-1}\cdot\det
M_{V}\in\mathcal{O}_{U}$ and, for any solution $Y_{0}=(y_{1},\dots,y_{n})^{t}$
to (96), $y_{j}=V_{n+1-j}y_{n}$, $j=1,\dots,n$ hold, where $M_{V}$ is a matrix
defined by
$(V_{n},\dots,V_{1})^{t}=M_{V}(\partial_{x_{1}},\dots,\partial_{x_{n}})^{t}$.

###### Proof.

First, we put $V_{1}=-\lambda_{n}^{-1}\sum_{i=1}^{n}z_{i}\partial_{z_{i}}$,
which is actually a vector field on $U$ since it is symmetric with respect to
$(z_{1},\dots,z_{n})$. Then $V_{1}y_{n}=y_{n}$ holds. Indeed,

$\displaystyle-\lambda_{n}^{-1}\sum_{i=1}^{n}z_{i}\frac{\partial
Y_{0}}{\partial z_{i}}$
$\displaystyle=-\lambda_{n}^{-1}\sum_{i,j}z_{i}\frac{\partial x_{j}}{\partial
z_{i}}\frac{\partial Y_{0}}{\partial x_{j}}$
$\displaystyle=-\lambda_{n}^{-1}\sum_{i,j}z_{i}\frac{\partial x_{j}}{\partial
z_{i}}T^{-1}\tilde{B}^{(j)}B_{\infty}Y_{0}$
$\displaystyle=\lambda_{n}^{-1}B_{\infty}Y_{0}$

since $T^{-1}$ and $\tilde{B}^{(j)}$ are commutative and
$\sum_{i,j}z_{i}\frac{\partial x_{j}}{\partial z_{i}}\tilde{B}^{(j)}=-T$
holds. We put $B^{(i)}_{0}=T^{-1}\tilde{B}^{(i)}B_{\infty}$ and
$Y_{0}^{\prime}=(\partial y_{n}/\partial x_{1},\dots,\partial y_{n}/\partial
x_{n})^{t}$. Then we have

$Y_{0}^{\prime}=\begin{pmatrix}(B_{0}^{(1)})_{n1}&\cdots&(B_{0}^{(1)})_{nn}\\\
{}&\vdots&{}\\\
(B_{0}^{(n)})_{n1}&\cdots&(B_{0}^{(n)})_{nn}\end{pmatrix}Y_{0}=:MY_{0}.$

It holds that

$M=\begin{pmatrix}\tilde{B}^{(1)}_{n1}&\cdots&\tilde{B}^{(1)}_{nn}\\\
{}&\vdots&{}\\\
\tilde{B}^{(n)}_{n1}&\cdots&\tilde{B}^{(n)}_{nn}\end{pmatrix}T^{-1}B_{\infty}.$

Take arbitrary vector fields $V_{2},\dots,V_{n}$ on $U$ and define a matrix
$M_{V}$ by

$(V_{n},\dots,V_{1})^{t}=M_{V}(\partial_{x_{1}},\dots,\partial_{x_{n}})^{t}.$

Put $Y^{\prime\prime}_{0}=M_{V}Y^{\prime}_{0}.$ Then it holds that
$Y^{\prime\prime}_{0}=(V_{n}y_{n},\dots,V_{1}y_{n})^{t}=(V_{n}y_{n},\dots,V_{2}y_{n},y_{n})^{t}$.

$\underline{({\rm i})\Rightarrow({\rm ii})}$ By the assumption, the Okubo type
system (96) has a flat structure. Take the flat coordinate system
$(t_{1},\dots,t_{n})=(\mathcal{C}_{n1},\dots,\mathcal{C}_{nn})$ as the
independent variables of (96). Then it holds that $M=T^{-1}B_{\infty}$ since
$\tilde{B}^{(i)}_{nj}=\partial\mathcal{C}_{nj}/\partial t_{i}=\delta_{ij}$.
Take
$(V_{n},\dots,V_{1})^{t}=M_{V}(\partial_{t_{1}},\dots,\partial_{t_{n}})^{t}$
as $M_{V}=B_{\infty}^{-1}T$. Then we have

$Y_{0}^{\prime\prime}=M_{V}Y_{0}^{\prime}=M_{V}MY_{0}=Y_{0}.$

$\underline{({\rm ii})\Rightarrow({\rm i})}$ Assume that there are logarithmic
vector fields along $D$
$(V_{n},\dots,V_{1})^{t}=M_{V}(\partial_{x_{1}},\dots,\partial_{x_{n}})^{t}$
such that $(\det(-T))^{-1}\det M_{V}\in\mathcal{O}_{U}$ and
$Y^{\prime\prime}_{0}=M_{V}MY_{0}=Y_{0}.$ Then we have $\det(M_{V}M)=1$, which
implies that

$\begin{vmatrix}\tilde{B}^{(1)}_{n1}&\cdots&\tilde{B}^{(1)}_{nn}\\\
{}&\vdots&{}\\\
\tilde{B}^{(n)}_{n1}&\cdots&\tilde{B}^{(n)}_{nn}\end{vmatrix}=\prod_{j=1}^{n}(\lambda_{n}-\lambda_{j}-1)\begin{vmatrix}\frac{\partial
T_{n1}}{\partial x_{1}}&\cdots&\frac{\partial T_{nn}}{\partial x_{1}}\\\
{}&\vdots&{}\\\ \frac{\partial T_{n1}}{\partial x_{n}}&\cdots&\frac{\partial
T_{nn}}{\partial x_{n}}\end{vmatrix}\neq 0.$

∎

######  Proposition 5.2.

Assume that the Okubo type system (96) has a flat structure on $U$ and take
the flat coordinate system
$(t_{1},\dots,t_{n})=(\mathcal{C}_{n1},\dots,\mathcal{C}_{nn})$ as the
independent variables of (96). Then there is a function $y=y(t)$ such that
$Y_{0}=(\partial y/\partial t_{1},\dots,\partial y/\partial t_{n})^{t}.$

###### Proof.

We consider the “contiguous” equation of (96):

$dY^{(-)}_{0}=B^{(-)}Y_{0}^{(-)}:=T^{-1}d\mathcal{C}(B_{\infty}-I_{n})Y^{(-)}_{0}.$
(97)

For any solution $Y^{(-)}_{0}=(y^{(-)}_{1},\dots,y^{(-)}_{n})^{t}$ to (97), we
put $Y^{\prime(-)}_{0}=(\frac{\partial y_{n}^{(-)}}{\partial
t_{1}},\dots,\frac{\partial y_{n}^{(-)}}{\partial t_{n}})^{t}$ and
$M^{(-)}=T^{-1}(B_{\infty}-I_{n})$. Then we have

$\displaystyle dY^{\prime(-)}_{0}$
$\displaystyle=\bigl{(}M^{(-)}B^{(-)}(M^{(-)})^{-1}+dM^{(-)}(M^{(-)})^{-1}\bigr{)}Y^{\prime(-)}_{0}$
$\displaystyle=\bigl{(}T^{-1}(B_{\infty}-I_{n})d\mathcal{C}-T^{-1}dT\bigr{)}Y^{\prime(-)}_{0}$
$\displaystyle=T^{-1}\bigl{(}(B_{\infty}-I_{n})d\mathcal{C}+d\mathcal{C}+[d\mathcal{C},B_{\infty}]\bigr{)}Y^{\prime(-)}_{0}=T^{-1}d\mathcal{C}B_{\infty}Y^{\prime(-)}_{0}.$

Hence we can take $y=y^{(-)}_{n}$ as the desired function. ∎

###  5.2 Flat structure with a potential

In this subsection, we treat the case where a flat structure has a potential.
In this case, we give alternative proofs of some results in [10].

Recall the notation $A^{*}=JA^{t}J$ where $J_{ij}=\delta_{i+j,n+1}$ for an
$n\times n$ matrix $A$, and that a flat structure has a potential $F$ if and
only if $\mathcal{C}=\mathcal{C}^{*}$ (or equivalently
$\mathcal{T}=\mathcal{T}^{*}$) by Proposition 4.5.

######  Lemma 5.3.

We assume that an Okubo type system (82) has a flat structure on $U$. If
$T=T^{*}$ holds, then $B_{\infty}$ satisfies
$B_{\infty}+B_{\infty}^{*}=-2rI_{n}$ for some $r\in\mathbb{C}$.

###### Proof.

The lemma can be proved by a way similar to the proof of $({\rm
i})\Rightarrow({\rm iii})$ in Proposition 4.5. ∎

In [10], Dubrovin presented a correspondence between massive (i.e.,
semisimple) Frobenius manifolds and the spaces of isomonodromic deformations
of linear differential equations with special conditions. Combining
Proposition 5.1 and the following proposition is equivalent to Dubrovin’s
result.

######  Proposition 5.3.

We consider an Okubo type system (82) on $W$. It holds that $T=T^{*}$ if and
only if $\bar{B}_{\infty}=P^{-1}B_{\infty}P$ satisfies
$\bar{B}_{\infty}+(\bar{B}_{\infty})^{t}=-2rI_{n}$ for some $r\in\mathbb{C}$.

###### Proof.

Assume $T=T^{*}$. We can choose $P$ such that

$P^{-1}TP=\mbox{diag}\,[z_{1},\dots,z_{n}],\ \ \ \ P^{t}J=P^{-1}$

by a consequence of an elementary linear algebra. Then the claim
$\bar{B}_{\infty}+(\bar{B}_{\infty})^{t}=-2rI_{n}$ is concluded by

$(\bar{B}_{\infty})^{t}=P^{t}B_{\infty}(P^{-1})^{t}=P^{-1}JB_{\infty}JP=P^{-1}B_{\infty}^{*}P=P^{-1}(-B_{\infty}-2rI_{n})P=-\bar{B}_{\infty}-2rI_{n}.$

Conversely, for $\bar{B}_{\infty}$ satisfying
$\bar{B}_{\infty}+(\bar{B}_{\infty})^{t}=-2rI_{n}$, we can take $Q$ such that
$Q^{-1}\bar{B}_{\infty}Q=B_{\infty}$ and $Q^{-1}=JQ^{t}$. Then it holds that

$\displaystyle T^{*}$
$\displaystyle=J(Q^{-1}\mbox{diag}[z_{1},\dots,z_{n}]Q)^{t}J=JQ^{t}\mbox{diag}[z_{1},\dots,z_{n}](Q^{-1})^{t}J$
$\displaystyle=Q^{-1}\mbox{diag}[z_{1},\dots,z_{n}]Q=T.$

∎

######  Lemma 5.4.

We consider an Okubo type system (82) on $W\subset U\setminus\Delta_{H}$.
Under the assumption $T=T^{*}$, the characteristic exponents of the
differential equation of Okubo type
$\frac{dY}{dz}=B^{(z)}Y=-(zI_{n}-T)^{-1}B_{\infty}Y$ satisfy the relations
$\lambda_{i}+\lambda_{n+1-i}=-2r$ and $r_{i}=\mbox{tr}\,B_{i}^{(z)}=r$
$(i=1,\dots,n)$ for some $r\in\mathbb{C}$, where $B_{i}^{(z)}$ is the residue
matrix of $B^{(z)}$ at $z=z_{i}$.

###### Proof.

It follows that $\lambda_{i}+\lambda_{n+1-i}=-2r$ obviously from
$B_{\infty}+B_{\infty}^{*}=-2rI_{n}$. From the equality

$B^{(z)}=-(zI_{n}-T)^{-1}B_{\infty}=-P\begin{pmatrix}z-z_{1}&{}&{}\\\
{}&\ddots&{}\\\ {}&{}&z-z_{n}\end{pmatrix}^{-1}\bar{B}_{\infty}P^{-1},$

we have

$B_{i}^{(z)}=-P\begin{pmatrix}0&\cdots&0\\\ {}&\vdots&{}\\\
(\bar{B}_{\infty})_{i1}&\cdots&(\bar{B}_{\infty})_{in}\\\ {}&\vdots&{}\\\
0&\cdots&0\end{pmatrix}P^{-1},$

where $B_{i}^{(z)}$ is the residue matrix of $B^{(z)}$ at $z=z_{i}$. Hence we
obtain $r_{i}=\mbox{tr}\,B_{i}^{(z)}=-(\bar{B}_{\infty})_{ii}=r$ by
Proposition 5.3. ∎

In the case of $n=3$, it is possible to characterize Okubo type systems with
Frobenius structures by only the numerical condition in Lemma 5.4.

######  Proposition 5.4.

We consider the case of $n=3$ and assume that an Okubo type system in several
variables (82) has a flat structure on $W$. Then the flat structure has a
prepotential (i.e., the flat structure becomes a Frobenius structure) if and
only if there is $r\in\mathbb{C}$ such that $\lambda_{i}+\lambda_{n+1-i}=-2r$,
$r_{i}=r$ $(i=1,2,3)$.

###### Proof.

We suppose the independent variables of (82) are flat coordinates, i.e.,
$x_{i}=t_{i}$, $i=1,\dots,n$. We assume $r_{i}=r,$
$\lambda_{i}+\lambda_{n+1-i}=-2r,$ $i=1,2,3.$ However, without loss of
generality, we may assume $r_{i}=0,$ $\lambda_{i}+\lambda_{n+1-i}=0,$
$i=1,2,3$ by replacing $\lambda_{i}+r$ by $\lambda_{i}$. By using a matrix $P$
such that $P^{-1}TP=\mbox{diag}[z_{1},\dots,z_{n}],$ the residue matrix
$B^{(z)}_{i}$ of $B^{(z)}=-(zI_{n}-T)^{-1}B_{\infty}$ at $z=z_{i},$ $i=1,2,3$
is written as

$B^{(z)}_{i}=-P^{-1}\begin{pmatrix}\delta_{i1}&{}&{}\\\ {}&\delta_{i2}&{}\\\
{}&{}&\delta_{i3}\end{pmatrix}PB_{\infty}=-\begin{pmatrix}(P^{-1})_{1i}P_{i1}&(P^{-1})_{1i}P_{i2}&(P^{-1})_{1i}P_{i3}\\\
(P^{-1})_{2i}P_{i1}&(P^{-1})_{2i}P_{i2}&(P^{-1})_{2i}P_{i3}\\\
(P^{-1})_{3i}P_{i1}&(P^{-1})_{3i}P_{i2}&(P^{-1})_{3i}P_{i3}\end{pmatrix}B_{\infty}.$

Then by the assumption $r_{i}=\mbox{tr}\,B^{(z)}_{i}=0$, we have

$(P^{-1})_{1i}P_{i1}=(P^{-1})_{3i}P_{i3},\ \ \ \ i=1,2,3,$ (98)

noting $\lambda_{1}+\lambda_{3}=0,\lambda_{2}=0$. From (98), it holds that

$\frac{(P^{-1})_{2i}P_{i3}}{(P^{-1})_{1i}P_{i2}}=\frac{(P^{-1})_{2i}P_{i3}P_{i1}}{(P^{-1})_{1i}P_{i1}P_{i2}}=\frac{(P^{-1})_{2i}P_{i3}P_{i1}}{(P^{-1})_{3i}P_{i3}P_{i2}}=\frac{(P^{-1})_{2i}P_{i1}}{(P^{-1})_{3i}P_{i2}}.$
(99)

We put $\kappa=\displaystyle\frac{(P^{-1})_{2i}P_{i3}}{(P^{-1})_{1i}P_{i2}}$.
We note that $\kappa$ does not depend on $i$, which can be verified by direct
computations using the cofactor matrix expression of $P^{-1}$. In turn, we
have

$-\sum_{k=1}^{3}\frac{\partial t_{k}}{\partial
z_{i}}\tilde{B}^{(k)}=P^{-1}\begin{pmatrix}\delta_{i1}&{}&{}\\\
{}&\delta_{i2}&{}\\\
{}&{}&\delta_{i3}\end{pmatrix}P=\begin{pmatrix}(P^{-1})_{1i}P_{i1}&(P^{-1})_{1i}P_{i2}&(P^{-1})_{1i}P_{i3}\\\
(P^{-1})_{2i}P_{i1}&(P^{-1})_{2i}P_{i2}&(P^{-1})_{2i}P_{i3}\\\
(P^{-1})_{3i}P_{i1}&(P^{-1})_{3i}P_{i2}&(P^{-1})_{3i}P_{i3}\end{pmatrix}.$

From (99), we find

$\Bigl{(}-\sum_{k=1}^{3}\frac{\partial t_{k}}{\partial
z_{i}}\tilde{B}^{(k)}\Bigr{)}^{*}=K\Bigl{(}-\sum_{k=1}^{3}\frac{\partial
t_{k}}{\partial z_{i}}\tilde{B}^{(k)}\Bigr{)}K^{-1},\ \ i=1,2,3,$

where we put $K=\mbox{diag}[\kappa,1,\kappa]$. Noting

$T=-E\mathcal{C}=-\sum_{i,k=1}^{3}z_{i}\frac{\partial t_{k}}{\partial
z_{i}}\tilde{B}^{(k)},\ \ \ \ \ \tilde{B}^{(i)}=-\sum_{j=1}^{3}\frac{\partial
z_{j}}{\partial t_{i}}\Bigl{(}-\sum_{k=1}^{3}\frac{\partial t_{k}}{\partial
z_{j}}\tilde{B}^{(k)}\Bigr{)},$

we have $T^{*}=KTK^{-1}$, $\mathcal{C}^{*}=K\mathcal{C}K^{-1}$ and
$\bigl{(}\tilde{B}^{(i)}\bigr{)}^{*}=K\tilde{B}^{(i)}K^{-1}$ $(i=1,\dots,n)$.
From $\tilde{B}^{(i)}=\frac{\partial\mathcal{C}}{\partial t_{i}}$, it follows
that

$\bigl{(}\tilde{B}^{(i)}\bigr{)}^{*}=\frac{\partial\mathcal{C}^{*}}{\partial
t_{i}}=\frac{\partial(K\mathcal{C}K^{-1})}{\partial
t_{i}}=K\tilde{B}^{(i)}K^{-1}+K\Bigl{[}K^{-1}\frac{\partial K}{\partial
t_{i}},\mathcal{C}\Bigr{]}K^{-1}.$

Therefore we have $[K^{-1}\frac{\partial K}{\partial t_{i}},\mathcal{C}]=O$
$(i=1,\dots,n)$, from which we find that $\kappa$ is a constant. Put
$T^{\prime}=K^{1/2}TK^{-1/2}$ for
$K^{1/2}=\mbox{diag}[\kappa^{1/2},1,\kappa^{1/2}]$. Then it holds that
$(T^{\prime})^{*}=T^{\prime}$. (This change of $T$ corresponds to the
normalization of the flat coordinate system in Proposition 4.5.) ∎

From this proposition, we obtain the equivalence in [10] between the $n=3$
WDVV equation and a one-parameter family of the Painlevé VI equations:

######  Corollary 5.2.

The WDVV equation

$\displaystyle\sum_{m=1}^{3}\frac{\partial^{3}F}{\partial t_{k}\partial
t_{i}\partial t_{m}}\frac{\partial^{3}F}{\partial t_{l}\partial t_{j}\partial
t_{4-m}}=\sum_{m=1}^{3}\frac{\partial^{3}F}{\partial t_{l}\partial
t_{i}\partial t_{m}}\frac{\partial^{3}F}{\partial t_{k}\partial t_{j}\partial
t_{4-m}},\ \ i,j,k,l=1,2,3,$ $\displaystyle\frac{\partial^{3}F}{\partial
t_{3}\partial t_{i}\partial t_{j}}=J_{ij},\ \ \ \ i,j=1,2,3,$ $\displaystyle
EF=(1-2r)F$

is equivalent to the one-parameter family of the Painlevé VI equations

$\displaystyle\frac{d^{2}y}{dt^{2}}=$
$\displaystyle\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)\left(\frac{dy}{dt}\right)^{2}-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)\frac{dy}{dt}$
$\displaystyle+\frac{y(y-1)(y-t)}{t^{2}(t-1)^{2}}\left(\frac{r^{2}}{2}-\frac{r^{2}}{2}\frac{t}{y^{2}}+\frac{r^{2}}{2}\frac{t-1}{(y-1)^{2}}+\frac{(1-r^{2})}{2}\frac{t(t-1)}{(y-t)^{2}}\right).$

##  6 Flat generator system for invariant polynomials of a complex reflection
group

In this section, we treat a problem on the existence of a flat coordinate
system for a complex reflection group. In the case of a real reflection group,
K. Saito [32] proved the existence of a flat coordinate system (see also
[33]). For a well-generated complex reflection group $G$, we construct a flat
structure on the affine space whose coordinate system consists of basic
$G$-invariant polynomials from an Okubo type system in several variables
called the $G$-quotient system (Theorem 6.1). As a consequence, we find that
the potential vector field of the flat structure for the complex reflection
group $G$ has polynomial entries, namely we can obtain a polynomial solution
of the extended WDVV equation from a complex reflection group $G$ (Corollary
6.1).

Let $G$ be a finite irreducible complex (unitary) reflection group acting on
the standard representation space

$U_{n}=\\{(u_{1},u_{2},\dots,u_{n})\,|\,u_{j}\in\mathbb{C}\\}$,

and let $F_{i}(u)$ of degree $d_{i}$, $1\leq i\leq n$ be a fundamental system
of $G$-invariant homogeneous polynomials. We define a coordinate functions on
$X:=U_{n}/G$ by, $x_{i}=F_{i}(u)$, $1\leq i\leq n$. Let $D\subset X$ be the
branch locus of $\pi_{G}:U_{n}\to X$. Let $h(x)$ be the (reduced) defining
function of $D$ in the coordinates $x=(x_{1},x_{2},\dots,x_{n})$. We assume
that $d_{1}<d_{2}<\dots<d_{n}$ and that $G$ is well generated (see e.g. [1]).
Then it is known that $h(x)$ is a monic polynomial in $x_{n}$ of degree $n$
([1]). We define a weight $w(\cdot)$ by $w(x_{i})=d_{i}/d_{n}$. Then $h(x)$ is
a weighted homogeneous polynomial in $x$. It is known that $D$ is free ([29],
[38]). Let

$B_{\infty}=\mbox{diag}\left[w(x_{1}),w(x_{2}),\dots,w(x_{n})\right]-\left(1+{1\over
d_{n}}\right)I_{n}.$ (100)

In this section, we prove the following theorem:

######  Theorem 6.1.

There is a uniquely defined (up to constant multiplications) special
$G$-invariant homogeneous polynomials $F^{\prime}_{i}(u)$ of degree $d_{i}$,
$1\leq i\leq n$ satisfying the following conditions:

$({\rm i})$ Let $t_{i}=F^{\prime}_{i}(u)$, and let $h(t)$ be the defining
function of $D$ in this coordinates $t=(t_{1},t_{2},\dots,t_{n})$. Let
$V_{i}^{(h)}(t)$, $1\leq i\leq n$ and $M_{V^{(h)}}(t)$ be defined by (48) with
respect to $h(t)$. Let $C(t)$ be the (weighted homogeneous) matrix satisfying
$V_{1}^{(h)}C(t)=M_{V^{(h)}}(t)$. Define $V^{\prime}_{i}$ by

$\displaystyle\begin{pmatrix}V^{\prime}_{n}&V^{\prime}_{n-1}&\dots&V^{\prime}_{1}\end{pmatrix}^{t}$
$\displaystyle=-B_{\infty}^{-1}\begin{pmatrix}V^{(h)}_{n}(t)&V^{(h)}_{n-1}(t)&\dots&V^{(h)}_{1}(t)\end{pmatrix}^{t}$
$\displaystyle=-B_{\infty}^{-1}M_{V^{(h)}}(t)\begin{pmatrix}\partial_{t_{1}}&\partial_{t_{2}}&\dots&\partial_{t_{n}}\end{pmatrix}^{t}.$

In particular
$V^{\prime}_{1}=d_{n}V_{1}^{(h)}(t)=\sum_{i=1}^{n}d_{i}\,t_{i}\,\partial_{t_{i}}$.
Then, for any homogeneous linear function $y(u)$ of $u$,

$Y_{0}=\begin{pmatrix}V^{\prime}_{n}&V^{\prime}_{n-1}&\dots&V^{\prime}_{1}\end{pmatrix}^{t}y(u)$
(101)

satisfies the Okubo type system

$dY_{0}=\left[-\left(V_{1}^{(h)}C(t)\right)^{-1}\,dC(t)\,B_{\infty}\right]Y_{0}.$
(102)

It holds that $(Y_{0})_{n,1}=V^{\prime}_{1}\,y(u)=y(u)$.

$({\rm ii})$ It holds that $C_{n,j}(t)=t_{j}$, $1\leq j\leq n$, and hence
$\\{t_{j}\\}$ gives a flat coordinate system on $X$ associated to (102) by
Theorem 5.1.

######  Definition 6.1.

We call $\\{F^{\prime}_{i}(u)\\}$ a flat generator system of $G$-invariant
polynomials.

######  Remark 6.1.

Roughly speaking, a generating system in [13] is introduced as a system of
differential equations for the coordinate $(u_{1},\dots,u_{n})$ as multi-
valued functions of $(x_{1},\dots,x_{n})$. In [21], the generating system is
rewritten as a Pfaffian system and called a $G$-quotient system. The equation
(102) is a $G$-quotient system in a form of Okubo type.

Again, we let $x_{i}=F_{i}(u)$, $1\leq i\leq n$ for arbitrarily given
fundamental system of $G$-invariant homogeneous polynomials. Let
$H_{i}:=\\{\ell_{i}(u)=0\\}$, $1\leq i\leq N$ be all the distinct reflecting
hyperplanes of $G$. If $H_{i}$ is the reflecting hyperplanes of $g_{i}\in G$
with the order $m_{i}$, then we have $g_{i}(H_{i})=H_{i}$, $g_{j}(H_{i})\neq
H_{i}$ for $j\neq i$. As stated in [36]

$h(x(u))=c_{1}\,\prod_{i=1}^{N}\ell_{i}(u)^{m_{i}},\quad\det\left[\left({\partial
x_{j}\over\partial
u_{i}}\right)_{i,j}\right]=c_{2}\,\prod_{i=1}^{N}\ell_{i}(u)^{m_{i}-1},$ (103)

for some non-zero constant numbers $c_{1},c_{2}$.

Let

$\tilde{h}(u)=\prod_{i=1}^{N}\ell_{i}(u),\quad\mbox{and}\quad\tilde{D}=\cup_{i=1}^{N}H_{i}.$

######  Lemma 6.1.

Fix $i\in\\{1,\dots,N\\}$ arbitrarily. Let $u^{(0)}$ be a generic point of
$H_{i}$, and $x^{(0)}=x(u^{(0)})$. Put $y_{1}(u)=l_{i}(u)$. Then
$y_{1}(u)^{m_{i}}$ is $g_{i}$-invariant, and $h(x)=c_{1}(x)y_{1}(u)^{m_{i}}$
for some non-vanishing holomorphic function $c_{1}(x)$ in a neighborhood of
$x^{(0)}$. There are $g_{i}$-invariant linearly independent homogeneous linear
functions $y_{k}(u)$, $2\leq k\leq n$. Then there are locally holomorphic
functions $\eta_{k}(x)$ at $x^{(0)}$, $2\leq k\leq n$ such that
$y_{k}(u)=\eta_{k}(x(u))$, and that $\\{h(x),\eta_{2}(x),\dots,\eta_{n}(x)\\}$
is a local coordinate system at $x^{(0)}$.

Let $V_{i}=\sum_{j=1}^{n}v_{i,j}(x)\,\partial_{x_{j}}$, $1\leq i\leq n$ be
logarithmic vector fields along $D$, and let
$M_{V}=(v_{i,j})\in\mathbb{C}[x]^{n\times n}$. Assume
$w(V_{n+1-i})=1-w(x_{i})$, and $\det(M_{V})=h(x)$. Let

$M_{\tilde{V}}=M_{V}\left({\partial x\over\partial
u}\right)^{-1},\quad\mbox{and}\quad\begin{pmatrix}\tilde{V}_{n}&\tilde{V}_{n-1}&\dots&\tilde{V}_{1}\end{pmatrix}^{t}=M_{\tilde{V}}\begin{pmatrix}\partial_{u_{1}}&\partial_{u_{2}}&\dots&\partial_{u_{n}}\end{pmatrix}^{t}.$

Then, from (103), it holds that

$\det(M_{\tilde{V}})=c\,\tilde{h}(u),$ (104)

for a constant number $c\neq 0$. The following lemma follows from the
definition of ${\tilde{V}}_{i}$:

######  Lemma 6.2.

Let $f(x)$ be a holomorphic function in an open set of $X$. Then we have

${\tilde{V}}_{i}((f\circ\pi_{G})(u))=(V_{i}(f)\circ\pi_{G})(u).$

######  Lemma 6.3.

$({\rm i})$ The entries of $M_{\tilde{V}}$ are polynomials in $u$.

$({\rm ii})$ $\tilde{V}_{k}$, $1\leq k\leq n$ are logarithmic vector fields
along $\tilde{D}$, and $\tilde{D}$ is free in the sense of Saito.

$({\rm iii})$ For any $k\in\\{1,2,\dots,n\\}$, all entries of
$(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}$ are $G$-invariant
polynomials in $u$, that is, polynomials in $x$.

###### Proof.

Proof of (i). By definition, $M_{\tilde{V}}\in\mathbb{C}(u)^{n\times n}$, and
$h(x(u))\,M_{\tilde{V}}\in\mathbb{C}[u]^{n\times n}$. Fix
$i\in\\{1,2,\dots,N\\}$ arbitrarily. All we have to show is that
$M_{\tilde{V}}$ is holomorphic at (generic point of) $H_{i}$.

Fix $k\in\\{1,2,\dots,n\\}$, and let $V=V_{k}$, $\tilde{V}=\tilde{V}_{k}$. Let
$u^{(0)}$, $y_{j}(u)$, $\eta_{j}(x)$, $m_{i}$ be as in Lemma 6.1. Let
$m=m_{i}$. Then
$\tilde{V}=\sum_{j=1}^{n}\tilde{V}(y_{j}(u))\,\partial_{y_{j}}$. It is
sufficient to prove that $\tilde{V}(y_{j}(u))$, $1\leq j\leq n$ are
holomorphic at $u^{(0)}$.

Let $j=1$. Then $\tilde{V}(y_{1}^{m})=m\,y_{1}^{m-1}\tilde{V}(y_{1})$. On the
other hand

$\displaystyle\tilde{V}(y_{1}^{m})=V(c_{1}^{-1}(x)h(x))=c_{1}^{-1}(x)V(h(x))+V(c_{1}^{-1}(x))\,h(x)=a(x)h(x),$

for a holomorphic function $a(x)$ at $x^{(0)}$. Consequently
$\tilde{V}(y_{1})$ is holomorphic at $u^{(0)}$.

Let $j\geq 2$. Then $\tilde{V}(y_{j}(u))=V(\eta_{j}(x))$, which implies that
$\tilde{V}(y_{j}(u))$ is holomorphic at $u^{(0)}$. This completes the proof of
(1).

Proof of (ii). Since $(V_{k}\,h(x))/h(x)\in\mathbb{C}[x]$, the equality (103)
implies

$\sum_{i=1}^{N}m_{i}\,(\tilde{V}_{k}\,l_{i}(u))/l_{i}(u)\in\mathbb{C}[u].$
(105)

From (i) proved above, we have $\tilde{V}_{k}\,l_{i}(u)\in\mathbb{C}[u]$. Then
(105) implies that

$(\tilde{V}_{k}\,l_{i}(u))/l_{i}(u)\in\mathbb{C}[u],\quad 1\leq k\leq n,\
1\leq i\leq N,$ (106)

which proves (ii).

Proof of (iii). Fix $i$ arbitrarily, and we will prove
$(\tilde{V}_{k}\,M_{\tilde{V}})M_{\tilde{V}}^{-1}$ is holomorphic along
$H_{i}$. Let $y_{1}(u)=l_{i}(u)$, and choose $y_{k}(u)=l_{i_{k}}(u)$, $2\leq
k\leq n$, so that $y_{1},y_{2},\dots,y_{n}$ are linearly independent. Then
(106) implies that

$\tilde{V}_{k}=\sum_{j=1}^{n}y_{j}\,v_{k,j}(y)\,\partial_{y_{j}},\quad 1\leq
k\leq n,$ (107)

for some $v_{k,j}(y)\in\mathbb{C}[y]$. This is equivalent to

$M_{\tilde{V}}=M^{\prime}_{\tilde{V}}(y)\cdot\mbox{diag}[y_{1},y_{2},\dots,y_{n}],$
(108)

for some $M^{\prime}_{\tilde{V}}(y)\in\mathbb{C}[y]^{n\times n}$. Then it also
holds that
$\tilde{V}_{k}\,M_{\tilde{V}}=M^{\prime\prime}_{\tilde{V}}(y)\cdot\mbox{diag}[y_{1},y_{2},\dots,y_{n}]$,
for some $M^{\prime\prime}_{\tilde{V}}(y)\in\mathbb{C}[y]^{n\times n}$. Thus
we have

$(\tilde{V}_{k}\,M_{\tilde{V}})M_{\tilde{V}}^{-1}=M^{\prime\prime}_{\tilde{V}}(y)\,(M^{\prime}_{\tilde{V}}(y))^{-1}.$
(109)

Since $\det(M_{\tilde{V}})=\prod_{i}l_{i}(u)$, the equality (108) implies that
$(M^{\prime}_{\tilde{V}}(y))^{-1}$ is holomorphic along $\\{y_{1}=0\\}$.
Consequently the equality (109) implies that
$(\tilde{V}_{k}\,M_{\tilde{V}})M_{\tilde{V}}^{-1}$ is holomorphic along
$H_{i}$. This proves that
$(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}\in\mathbb{C}[u]^{n\times
n}$.

We next prove that $(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}$ is
$G$-invariant. Let $g\in G$, and $\gamma\in\pi_{1}(X\setminus D,*)$ be a loop
such that $\gamma_{*}u=u\,g$. Then we have

$\displaystyle\gamma_{*}\left({\partial u\over\partial x}\right)$
$\displaystyle=\gamma_{*}\begin{pmatrix}{\partial u_{1}\over\partial
x_{1}}&{\partial u_{2}\over\partial x_{1}}&\dots&{\partial u_{n}\over\partial
x_{1}}\\\ {\partial u_{1}\over\partial x_{2}}&{\partial u_{2}\over\partial
x_{2}}&\dots&{\partial u_{n}\over\partial x_{2}}\\\ ...&...&\dots&...\\\
{\partial u_{1}\over\partial x_{n}}&{\partial u_{2}\over\partial
x_{n}}&\dots&{\partial u_{n}\over\partial x_{n}}\end{pmatrix}=\left({\partial
u\over\partial x}\right)\,g,$ $\displaystyle\gamma_{*}M_{\tilde{V}}$
$\displaystyle=\gamma_{*}M_{V}\,\left({\partial u\over\partial
x}\right)=M_{\tilde{V}}\,g,$
$\displaystyle\gamma_{*}\tilde{V}_{k}\,M_{\tilde{V}}$
$\displaystyle=\gamma_{*}V_{k}\,M_{\tilde{V}}=V_{k}\,M_{\tilde{V}}\,g=\tilde{V}_{k}\,M_{\tilde{V}}\,g.$

Consequently we have
$\gamma_{*}(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}=(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}$,
which means that $(\tilde{V}_{k}\,M_{\tilde{V}})\,M_{\tilde{V}}^{-1}$ is
$G$-invariant. This completes the proof of (3). ∎

Let

$(dM_{\tilde{V}})M_{\tilde{V}}^{-1}=\sum_{k=1}^{n}\hat{B}^{(k)}(x)\,dx_{k},$

and put

$\hat{Y}=\begin{pmatrix}\hat{y}_{1}&\hat{y}_{2}&\dots&\hat{y}_{n}\end{pmatrix}^{t}=\begin{pmatrix}V_{n}&V_{n-1}&\dots&V_{1}\end{pmatrix}^{t}y(u),$

for any homogeneous linear function $y(u)$ of $u$. Then
$\hat{y}_{n}=w(y)y=(1/d_{n})\,y$, and

$w(\hat{y}_{i})=w(V_{n-i+1})+(1/d_{n})=(d_{n}-d_{i}+1)/d_{n},\quad 1\leq i\leq
n.$ (110)

######  Lemma 6.4.

All entries of $h(x)\hat{B}^{(k)}(x)$ are weighted homogeneous polynomials in
$x$, and $\hat{Y}$ satisfies the differential equation