Funkcialaj Ekvacioj, 19 (1976) 175-189
On a Linear System of Pfaffian Equations
with Regular Singular Points
By
Masaaki YOSHIDA and Kyoichi TAKANO*
(Kobe University, Japan)
Introduction.
The purpose of this paper is to extend some well known theorems on ordinary
differential equations to Pfaffian systems.
We consider a completely integrable system
(1)
$dX=(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{i})X$
where $X$ is an unknown $m¥times m$ matrix and $P_{i}(x)$ is an
at $x=0$ , i.e. $P_{i}(x)$ permits a power series expansion
$P_{i}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
$m¥times m$
matrix holomorphic
,
convergent in a neighborhood of $x=0$ . Here denotes a multi-index
,
nonnegative integer, $0=(0, ¥cdots, 0)$ and
. For two multi-indices
and , $k¥geq l’$ means
for all
and $k>l’$ means $k¥geq l$ and
for some
. We propose to construct the solutions of (1) concretely and to find out the
dominant coefficients of
which determine the local behavior of the solutions.
As in the local theory of ordinary differential equations, our study is divided
into two parts:
(i) to find out a formal transformation which reduces (1) to a solvable equa$k$
$(k_{1^{ }},¥cdots, k_{n})$
$k$
$x^{k}=x_{l^{l}}^{k}¥cdots x_{n^{n}}^{k}$
$l$
“
’
“
$k_{i}¥geq l_{i}$
$i’$
’
“
’
$k_{i}$
“
$k_{i}>l_{i}$
$i’’$
$¥{P_{i,k}¥}$
tion,
to prove the convergence of the formal transformation. We call the
former the formal part and the latter the analytic part. The analytic part for
system (1) is easily achieved; in fact, the technique is the same as in that for ordinary
differential equations. However, the formal part for system (1) contains an essential difficulty, which does not appear in that for ordinary differential equations,
because the recursion formulas to determine the coefficients of a formal tronsforma-
(ii)
*
This paper contains the complete proofs of the results published in the Proceedings
of the Japan Academy, 51 (1975), 219-223.
176
M. YOSHIDA and K. TAKANO
tion and the integrability condition are extremely complicated for our case. Thus
our effort is concentrated on making clear the effect of a transformation on the consequent equation and the role of integrability condition. We show that the complication can be resolved by decomposing a transformation into a sequence of
simple transformations of special form. As far as the authors know, the idea of
decomposing a transformation in the formal theory is due to Professor M. Hukuhara.
In Section 1, we prepare, in advance, a convergence theorem guaranteeing
that the formal transformations considered in this paper are actually convergent.
,
In Section 2, we state the integrability condition in terms of the coefficients
since the expression of the integrability condition in this form is required to achieve
the calculations below. In Section 3, first we give the definition of the notion that
a system of the form (1) has ’reduced’ form. Next we prove that by a suitable
change of variables $X=U(x)Y$ with $U(x)$ invertible and holomorphic at $x=0$ ,
system (1) is transformed to a system
$¥{P_{t,k}¥}$
(2)
$d¥mathrm{Y}=(¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i})¥mathrm{Y}$
which has ’reduced’ form. We determine $U(x)$ as a product of infinitely many
simple matrices of special form. This section is the main part of our paper. Indeed, once the reduction to ’reduced’ form is completed, there remains no difficulty.
In Section 4, we show that a substitution
$¥mathrm{Y}=x_{1}^{L_{1}}¥cdots x_{n^{n}}^{L}¥cdot Z$
, being a nonnegative integer, changes system (2), which
with $L_{i}=$ diag
has ’reduced’ form, to a system
$(l_{i}^{1_{ }},¥cdots, l_{i}^{m})$
$l_{i}^{a}$
(3)
$dZ=(¥sum_{i=1}^{n}¥frac{B_{i}}{x_{i}}dx_{i})Z$
, . Combining the results in Sections 3 and 4, we
with constant matrices ,
can establish one of our two main theorems, which is stated in the last section.
In the arguments above, it was not the question if some $P_{i}(x)/x_{i}$ is holomorphic at $x=0$ or not. However, in the first alternative, we had better eliminate the
variable , before making the change of variables studied in Sections 3 and 4.
Thus Section 5 is devoted to this elimination problem; in more detail, it shows that
if $P_{i}(x)/x_{i}$ is holomorphic at $x=0$ for $¥nu<i¥leq n$ , system (1) can be changed to the
system
$B_{1}$
$¥cdots$
$B_{n}$
$x_{i}$
(4)
$d¥mathrm{Y}=(¥sum_{i=1}^{¥nu}¥frac{P_{i}(x_{1},,x_{¥nu},0,,0)}{x_{i}}¥cdots¥cdots dx_{i})¥mathrm{Y}$
On a Linear System of
Pfaffian
Equations
177
$U(x)Y$ with $U(x)$ invertible and holomorphic at $x=0$ .
by a suitable substitution
Combining the results of Sections 3, 4 and 5, we have the other main theorem,
which is also stated in the last section.
Convergence theorem.
§1.
In this section, as was announced in the introduction, we shall prove the following convergence theorem. It is an extension of a well known theorem for one
variable (cf. [4]) to several variables.
Theorem 1.
Let
(5)
$du=(¥sum_{i=1}^{n}¥frac{F_{i}(x)}{x_{i}}dx_{i})u$
be a completely integrable system, where
$F_{i}(x)=¥sum_{k¥geq 0}F_{i,k}x^{k}$
$u$
,
is a matrix convergent and holomorphic for
solution of (5) written as
(6)
is a vector and
$i=1$ ,
$|x|<_{¥epsilon}$
$¥cdots$
,
$n$
Then any formal power series
.
$u(x)=¥sum_{k¥geq 0}u_{k}x^{k}$
converges
for
$|x|<_{¥epsilon}$
and represents a holomorphic solution
of (5).
Proof. Since (6) formally satisfies (5), we have
$F_{i,0}u_{0}=0$
and
$(F_{i,0}-k_{i}I)u_{k}=-¥sum_{0<l¥leq k}F_{i,l}u_{k-l}$
, . There exists a positive integer
for all $i=1$ ,
such that $¥det(F_{i,0}-k_{i}I)¥neq 0$
.
$||(F_{i,0}-k_{i}I)^{-1}||
¥
leq
1$
for all
. Here
and
denotes the usual Euclidian norm.
Hence
$¥cdots$
$n$
$k_{i}^{0}$
$k_{i}>k_{i}^{0}$
(7)
for
$k¥not¥leq k^{0}$
$||$
$||$
$||u_{k}||¥leq¥sum_{0<l¥leq k}f_{l}||u_{k-l}||$
, where
$f_{l}=¥max_{1¥leq i¥leq n}||F_{i,l}||$
and so is holomorphic for
$v(x)$ defined by
.
$|x|<_{¥epsilon}$
We note that the series
$f(x)=¥sum_{¥iota>0}f_{l}x^{¥iota}$
converges
. Now we introduce the scalar majorizing function
M. YOSHmA and K. TAKANO
178
(8)
$v(x)=f(x)v(x)+||u_{0}||+¥sum_{0<k¥leq k^{0}}¥{||u_{¥mathrm{k}}||-¥sum_{0<¥iota¥leq k}f_{l}||u_{k-l}||¥}x^{k}$
Since
for
there exists a positive number
Let
$f(0)=0$ ,
$|x|<_{¥epsilon^{¥prime}}$
.
(9)
$v(x)=¥sum_{k¥underline{>}0}v_{k}x^{k}$
,
$6^{¥prime<_{¥epsilon}}$
such that
$|x|<_{¥epsilon^{¥prime}}$
$v(x)$
.
is holomorphic
.
Substitution of (9) in (8) leads to the recursion formulas
$v_{k}=||u_{k}||$
,
$k¥leq k^{0}$
and
$v_{k}=¥sum_{0<¥iota¥leq k}f_{l}||u_{k-l}||$
,
$k¥not¥leq k^{0}$
for all
Comparison with (7) shows
series (9) implies the convergence of (6) in
is now immediate.
$||u_{k}||¥leq v_{k}$
$k$
.
$|x|<_{¥epsilon^{¥prime}}$
§2.
.
Hence the convergence of the
. The completion of the proof
Integrability condition.
The integrability condition for a system
(10)
$dX=¥Omega(x)X$
$d¥Omega(x)-¥Omega(x)¥wedge¥Omega(x)=0$
In case
$¥Omega(x)=¥sum_{i=1}^{n}¥frac{P_{i}(x_{i})}{x_{i}}dx_{i}$
is
.
where
$P_{i}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
is convergent and holomorphic in a neighborhood of
equivalent to
(10)
, and $k=(k_{1^{ }},¥cdots, k_{n})$ .
for all , $j=1$ ,
matrices: $[A, B]=AB-BA$ .
§3.
the condition (10) is
$k_{j}P_{i,k}-k_{i}P_{j,k}+¥sum_{k^{¥prime}+k^{¥prime¥prime}=k}[P_{i,k^{¥prime}}, P_{j,k^{¥prime¥prime}}]=0$
$i,j;k$
$i$
$x=0$ ,
$¥cdots$
$n$
Here
$[, ]$
denotes the usual bracket of
Reduction to a ‘reduced’ form.
The purpose of this section is to reduce system (1) to a system which has a
form as simple as possible by a change of variables $X=U(x)Y$ with $U(x)$ invertible
and holomorphic at $x=0$ . We shall begin with giving the following definition.
Definition.
We say the equation
On
a Linear
System
of Pfaffian
179
Equations
$dX=(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{t})X$
is ‘reduced’ with respect to
, , where
for all $j=1,2$ ,
$(k, (¥alpha, ¥beta))$
$¥cdots$
if
$a_{i}^{aa}-a_{i}^{¥beta¥beta}-k_{i}¥neq ¥mathit{0}$
$k=(k_{1^{ }},¥cdots, k_{n})$
$n$
,
for some
$P_{t}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
,
$i$
implies
$p_{jk}^{a¥beta}=0$
$P_{i,k}=(p_{ik}^{**})$
and
Furthermore we say the equation has a‘reduced’ form if it is reduced
.
with respect to all
First we shall show that by a suitable formal transformation $X=U(x)¥mathrm{Y}$ , $U(x)$
being a formal power series of , (1) can be changed to a system which
$P_{i,0}=(a_{i}^{**})$
.
$(k, (¥alpha, ¥beta))$
$x$
$=¥sum_{k¥geq 0}U_{k}x^{k}$
has ‘reduced’ form. Next we shall prove the convergence of the formal power
. We remark that the formal reduction will be carried out by deseries
$a$
$¥sum_{k¥geq 0}U_{k}x^{k}$
composing a transformation
$X=(¥sum_{h¥underline{>}0}U_{k}x^{k})¥mathrm{Y}$
into a product of infinitely many
transformations.
3.1.
Before exposing the process to decompose
$X=$
, we explain the reason why such a decomposition is necessary.
Let
Formal reduction.
$(¥sum_{k¥geq 0}U_{k}x^{k})¥mathrm{Y}$
(2) be the system changed from (1) by
$X=U(x)¥mathrm{Y}$
, then we have
$Q_{i}(x)=U(x)^{-1}P_{t}(x)U(x)-U(x)^{-1}x_{i}¥frac{¥partial U(x)}{¥partial x_{i}}$
In the usual way, we substitute
$U(x)=¥sum_{k¥geq 0}U_{k}x^{k}$
finitely many relations between the coefficients
in
$Q_{i}(x)=¥sum_{k¥geq 0}Q_{i,k}x^{k})$
$i=1$ ,
,
⃝,
n.
in the above equations and get in$¥{U_{k}¥}$
, and determine the values of
,
$¥{P_{i,k}¥}$
$¥{U_{k}¥}$
,
$¥{Q_{i,k}¥}(¥mathrm{t}¥mathrm{h}¥mathrm{e}$
coefficients
such that almost all of
vanish. Considering these relations as recursion formulas to determine
successively, we can easily verify that each
must satisfy n relations at the same
time. The integrability condition must assure that
can be determined compatibly. However, the recursion formulas and the integrability condition are so
comph.cated that it is very difficult to see to what extent system (1) can be simplified
and that even if a canonical form is incidentally found out, it is hard to show that
can be determined compatibly.
Thus, in order to make clear the effect of the coefficients
on the consequent equation and the role of integrability condition, we decompose the trans$¥{Q_{i,k}¥}$
$¥{U_{k}¥}$
$U_{k}$
$U_{k}$
$U_{k}$
$¥{U_{k}¥}$
formation
$X=(¥sum_{k¥geq 0}U_{k}x^{k})¥mathrm{Y}$
into a product of simple transformations.
Now we return to our subject. It is easy to see that every formal power
series $U(x)=¥sum_{k¥geq 0}U_{k}x^{k}(¥det U_{0}¥neq 0)$ can be uniquely decomposed as follows
180
M. YOSHiDA and K. TAKANO
$ U(x)=U_{0}¥cdot U_{1}(x)¥cdots U_{N}(x)¥cdots$
where
$U_{0}$
is the constant matrix of the given
$¥sum_{k¥geq 0}U_{k}x^{k}$
and
$U_{N}(x)=U_{l¥backslash ¥Gamma}^{(1,m)}(x)¥cdot U_{N}^{(2,m)}(x)¥cdots U_{N}^{(m,m)}(x)$
$¥times U_{N}^{(1,m-1)}(x)¥cdot U_{N}^{(2,m-1)}(x)¥cdots U_{N}^{(m,m-1)}(x)$
$¥times U_{N}^{(1,1)}(x)¥cdot U_{N}^{(2,1)}(x)¥cdots U_{N}^{(m,1)}(x)$
$U_{N}^{(a,¥beta)}(x)=I+¥sum_{|k|=N}U_{N,k}^{(a,¥beta)}x^{k}$
Here
$|k|=¥sum_{i=1}^{n}k_{i}$
zero except for
, and
$U_{N}^{(a,¥beta)}$
,
.
is a constant matrix of which the
component is
$(¥gamma, ¥delta)$
will be denoted by
component of
. (The
of the special form as above
Conversely the infinite product of
can be expanded into a formal power series of . In other words, the formal
$(¥gamma, ¥delta)=(¥alpha, ¥beta)$
$(¥alpha, ¥beta)$
$U_{N,k}^{(a,¥beta)}$
$¥{U_{N}^{(a,¥beta)}(x)¥}$
$u_{Nk}^{a¥beta}.)$
$x$
transformation
transformations
$¥{X=U_{N}^{(a,¥beta)}(x)¥mathrm{Y}¥}$
$X=(¥sum_{k¥geq 0}U_{h}x^{k})¥mathrm{Y}$
can be uniquely decomposed into a sequence of
$¥{X=U_{N}^{(a,¥beta)}(x)¥mathrm{Y}¥}$
and conversely a sequence of transformations
determines a formal transformation
$X=(¥sum_{k¥geq 0}U_{k}x^{k})¥mathrm{Y}$
.
If we decompose $X=U(x)Y$ in this way, then we can easily see the effect of
each
on the consequent equation and the role of the integrability condition.
,
,
,
,
,
,
We shall show how to determine
successively.
$U_{N}^{(a,¥beta)}(x)$
$U_{0}$
Determination of
in
matrices
$P_{i,0}$
$U_{0}$
.
$U_{1}^{(1,m)}(x)$
$¥cdots$
$U_{1}^{(m,1)}(x)$
$¥cdots$
$U_{N}^{(a,¥beta)}(x)$
?
, the constant
By the integrability condition (11)
in (1) are commutin
and
$i,j:0$
$P_{i}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
$P_{j,0}$
$P_{j}(x)=¥sum_{k¥geq 0}P_{j,k}x^{k}$
, . Therefore, by the elementary matrix
ative: $[P_{i,0}, P_{j,0}]=0$ for all , $j=1$ ,
such that
matrix
constant
nonsingular
theory, we can choose a
$¥cdots$
$i$
$n$
$U_{0}$
$¥mathrm{e}¥cdots i,’ ¥mathrm{t}n¥mathrm{i}¥mathrm{s}1¥mathrm{o}¥mathrm{w}¥mathrm{e}¥mathrm{r}¥mathrm{t}¥mathrm{r}¥mathrm{i}¥mathrm{a}¥mathrm{n}¥mathrm{g}¥mathrm{u}1¥mathrm{a}¥mathrm{r}¥mathrm{h}¥mathrm{e}¥mathrm{n}a_{j}^{a¥beta}=0¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{a}11j’=1$
(12)
$¥{((¥mathrm{i})¥mathrm{i}¥mathrm{i})$
,
$¥mathrm{i}A_{i}=(a_{i}^{¥alpha¥beta})=U_{0}^{-1}.P_{i,0}U_{0},i=¥mathrm{l}¥mathrm{f}a_{i}^{aa}-a_{i}^{¥beta¥beta}¥neq 0¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{s}¥mathrm{o}’ ¥mathrm{m}$
We note that the equation transformed by
.
for all
$(0, (¥gamma, ¥delta))$
$X=U_{¥mathit{0}}Y$
$¥cdots$
, , where
$n$
$A_{i}$
’reduced’ with respect to
$(¥gamma, ¥delta)$
namely the
,
,
,
,?.
Determination of
described
process
is determined successively by the induction
coefiBcient
, regarding as
below. We apply the following proposition to determine
the equation in $X$ in the proposition the equation transformed from (1) by $X=$
$U_{1}^{(1,m)}(x)$
$U_{1}^{(2,m)}(x)$
$¥cdots$
$U_{N}^{(a,¥beta)}(x)$
$U_{N}^{(a,¥beta)}(x)$
$u_{Nk}^{a¥beta}(|k|=N)$
$U_{N}^{(a,¥beta)}(x)$
On a Linear System
$U_{0}¥cdot U_{1}^{(1,m)}(x)¥cdots U_{N}^{(a-1,¥beta)}(x)¥mathrm{Y}$
ing the notation
scribed as follows.
of Pfaffian
181
Equations
which is obviously completely integrable. By introducor
for
,
, the induction process is de-
$(¥gamma, ¥delta)<$ $(¥alpha, ¥beta)$
$¥delta>¥beta$
$¥delta=¥beta$
Proposition 1 (Induction process).
$¥gamma¥backslash ^{/}¥alpha$
Assume that the completely integrable
system
$dX=(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{i})X$
with
$P_{i}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
,
$P_{i,k}=(p_{ik}^{**})$
,
is ’reduced’ with respect to
both for the cases when $|k|<N$ ,
arbitrary
$P_{i,0}=A_{i}=(a_{i}^{**})$
and when $|k|=N$ ,
. Assume further
satisfies the condition (12). Then one finds a transformation
$(k, (¥gamma, ¥delta))$
$(¥gamma, ¥delta)$
$(¥gamma, ¥delta)<$ $(¥alpha, ¥beta)$
$X=U_{N}^{(a,¥beta)}(x)¥mathrm{Y}$
such that the consequent equation
$d¥mathrm{Y}=(¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i})¥mathrm{Y}$
with
$Q_{t}(x)=¥sum_{k¥geq 0}Q_{i,k}x^{k}$
is ’reduced’ with respect to
and when $|k|=N$ ,
Furthermore
$(k, (¥gamma, ¥delta))$
$(¥gamma, ¥delta)¥leq(¥alpha, ¥beta)$
(13)
(14)
If
$Q_{i,k}=P_{i,k}$
$q_{ih}^{¥gamma¥delta}=p_{ik}^{¥gamma¥delta}$
$a_{i_{0}}^{aa}-a_{i_{0}}^{¥beta¥beta}-k_{i_{0}}¥neq 0$
for
,
$Q_{i,k}=(q_{ik}^{**})$
both for the cases when
$|k|<N$ ,
$(¥gamma, ¥delta)$
arbitrary
.
for
$k$
$(|k|<N)$ , $i=1$ ,
$k(|k|=N)$ ,
$(¥gamma, ¥delta)<(¥alpha, ¥beta)$
for some , then the value of
$i_{0}$
,
,
$¥cdots$
, ,
$i=1$ ,
$u_{Nk}^{¥alpha¥beta}(|k|=N)$
$(a_{i_{0}}^{¥alpha a}-a_{i_{0}}^{¥beta¥beta}-k_{i_{0}})u_{Nk}^{a¥beta}+p_{i_{0}k}^{¥alpha¥beta}=0$
$n$
$¥cdots$
, .
$n$
is determined by
.
Proof. It is easy to see
$Q_{i}(x)=(I+¥sum_{|k|=N}U_{N,k}^{(a,¥beta)}x^{k})^{-1}P_{i}(x)(I+¥sum_{|k|=N}U_{N,k}^{(a,¥beta)}x^{k})$
$-(I+¥sum_{|k|=N}U_{N,k}^{(a,¥beta)}x^{k})^{-1}(¥sum_{|k|=N}k_{i}U_{N,k}^{(a,¥beta)}x^{k})$
.
182
M. YOSHIDA and K. TAKANO
Comparing the coefficients of
(15)
with
and
$x^{k}$
,
$|k|<N$
or
$|k|=N$ ,
we obtain (13) and
$Q_{i,k}=P_{i,k}+[A_{t}, U_{N,k}^{(a,¥beta)}]-k_{i}U_{N,k}^{(a,¥beta)}$
$|k|=N$ , $i=1$ ,
$¥cdots$
, .
Since
$n$
$A_{i}$
is lower triangular, we can easily derive (14)
$q_{ik}^{a¥beta}=p_{ik}^{a¥beta}+(a_{i}^{aa}-a_{i}^{¥beta¥beta}-k_{i})u_{Nk}^{a¥beta}$
from (15).
Now suppose
$k(|k|=N)$
(16)
,
$|k|=N$ ,
$i=1$ ,
$¥cdots$
, ,
$n$
satisfies
$a_{i_{0}}^{aa}-a_{i_{0}}^{¥beta¥beta}-k_{i_{0}}¥neq 0$
for some .
In order to prove that
can be chosen so that
sufficient to show the relation
$u_{Nk}^{a¥beta}$
(17)
,
$i_{0}$
$q_{ik}=0$
for all
,
$i=1$ ,
$(a_{i_{¥mathit{0}}}^{aa}-a_{i_{¥mathit{0}}}^{¥beta¥beta}-k_{i_{¥mathit{0}}})p_{ik}^{a¥beta}-(a_{i}^{aa}-a_{i}^{¥beta¥beta}-k_{i})p_{i_{¥mathit{0}}k}^{a¥beta}=¥mathit{0}$
$i=1$ ,
$¥cdots$
,
$¥cdots$
, , it is
$n$
$n$
holds, which will be called the compatibility condition. It will be shown that the
condition (17) can be derived from the integrabih.ty condition, which will complete
the proof of Proposition 1.
Let us calculate the
component of the integrability condition
$(¥alpha, ¥beta)$
(11)
$k_{i}P_{i_{0},k}-k_{i}{}_{0}P_{i,k}+¥sum_{k^{¥prime}+k^{¥prime¥prime}=k}[P_{t_{0},k^{¥prime}}, P_{t,k^{¥prime¥prime}}]=0$
$i_{0},t;k$
For the third term of (11)
(18)
$i_{0},i;k$
.
, we have
$¥sum_{k^{¥prime}+k^{¥prime¥prime}=k}[P_{i_{0},k^{¥prime}}, P_{i,k^{¥prime¥prime}}]=[A_{i_{0}}, P_{i,k}]-[A_{i}, P_{i_{0},k}]+h^{¥prime}k^{¥prime},+k^{¥prime¥prime}-kk^{¥prime¥prime}>0¥sum_{-}[P_{i_{0},k^{¥prime}}, P_{i,k^{¥prime¥prime}}]$
The
$(¥alpha, ¥beta)$
.
component of the first term of (18) is expressed as
$[A_{i_{0}},, P_{i,k}]^{a¥beta}=(a_{i_{0}}^{aa}-a_{i_{0}}^{¥beta¥beta})p_{ik}^{a¥beta}+¥sum_{¥xi<a}a_{i_{0}}^{a¥xi}p_{ih}^{¥xi¥beta}-¥sum_{¥xi>¥beta}a_{i_{0}}^{¥xi¥beta}p_{ik}^{a¥xi}$
,
is lower triangular. In this relation, if
because
for some
, then
for all and
we have
for all /, by the assumption of
the proposition that the system considered is ’reduced’ with respect to
both
for the cases when $k=0$ ,
. Hence
arbitrary and when $|k|=N$ ,
for all /, which is a contradiction to (16). Therefore,
for all
. For the same reason,
for all
. Thus we have
$A_{i_{0}}$
$a_{j}^{aa}-a_{j}^{¥xi¥xi}=0$
$a_{i_{0}}^{a¥xi}p_{ik}^{¥xi¥beta}¥neq 0$
$j$
$¥xi<_{¥alpha}$
$a_{j}^{¥xi¥xi}-a_{j}^{¥beta¥beta}-k_{J}=0$
$(k, (¥gamma, ¥delta))$
$(¥gamma, ¥delta)<(¥alpha, ¥beta)$
$(¥gamma, ¥delta)$
$a_{j}^{aa}-a_{j}^{¥beta¥beta}-k_{f}=¥mathit{0}$
$¥xi<_{¥alpha}$
(19)
$a_{i_{0}}^{a¥xi}p_{ik}^{¥xi¥beta}=0$
$a_{i_{0}}^{¥xi¥beta}p_{ik}^{a¥xi}=0$
$¥xi>¥beta$
$[A_{i_{0}}, P_{i,k}]^{a¥beta}=(a_{i_{0}}^{aa}-a_{i_{0}}^{¥beta¥beta})p_{ik}^{a¥beta}$
,
$[A_{i}, P_{i_{0},k}]^{a¥beta}=(a_{i}^{aa}-a_{i}^{¥beta¥beta})p_{i_{0}k}^{a¥beta}$
.
and analogously
(20)
On a Linear System of
Finally, it will be shown that the
Pfaffian
183
Equations
component of the third term of (18),
$(¥alpha, ¥beta)$
i.e.
$[P_{i_{0},k^{¥prime}}, P_{i,k^{¥prime¥prime}}]^{a¥beta}=¥sum_{¥xi=1}^{m}p_{i_{0}k^{¥prime}}^{a¥xi}p_{ik}^{¥xi¥beta},,-¥sum_{¥xi=1}^{m}p_{i_{0}k^{¥prime}}^{¥xi¥beta}p_{ik^{¥prime¥prime}}^{a¥xi}$
vanishes.
If
$p_{i_{0}h^{¥prime}}^{a¥xi}p_{ik^{¥prime¥prime}}^{¥xi¥beta}¥neq 0$
,
$k^{¥prime}+k^{¥prime¥prime}=k$
,
$k^{¥prime}$
,
$k^{¥prime¥prime}>0$
for some , then we have
$¥xi$
$a_{j}^{aa}-a_{j}^{¥xi¥xi}-k_{j}^{¥prime}=¥mathit{0}$
for all
$j$
and
$a_{j}^{¥xi¥xi}-a_{j}^{¥beta¥beta}-k_{j}^{¥prime¥prime}=0$
for all ,
$j$
by the assumption of the proposition that our system is ’reduced’ with respect to
for $|k|<N$ ,
arbitrary. Hence
for all /, which is a
contradiction to (16). Therefore
for all . In the same way, we have
for all . Thus
$(k, (¥gamma, ¥delta))$
$a_{j}^{aa}-a_{j}^{¥beta¥beta}-k_{j}=¥mathit{0}$
$(¥gamma, ¥delta)$
$¥xi$
$p_{i_{0}k^{¥prime}}^{a¥xi}p_{ik^{¥prime¥prime}}^{¥xi¥beta}=0$
$¥xi$
$p_{i_{0}k^{¥prime}}^{¥xi¥beta}p_{ik^{¥prime¥prime}}^{a¥xi}=0$
(21)
$[P_{i_{0},k^{¥prime}}, P_{i,k^{¥prime¥prime}}]^{a¥beta}=0$
. Therefore, from (11) $i_{0},i:k’(18)$ , (19), (20)
, ,
for all ,
with
and (21), we obtain the compatibility condition (17). Thus we have completed the
proof of Proposition 1.
thus determined, we have
By expanding the infinite product of
$k^{¥prime}$
$k^{¥prime}+k^{¥prime¥prime}=k$
$k^{¥prime¥prime}$
$k^{¥prime}$
$k^{¥prime¥prime}>0$
$¥{U_{N}^{(a,¥beta)}(x)¥}$
Theorem 2.
One can find a formal power series
that the formal substitution
has
$a$
$X=(¥sum_{k¥geq 0}U_{k}x^{k})¥mathrm{Y}$
with
$¥sum_{k¥geq 0}U_{k}x^{k}$
$¥det U_{0}¥neq 0$
, such
changes system (1) to a system which
’reduced’ form.
Remark 1. In the induction process above, for sufficiently large $N$ ,
in Theorem 2 is not uniquely
is uniquely determined. Therefore, although
$U_{N}^{(a,¥beta)}(x)$
$¥sum_{k¥geq 0}U_{k}x^{k}$
determined, it contains only a finite number of undetermined parameters.
Remark 2.
In Theorem 2, if no two eigenvalues of
for every , we can choose
$i$
$¥sum_{k¥geq 0}U_{k}x^{k}$
with
$U_{0}=I$
$P_{i,0}$
such that, by
differ by an integer
$X=(¥sum_{h¥geq 0}U_{k}x^{k})¥mathrm{Y}$
, (1)
is changed to the equation
$d¥mathrm{Y}=(¥sum_{i=1}^{n}¥frac{P_{i,0}}{x_{i}}dx_{i})¥mathrm{Y}$
In this case,
3.2,
$¥sum_{k¥geq 0}U_{k}x^{k}$
$U_{k}(k>0)$
.
is uniquely determined.
Analytic reduction. As is easily seen, the formal power series
in Theorem 2 satisfies the equation
$U(x)=$
184
M. YOSHIDA and K. TAKANO
(22)
$dU(x)=(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{i})U(x)-U(x)(¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i})$
in the formal sense.
(23)
If we regard
du(x)
$U(x)$
as an
$¥mathrm{m}^{2}$
-vector
,
, (22) is equivalent to
$u(x)$
$=(¥sum_{i=1}^{n}¥frac{1}{x_{i}}(P_{i}(x)¥otimes I-I¥otimes^{¥iota}Q_{i}(x))dx_{i})u(x)$
.
Since $Q_{i}(x)$ is a polynomial of , we can apply Theorem 1 to equation (23) to prove
. Thus, by the definition of a‘reduced’ form, we have
the convergence of
$x$
$¥sum_{k¥geq 0}U_{k}x^{k}$
Given a completely integrable system (1), we have a convergent
with $¥det U_{0}¥neq 0$ , such that the transformation $X=U(x)Y$ takes
Theorem 3.
series
¥
¥
$U(x)= sum_{h geq 0}U_{k}x^{k}$
(1) to
(24)
$d¥mathrm{Y}=(¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i})¥mathrm{Y}$
,
where
(i)
$Q_{i}(x)=A_{i}+¥sum_{k>0}Q_{i,k}x^{k}$
$A_{i}=(a_{i}^{**})$
(ii)
the
(finite sum),
being lower triangular,
$(¥alpha, ¥beta)$
component
$q_{i}^{a¥beta}(x)$
of
$Q_{i}(x)$
is a monomial of ,
$x$
$q_{i}^{a¥beta}(x)=q_{i}^{a¥beta}x_{1}^{h_{1}}¥cdots x_{n^{n}}^{k}$
with
$k_{¥mu}=a_{¥mu}^{aa}-a_{¥mu}^{¥beta¥beta}$
integer
§4.
$k_{¥mu}$
for
all
,
$¥mu=1$
$¥mu=1$
,
,
$¥cdots$
$¥cdots$
, .
$n$
$q_{i}^{a¥beta}(x)¥neq 0$
,
implies that
$a_{¥mu}^{aa}-a_{¥mu}^{¥beta¥beta}$
$n$
Singular transformation.
Consider the ’reduced’ equation (24) in Theorem 3.
. A singular transformation
matrix diag
$(l_{i^{ }}^{1},--, l_{i}^{m})$
$¥mathrm{Y}=x_{1}^{L_{1}}-x_{n}^{L_{n}}Z$
changes (24) to
$dZ=(¥sum_{i=1}^{n}¥frac{B_{i}(x)}{x_{i}}dx_{i})Z$
where
is a nonnegative
, .
Let
$L_{i}$
be a diagonal
On a Linear System of
Pfaffian
$B_{i}(x)=x_{l}^{-L_{l}}¥cdots x_{n}^{-L_{n}}Q_{i}(x)x_{l}^{L_{l}}$
185
Equations
?
$x_{n^{n}}^{L}-L_{i}$
,
or equivalently
.
$b_{i}^{a¥beta}(x)=q_{i}^{a¥beta}(x)x_{1}^{¥iota_{1}^{¥beta}-¥iota_{1}^{a}}¥cdots x_{n^{n}}^{l^{¥beta}-¥iota_{n}^{a}}-¥delta_{¥rho}^{a}l_{i}^{a}$
¥
¥
,
Here,
, and denotes the Kronecker symbol.
We shall show that
becomes constant by choosing nonnegative integers
suitably. We classify
and
belong to the same class
, so that
iff
is an integer. We denote by
the class of
. For every
, we
denote by
a member of
which has the minimum real part. Then by tak$ing$
,
becomes a constant
, by virtue of the properties (i) and
(ii) in Theorem 3. Thus we have proved
$Q_{i}(x)=(q_{i}^{a beta}(x))$
$B_{t}(x)=(b_{i}^{a beta}(x))$
$¥delta_{¥beta}^{a}$
$b_{i}^{a¥beta}(x)$
$¥{a_{i}^{aa}¥}_{a=1,m}¥cdots$
$¥{l_{i}^{a}¥}$
$a_{i}^{aa}-a_{i}^{¥beta¥beta}$
$a_{i}^{¥beta¥beta}$
$a_{i}^{aa}$
$[a_{i}^{aa}]$
$a_{i}^{aa}$
$a_{i}^{aa}$
$[a_{i}^{aa}]$
$a_{i^{00}}^{a¥alpha}$
$b_{i}^{a¥beta}(x)$
$l_{i}^{a}=a_{i}^{aa}-a_{i^{¥mathit{0}¥mathit{0}}}^{aa}$
$b_{i}^{a¥beta}$
By a change
Theorem 4.
of variables
$¥mathrm{Y}=x_{1}^{L_{1}}¥cdots x_{n^{n}}^{L}Z$
with
, being nonnegative integer, the ’reduced: equation (24)
in Theorem 3 is transformed to
$¥mathrm{L}_{¥mathrm{i}}=¥mathrm{d}¥mathrm{i}¥mathrm{a}¥mathrm{g}$
$(l_{i^{ }}^{1},¥cdots, l_{i}^{m})$
$l_{i}^{a}$
$dZ=(¥sum_{i=1}^{n}¥frac{B_{i}}{x_{i}}dx_{i})Z$
where
$B_{i}$
is a constant matrix given by
(finite sum)
$B_{i}=A_{i}-L_{i}+¥sum_{k>0}Q_{i,h}$
and
$[B_{t}, B_{j}]=0$
, , $j=1$ ,
$i$
$¥cdots$
, .
$n$
in Theorem 4 is not uniquely determined but is unique up to
Remark 3.
integers in the following sense: Let
and
be two diagonal matrices satisfying
the conditions of Theorem 4, then
for all , with
.
$L_{¥ell}$
$L_{i}$
$L_{i}^{¥prime}$
$l_{i}^{a}-l_{i}^{¥beta}=l_{i}^{¥prime_{a}}-l_{i}^{¥prime¥beta}$
§5.
$¥alpha$
$¥beta$
$[a_{i}^{aa}]=[a_{i}^{¥beta¥beta}]$
Elimination of nonessential variables.
In this section, we consider the special case when $P_{i}(x)/x_{i}$ is holomorphic at
$x=0$ for $¥nu<i¥leq n$ . In order to eliminate the variables
,
, , we consider a
sequence of transformations $¥{X=U_{N}(x)¥mathrm{Y}¥}_{N=1,2},¥ldots$ where
$x_{¥nu+1}$
$U_{N}(x)=I+¥sum_{|h|=N,¥hat{k}>0}U_{N,k}x^{k}$
with
$¥hat{k}=(k_{¥nu+1^{ }},¥cdots, k_{n})$
.
$x_{n}$
,
As an induction process we have
Proposition 2 (Induction process).
system
$¥cdots$
Assume that the completely integrable
186
M. YOSHIDA and K. TAKANO
$dX=(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{i})X$
,
$P_{i}(x)=¥sum_{k¥geq 0}P_{i,k}x^{k}$
,
satisfies the conditions:
is holomorphic at $x=0$ for $¥nu<i¥leq n$ ,
(i)
, $0¥leq|k|<N$ .
, $0<|k|<N,¥hat{k}>0$ and for
(u) $P_{i,k}=0$ for
$X=U_{N}(x)Y$
with
Then we can choose a suitable substitution
$P_{i}(x)/x_{i}$
$i>_{¥nu}$
$ i¥leq¥nu$
$U_{N}(x)=I+¥sum_{|k|=N,¥hat{k}>0}U_{N,k}x^{h}$
,
such that the consequent system
$d¥mathrm{Y}=(¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i})¥mathrm{Y}$
,
$Q_{i}(x)=¥sum_{k¥geq 0}Q_{i,k}x^{k}$
satisfies the conditions:
(i)’
$(¥mathrm{i}¥mathrm{i})$
’
(iii)’
$Q_{i}(x)/x_{i}$
$Q_{i,h}=0$
is holomorphic at $x=0$ for $¥nu<i¥leq n$ ,
, $0<|k|¥leq N,¥hat{k}>0$ and for
for
$i>_{¥nu}$
$ i¥leq¥nu$
,
$Q_{i}(x_{1^{ }},¥cdots, x_{¥nu}, 0, ¥cdots, 0)=P_{i}(x_{1^{ }},¥cdots, x_{¥nu}, 0, ¥cdots, 0)$
$0¥leq|k|¥leq N$
for
$ i¥leq¥nu$
and the condition (ii)’ for
,
Proof, The conditions
, $0¥leq k<N$ are immediately verified by
and for
$(¥mathrm{i})’$
$¥hat{k}>0$
$(¥mathrm{i}¥mathrm{i}¥mathrm{i})^{¥prime}$
,
.
$ i¥geq¥nu$
, $0<|k|<N$ ,
$i>_{¥nu}$
(25)
$¥sum_{i=1}^{n}¥frac{Q_{i}(x)}{x_{i}}dx_{i}=U_{N}(x)^{-1}(¥sum_{i=1}^{n}¥frac{P_{i}(x)}{x_{i}}dx_{i})U_{N}(x)-U_{N}(x)^{-1}dU_{N}(x)$
Therefore, we have only to show that by choosing the values of
, $|k|=N$ . By (25), we have
, $|k|=N,¥hat{k}>0$ and for
for
$Q_{i,k}=P_{i,k}+[P_{i,0}, U_{N,k}]-k_{i}U_{N,k}$
$k$
,
$(¥mathrm{i}¥mathrm{i})^{¥prime}$
holds
$i>_{¥nu}$
$ i¥leq¥nu$
For all
by
$¥{U_{N,k}¥}$
.
$(|k|=N,¥hat{k}>0)$
let
$i(k)$
,
$i=1$ ,
be an index satisfying
$U_{N,k}=-¥frac{1}{k_{i(k)}}P_{i(k),k}$
For
thus determined, we can prove
by the integrability condition (11).
$U_{N,k}$
$¥cdots$
$Q_{i,k}=0$
, ,
$n$
$|k|=N,¥hat{k}>0$
$k_{i(k)}¥neq 0$
.
, and determine
$U_{N,k}$
.
for
$i=1$ ,
$¥cdots$
, ,
$n$
$|k|=N,¥hat{k}>0$ ,
Q.E.D.
By this proposition, one determines a sequence of transformations
which formally reduces (1) to
$ U_{N}(x)¥mathrm{Y}¥}_{N=1,2},¥ldots$
.
$d¥mathrm{Y}=(¥sum_{i=1}^{¥nu}¥frac{P_{i}(x_{1},,x_{¥nu},0,,0)}{x_{i}}¥cdots¥cdots dx_{i})¥mathrm{Y}$
$¥{X=$
On a Linear System of
Pfaffian
187
Equations
The convergence of the formal power series
$U(x)=U_{1}(x)U_{2}(x)¥cdots U_{N}(x)¥cdots=I+¥sum_{k>0_{¥hat{k}}>0},U_{k}x^{k}$
can be proved by Theorem 1 in an analogous way as in Section 3.
Thus we have
Theorem 5. Let a completely integrable system (1) be given. If $P_{i}(x)/x_{i}$ is
holomorphic at $x=0$ for $¥nu<i¥leq n$ , then (1) can be changed to the system
$d¥mathrm{Y}=(¥sum_{i=1}^{¥nu}¥frac{P_{i}(x_{1},,x_{¥nu},0,,0)}{x_{i}}¥cdots¥cdots dx_{i})¥mathrm{Y}$
by a substitution $X=U(x)Y$ where
the form
$U(x)$
is invertible and holomorphic at
$U(x)=I+k_{¥nu+1}+.¥cdot¥sum_{k>0}.+k_{n}>0U_{k}x^{k}$
$x=0$
of
.
Remark 4. It must be noted that, in the above substitution $U(x)$ , we have
for $k_{v+1}=¥cdots=k_{n}=0$ . This fact plays an important role in applications.
$U_{k}=0$
§6.
Main theorems.
Combining Theorems 3 and 4, we have
Theorem . Let a completely integrable system (1) be given, where
,
$=1$ ,
, , is holomorphic at $x=0$ . Then we have a matrix $U(x)$ invertible and
holomorphic at $x=0$ and a diagonal matrix , $i=1$ ,
, , whose components are
nonnegative integers, such that the transformation
$P_{i}(x)$
$¥epsilon$
$¥cdots$
$i$
$n$
$L_{i}$
$¥cdots$
$n$
$X=U(x)x_{l}^{L_{l}}¥cdots x_{n^{n}}^{L}Z$
changes (I) to
$dZ=(¥sum_{i=1}^{n}¥frac{B_{i}}{x_{i}}dx_{i})Z$
, , is a constant matrix satisfying $[B_{i},B_{j}]=0$ for all $i,i=1$ , $¥cdots,n$ .
where , $i=1$ ,
and the coefficients in the power series for
,
The calculation of the matrices
$U(x)$ can be concretely performed by algebraic operations.
$¥cdots$
$B_{i}$
$n$
$B_{i}$
$L_{i}$
Combining Theorems 3, 4 and 5, we have
Theorem 7. Let a completely integrable system (1) be given, where
,
$=1$ ,?, , is holomorphic at $x=0$ .
$P_{i}(x)/x_{i}$ , $i=¥nu+1$ ,
, , is holoIf further
morphic at $x=0$ , then we have two matrices $V(x)$ ,
invertible and
$P_{i}(x)$
$¥cdots$
$n$
$W(x_{1^{ }},¥cdots, x_{¥nu})$
$n$
$i$
188
M. YOSHIDA and K. TAKANO
holomorphic at $x=0$ having the
form
$V(x)=I+¥sum_{h>0,k_{p+1}++h_{n}>0}¥cdots V_{h}x^{k}$
,
$W(x_{1^{ }},¥cdots, x_{¥nu})=W_{0}+¥ldots¥sum_{k_{1}++k_{y}>0}W_{k_{1},k_{p}}¥cdots,x_{1^{1}}^{h}¥cdots x_{¥nu}^{k_{y}}$
and a diagonal matrix , $i=1$ ,
such that the transformation
$L_{i}$
$¥cdots$
, , whose components are nonnegative integers,
$¥nu$
$X=U(x)W(x_{1^{ }},¥cdots, x_{¥nu})x_{l}^{L_{l}}¥cdots x_{¥nu}^{L_{¥nu}}Z$
changes (1) to
$dZ=(¥sum_{i=1}^{¥nu}¥frac{B_{i}}{x_{i}}dx_{i})Z$
.
Here, , $i=1$ ,
, , is a constant matrix satisfying $[B_{i}, B_{j}]=0$ for all , $i=1$ ,
, .
The calculation of
,
and the coefficients
,
can be concretely
carried out by algebraic operations.
$¥cdots$
$B_{i}$
$i$
$¥nu$
$B_{i}$
$¥{V_{h}¥}$
$L_{i}$
$¥cdots$
$¥nu$
$¥{W_{k_{1},k_{¥nu}}¥cdots,¥}$
Remark 5. As an immediate consequence of Theorem 1, we have the following theorem which is equivalent to a result obtained by R. Gerard in an abstract
way ([3]).
Theorem.
A completely integrable system
$dX=(¥sum_{i=1}^{¥nu}¥frac{P_{i}(x)}{x_{i}}dx_{i}+¥sum_{i=¥nu+1}^{n}¥hat{P}_{i}(x)dx_{i})X$
with
, $i=1$ ,
,
, $i=¥nu+1$ ,
mental matrix solution of the form
$P_{t}(x)$
$¥cdots$
$¥nu,¥hat{P}_{i}(x)$
$¥cdots$
,
, , holomorphic at
$n$
$U(x)x_{l}^{L_{l}}¥cdots x_{¥nu}^{L_{y}}x_{l}^{B_{l}}$
?
$x_{¥nu}^{B_{p}}$
$x=0$ ,
has a
funda-
.
Here $U(x)$ is a matrix invertible and holomorphic at $x=0$ ,
is a diagonal matrix
whose components are nonnegative integers and
is a constant matrix with
$=0$ for all , $j=1$ ,
, .
$L_{i}$
$B_{i}$
$i$
$¥cdots$
$[B_{i}, B_{j}]$
$¥nu$
References
a
Deligne, P., Equations differentielles points reguliers singuliers, Lee. Notes in Math.,
163. Springer (1970).
[2] Gantmacher, F. R., The Theory of Matrices, Vol. I. Chelsea Publishing Company,
New York (1964).
[3] Gerard, R., Theorie de Fuchs sur une variete analytique complexe, J. Math. Pures
Appl., 47 (1968), 321-404.
[1]
On a Linear System
[4]
[5]
of Pfaffian
Equations
189
Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Interscience
Publishers (1965).
Yoshida, M. and Takano, K., Local theory of Fuchsian systems I, Proc. Japan Acad.,
51 (1975), 219-223.
nuna adreso:
Department of Mathematics
Kobe University
Rokko, Kobe
Japan
(Ricevita la 13-an de junio, 1975)