Funkcialaj Ekvacioj, 37 (1994) 59-64
Painleve-Umemura Extensions
By
K. NISHIOKA
(Keio University at SFC, Japan)
1.
Introduction
be a differential field of characteristic 0 and $U$ be a universal
extension of $K$ . In what follows, unless otherwise stated, differential field
extensions of $K$ concerned will be assumed to be finitely generated, hence
have $U$ as a universal extension. For a differential field extension
of $K$
by
we denote the field of constants of .
Let
be a differential field extension of $K$ . We say that
depends
rationally on arbitrary constants over $K$ if there exists a differential field
of $K$ such that
extension
and $R$ are free over $K$ and the equality
$ER=EC_{ER}$ holds.
Recent discussions on Painleve’s first transcendent motivate the following
[9],
[10], [11]).
(cf.
Let
$K$
$L$
$L$
$C_{L}$
$R$
$R$
$E$
$E$
Definition. A differential field extension
of $K$ is called a Painleve$K$
Umemura (PU-) extension of
if there exists a finite chain of differential field
$K:K=R_{0}
¥
subset
R_{1}
¥
subset
¥
cdots¥subset R_{m}$
extensions of
such that $R_{m}=R$ and
depends rationally on arbitrary constants for each . For instance, liouville
extensions, or more widely, //-extensions defined in [4] are PU-extensions.
$R$
$R_{i+1}/R_{i}$
$i$
The aim of this note is to prove the following.
Theorem. Suppose that a differential field extension $R$ of $K$ is contained
in a certain PU-extension of K. Then
is itself a PU-extension of $K$ .
$R$
2. Invariant subfields
be a field of characteristic 0 and be an indeterminate over . By
$F[[t]]$ we denote the ring of formal power series in
with coefficients from
$F[[t]]$
. Every ring homomorphism
of to
is always assumed to assign
the unity to the unity. Let
be a set of such isomorphisms. For a subfield
$¥subset E[[t]]$ , by
of
with
we denote the set of all in
that are
left invariant under every isomorphism in . This set is a subfield of E. (cf.
Kolchin [2, pp. 86-88].)
Let
$F$
$F$
$t$
$t$
$F$
$¥sigma$
$F$
$¥Sigma$
$E$
$F$
$¥Sigma E$
$E^{¥Sigma}$
$e$
$¥Sigma$
$E$
60
K. NISHIOKA
Lemma 1. Let
be a subfield
are linearly disjoint over
.
of
$E$
$E$
with LE
$F$
$¥subset E[[t]]$
.
Then
$F^{¥Sigma}$
and
$E^{¥Sigma}$
Proof. Suppose a finite number
of elements of
to be linearly dependent over . There exists a nontrivial linear combination of the
over $E$
which vanishes. Among such linear combinations pick up the shortest one,
say $¥sum e_{i}f_{i}=0$, where the
are nonzero elements of . We may assume
$e_{i}=1$ for some .
Applying
, through subtraction, $¥sum(e_{i}-¥sigma e_{i})f_{i}=0$ . By
our assumption on the sum, the coefficients of the linear combination must
be all zero. It thus follows every
lies in
. This is our assertion.
$f_{i}$
$F^{¥Sigma}$
$E$
$f_{i}$
$E$
$e_{i}$
$¥sigma¥in¥Sigma$
$i$
$E^{¥Sigma}$
$e_{i}$
Lemma 2. Let
be a subfield of F. Suppose $F=EF^{¥Sigma}$ . Then any inter$¥subset H[[t]]$ has the property
mediate field $H$ between
and
satisfying
$E$
$E$
$H=EH^{¥Sigma}$
$F$
$¥mathrm{E}¥mathrm{H}$
.
We know that
and $H$ are linearly disjoint over
. Hence
$H$
and
are also linearly disjoint over
since
. Let
be in $H$ . If
, then
. This contradicts the assumption $F=EF^{¥Sigma}$.
Thus
, therefore $H=EH^{¥Sigma}$.
Proof.
$F^{¥Sigma}$
$H^{¥Sigma}$
$EF^{¥Sigma}$
$EH^{¥Sigma}$
$h¥not¥in EH^{¥Sigma}$
$H^{¥Sigma}¥subset EH^{¥Sigma}¥subset H$
$h$
$h¥not¥in EF^{¥Sigma}$
$H¥subset EH^{¥Sigma}$
Lemma 3. Let I be a subfield of and , be intermediate fields between
$¥subset E[[t]]$ and
I and F. Suppose that
and are linearly disjoint over /,
¥
. Then every isomorphism in
can be extended to an isomorphism of
JE to JE[[t]] which is trivial on
and in this sense (JE) $¥Sigma=J$ holds.
$F$
$E$
$E$
$J$
$J$
$¥Sigma E$
$E^{ Sigma}=I$
$¥Sigma$
$J$
Proof.
Clearly
element of (JE) .
$¥Sigma$
To show the converse relation let
It has the expression
$J¥subset(JE)^{¥Sigma}$
.
$x=¥frac{¥sum a_{i}x_{i}}{¥sum b_{j}y_{j}}$
be an
$x$
,
where the and the are in , the and the are in , and
. We
may assume without losing generality that the
are linearly independent over
I therefore
and $b_{j}=y_{j}=1$ for some . Among such expressions pick one
such that the occurrence of nonzero
is the minimum in number. Applying
, we have
$b_{j}$
$a_{i}$
$E$
$x_{i}$
$y_{j}$
$¥sum b_{j}y_{j}¥neq 0$
$J$
$x_{i}$
$J$
$j$
$b_{j}$
$¥sigma¥in¥Sigma$
$x¥sum(a_{i}-¥sigma a_{i})x_{i}-¥sum(b_{j}-¥sigma b_{j})y_{j}=0$
The minimality implies
and then
of the . This completes the proof.
$b_{j}=¥sigma b_{j}$
$a_{i}=¥sigma a_{i}$
.
by the linear independence
$x_{i}$
The results apply to the proof of the following known fact: Let
be a
differential field extension of $K$ and $M$ be a differential field extension of
. Assume $M¥subset LC_{U}$ . Then $M=LC_{M}$ . In fact let $¥sigma=¥exp tD$ , where $D$
$L$
$L$
61
Pain/eve-Umemura Extensions
denotes the differentiation of $M$ , and
and Lemma 1 implies our assertion.
$¥Sigma=$
$¥{¥sigma¥}$
.
By Lemma 2
$(LC_{U})^{¥Sigma}=C_{U}$
3. Rational dependence
be a differential field of characteristic 0 and $R$ be a differential
field extension of $K$ . Denote by $¥Pi_{K}(R)$ the set of all intermediate differential
fields between $K$ and $R$ that depend rationally on arbitrary constants over
$K$ .
According to the last sentence in the previous section, the element
of
this set is characterized by the property that there exists a differential field
extension
of $K$ such that
and
are free over $K$ and the compositum
$EP$ is generated by constants over
. We shall here prove that this set has
the maximum element in the inclusion order and discuss its fundamental property. At the end of this section a proof of the Theorem will be given.
Let
$K$
$P$
$E$
$P$
$E$
$E$
$R$
Proposition 1. For $P¥in¥Pi_{K}(R)$ there is a
and
are free over $K$ and $EP$ $=EC_{EP}$ .
differential field
of
$E$
$K$
such that
$E$
By definition there is a differential field extension
of $K$ such
that
and
are free over $K$ and $FP/F$ is generated by constants. According
to the following lemma there is a differential field extension
of
such that
$FR$ and
are free and differentially isomorphic over . As
we may take
the image of .
Proof.
$F$
$P$
$F$
$P$
$Q$
$P$
$Q$
$E$
$F$
$S$
Lemma 4. There is a differential field extension
are free and differentially isomorphic over $K$ .
Proof.
$S$
of
$K$
such that
$R$
and
Let
denote the algebraic closure of $K$ in $R$ . Then $R/K$ is
regular. The quotient field
of the differential domain ¥
is
expressible as the compositum of two differential fields
and
.
$R$
Identifying the former with
and denoting the latter by , we have the
desired conclusion.
$L$
$¥langle R¥otimes_{L}R¥rangle$
$R otimes_{L}R$
$R¥otimes_{L}1$
$1¥otimes_{L}R$
$S$
Proposition 2.
The set
$¥Pi_{K}(R)$
has the maximum element.
Let
and
be in ¥
. Then there is a differential field
extension
of $K$ such that $R$ and
are free over $K$ and $EP/E$ is generated
by constants. Since $Q¥in¥Pi_{K}(ER)$ also holds there exists a differential field
extension of $K$ such that ER and
are free over $K$ and $FQ/F$ is generated
by constants. Hence $EFPQ/EF$ is generated by constants, therefore $PQ$
Proof.
$P$
$ Pi_{K}(R)$
$Q$
$E$
$E$
$F$
$F$
$¥in$
$¥Pi_{K}(R)$
.
In the sequel this maximum element of
Proposition 3.
Let
$E$
be a
$¥Pi_{K}(R)$
differential field
will be denoted
extension
of
$K$
$P_{K}(R)$
.
for which
$R$
K. NISHIOKA
62
and
are free over K.
$E$
for any
$¥Pi_{K}(R)$
Then
$EP$ $¥in¥Pi_{E}(ER)$
for any
$P¥in¥Pi_{K}(R)$
and
$ R¥cap Q¥in$
$Q¥in¥Pi_{E}(ER)$ .
Since $PIT_{K}(ER)$ , by Proposition 1, there exists
a differential field extension of $K$ such that ER and are free over $K$ and
$FP/F$ is generated by constants. This shows that $EFP/EF$ is generated by
constants, therefore $EP$ $¥in¥Pi_{E}(ER)$ . Assume next $Q¥in¥Pi_{E}(ER)$ and let $P=Q¥cap R$ .
are free over
of
such that ER and
Then we have a differential field
the proof.
completes
This
by
constants.
$FP/F$
$FQ/F$
generated
is
,
, hence
and
Proof. Assume
$P¥in¥Pi_{K}(R)$
.
$F$
$F$
$F$
$E$
$F$
$E$
Proposition 4.
into U. Then
$¥mathrm{a}¥mathrm{P}$
Let
$P¥in¥Pi_{K}(R)$
$¥in¥Pi_{¥sigma K}(¥sigma R)$
and
$¥sigma$
be a
differential
isomorphism
of
$R$
.
are
of $K$ such that $R$ and
Take a differential field extension
free over $K$ and $EP/E$ is generated by constants. It is known that there
which is a differential isomorphism of ER into
of
exists an extension
$=¥sigma K$
¥
$U$ .
. Since $EP/E$ is generated
are free over
and
Clearly
¥
¥
¥
is generated by constants as well. This
by constants it follows that
completes the proof.
Proof.
$E$
$E$
$¥sigma$
$¥tau$
$¥mathrm{x}¥mathrm{R}$
$= sigma R$
$¥tau K$
$¥tau E$
$ tau(E) sigma(P)/ tau E$
Proposition 5. Let
over K. Then
from
$R$
$E$
$R$
be a
and
extension of $K$ which is
are linearly disjoint over $P_{K}(R)$ .
differential field
$P_{E}(ER)$
free
In
particular
$P_{E}(ER)=EP_{K}(R)$ , $P_{K}(R)=R¥cap P_{E}(ER)$
.
of $R$ are linearly
Now suppose that nonzero elements
and
denote the algebraic closure of $K$ in
dependent over $P_{E}(ER)$ . Let
is
free
over
which
from
of
be the set of all differential field extension
. For each $ A¥in¥Phi$ by $n(A)$ we denote the dimension of the $P_{A}(AR)$ -vector
. There is an $ F¥in¥Phi$ with $n=n(F)$ being the
space generated by the
constitute the base
minimum. We may suppose that the elements
element
Any
$a=a_{i}$ has the
.
other
by
the
generated
space
for the vector
Proof.
$(a_{i})_{1¥leq i¥leq m}$
$R$
$L$
$R$
$L$
$¥Phi$
$L$
$a_{i}$
$(a_{i})_{1¥leq i¥leq n}$
$a_{i}$
expression
$a=¥sum_{i=1}^{n}x_{i}a_{i}$
,
$x_{i}¥in P_{F}(FR)$
.
denote the algebraic closure of in . The quotient field
is considered as a differential subfield of $U$ , and written $F_{1}F_{2}R$ , where
. The mappings
stands for
and
stands for
defined by
Let
$M$
$L$
$F$
$¥langle F¥otimes_{M}F¥otimes_{M}$
$ MR¥rangle$
$F_{1}$
$F¥otimes_{M}1$
$1¥otimes_{M}F$
$F_{2}$
$¥sigma_{1}(x)=x¥otimes 1$
are differential isomorphisms over
$M$ .
,
$¥sigma_{j}:F¥rightarrow F_{j}$
$¥sigma_{2}(x)=1¥otimes x$
These are also regarded as differential
63
Pain/eve-Umemura Extensions
isomorphisms of $FR$ into
through subtracting, we have
$F_{l}F_{¥mathit{2}}R$
over
Applying
$MR$ .
$¥sum_{i=1}^{n}(¥sigma_{1}x_{i}-¥sigma_{2}x_{i})a_{i}=0$
Since the elements
$x_{i}$
are all contained in
$P_{F}(FR)$
$¥sigma_{j}x_{i}¥in P_{F_{j}}(F_{j}R)$
$¥sigma_{j}$
to the above equality,
.
, by Proposition 4, it follows
,
whence
$¥sigma_{1}x_{i}-¥sigma_{2}x_{i}¥in P_{F_{1}F_{2}}(F_{1}F_{2}R)$
.
are linearly independent over
By our assumption we see the
, whence $x_{i}¥in MR$ , and $¥in P_{M}(MR)$ . (See
. This implies
Sublemma (25.5) in [10].) We must show $x_{i}¥in P_{K}(R)$ . To this end let $N$ be
including $M$ . Since $N$ and $R$ are linearly disjoint
a Galois extension of
of $N/L$ is regarded as a differential
over $L$ , every automorphism
automorphism of $NR/R$ . By Proposition 4, we see $¥sigma x_{i}¥in P_{N}(NR)$ . From the
equality
$(a_{i})_{1¥leq i¥leq n}$
$P_{F_{1}F_{2}}(F_{1}F_{2}R)$
$¥sigma_{1}x_{i}=¥sigma_{2}x_{i}$
$L$
$¥sigma$
$¥sum_{i=1}^{n}(x_{i}-¥sigma x_{i})a_{i}=0$
we find that
$x_{i}=¥sigma x_{i}$
for each , hence
$¥sigma$
$x_{i}¥in R$
, using Lemma 3.
This completes
the proof.
$K$
Proposition
and for each
$¥epsilon$
.
Let
$S$
be a
differential field
extension
,
.
$R_{i}=P_{R_{¥mathrm{i}-1}}(R)$
Then
for any
$i$
,
of
R.
Set
$R_{0}=S_{0}=$
$i¥geq 1$
$R$
and
$S_{i}$
$S_{i}=P_{S_{¥mathrm{i}-1}}(S)$
are linearly disjoint over
$R_{i}$
.
The proof proceeds by induction on . In case $i=0$, it is trivial.
Let us first show the assertion in case $i=1$ . Let $M$ denote the algebraic
and take
to be a differential field extension of $M$ such
closure of $K$ in
that and are linearly disjoint over $M$ and $ES_{l}=EC_{ES}$ . Then $ER_{l}=EC_{ER}$ .
are linearly disjoint over
, it follows that ER and
Since ER and
$ES_{l}=ER_{l}C_{ES}$ are linearly disjoint over
. By the way, $MR$ and
are
are linearly disjoint over $MR_{1}$ .
linearly disjoint over $M$ , whence $MK$ and
$R$
and $MR_{1}$ are linearly disjoint over
includes
, since
We also know that
$R$
$K$
in . By this we conclude that
are
and
the algebraic closure of
. Now assume is positive and the assertion in case
linearly disjoint over
$i-1$ holds validly. Since $R$ and
are linearly disjoint over
, by
$R$
are linearly disjoint over
Proposition 5,
. By
and
Proof.
$i$
$E$
$S$
$S$
$E$
$C_{ER}$
$C_{ES}$
$E$
$ER_{1}$
$ES_{1}$
$R_{1}$
$R_{1}$
$R$
$i$
$R_{1}$
$S_{i-1}$
$P_{S_{¥mathrm{i}-1}}(S_{i-1}R)$
$ES_{1}$
$R_{i-1}$
$R_{i}=P_{R_{¥mathrm{i}-1}}(R)$
64
K. NISHIOKA
the fact obtained just above, noting
.
are linearly disjoint over
$P_{S_{¥mathrm{i}-1}}(S_{i1}¥_ R)$
, we have $S_{i-1}R$ and
These verify the assertion in case .
$S_{i1}¥_¥subset S_{i1}¥_ R¥subset S$
of $K$
By hypothesis there exists a PU-extension
As above we can construct the differential subfields of $R$ and :
Proof of Theorem.
containing
$R$
.
$S_{i}$
$i$
$S$
$S$
$R_{i}=P_{R_{¥mathrm{i}-1}}(R)$
,
$S_{i}=P_{S_{¥mathrm{i}-1}}(S)(i¥geq 1);R_{0}=S_{0}=K$
.
According to the above proposition $R_{i}=R¥cap S_{i}$ . Since $S=S_{m}$ for some integer
$m$ it follows that $R_{m}=R¥cap S$ , which shows that
is a PU-extension of $K$ .
$R$
References
Buium, A., Differential function fields and moduli of algebraic varieties, Lecture Notes in
Math., 1226, Springer, Berlin, 1986.
[2] Kolchin, E. R., Differential algebra and algebraic groups, Academic Press, New York, 1973.
[3] Matsuda, M., First order algebraic differential equations, Lecture Notes in Math., 804,
Springer, Berlin, 1980.
[4] Nishioka, K., A class of transcendental functions containing elementary and elliptic ones,
Osaka J. Math., 22 (1985), 743-753.
[5]?, A note on the transcendency of Panileve’s first transcendent, Nagoya Math. J.,
109 (1988), 63-67.
[6]?, General solutions depending algebraically on arbitrary constants, Nagoya Math.
J., 113 (1989), 1-6.
[7]?, Differential algebraic function fields depending rationally on arbitrary constants,
Nagoya Math. J., 113 (1989), 173-179.
[8] Painleve, P., Leqons de Stockholm, OEuvres de P. Painleve I, 199-818, Editions du C.N.R.S.,
Paris, 1972.
[9] Umemura, H., On the irreducibility of the first differential equation of Painleve, Algebraic
Geometry and Commutative Algebra in honor of Masayosi NAGATA, Kinokuniya, Tokyo,
1987, 771-789.
[10]?, Second proof of the irreducibility of the first differential equation of Painleve,
Nagoya Math. J., 117 (1990), 125-171.
[11]?, Birational automorphism groups and differential equations, Nagoya Math. J., 119
(1990), 1-80.
[1]
nuna adresso:
Faculty of Environmental Information
5322 Endoh, Fujisawa 252
Japan
(Ricevita la 11-an de oktobro, 1991)