Singer M., Put M. van der - Galois Theory of Linear Differential Equations.pdf

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Galois Theory of Linear
Differential Equations
Marius van der Put
Department of Mathematics
University of Groningen
P.O.Box 800
9700 AV Groningen
The Netherlands
MichaelF.Singer
Department of Mathematics
North Carolina State University
Box 8205
Raleigh, N.C. 27695-8205
USA
July 2002
ii
Preface
This book is an introduction to the algebraic, algorithmic and analytic aspects
of the Galois theory of homogeneous linear differential equations. Although the
Galois theory has its origins in the 19 th Century and was put on a firm footing
by Kolchin in the middle of the 20 th Century, it has experienced a burst of
activity in the last 30 years. In this book we present many of the recent results
and new approaches to this classical field. We have attempted to make this
subject accessible to anyone with a background in algebra and analysis at the
level of a first year graduate student. Our hope is that this book will prepare
and entice the reader to delve further.
In this preface we will describe the contents of this book. Various researchers
are responsible for the results described here. We will not attempt to give
proper attributions here but refer the reader to each of the individual chapters
for appropriate bibliographic references.
The Galois theory of linear differential equations (which we shall refer to simply
as differential Galois theory) is the analogue for linear differential equations of
the classical Galois theory for polynomial equations. The natural analogue of a
field in our context is the notion of a differential field. This is a field k together
with a derivation ∂ : k
→
k , that is, an additive map that satisfies ∂ ( ab )=
k as a ). Except
for Chapter 13, all differential fields will be of characteristic zero. A linear
differential equation is an equation of the form ∂Y = AY where A is an n
∈
k (we will usually denote ∂a for a
∈
n
matrix with entries in k although sometimes we shall also consider scalar linear
differential equations L ( y )= ∂ n y + a n− 1 ∂ n− 1 y +
×
+ a 0 y = 0 (these objects
are in general equivalent, as we show in Chapter 2). One has the notion of a
“splitting field”, the Picard-Vessiot extension, which contains “all” solutions of
L ( y ) = 0 and in this case has the additional structure of being a differential field.
The differential Galois group is the group of field automorphisms of the Picard-
Vessiot field fixing the base field and commuting with the derivation. Although
defined abstractly, this group can be easily represented as a group of matrices
and has the structure of a linear algebraic group, that is, it is a group of invertible
matrices defined by the vanishing of a set of polynomials on the entries of these
matrices. There is a Galois correspondence identifying differential subfields with
···
iii
∂ ( a ) b + a∂ ( b ) for all a, b
iv
PREFACE
linear algebraic subgroups of the Galois group. Corresponding to the notion of
solvability by radicals for polynomial equations is the notion of solvability in
terms of integrals, exponentials and algebraics, that is, solvable in terms of
liouvillian functions and one can characterize this in terms of the differential
Galois group as well.
Chapter 1 presents these basic facts. The main tools come from the elementary
algebraic geometry of varieties over fields that are not necessarily algebraically
closed and the theory of linear algebraic groups. In Appendix A we develop the
results necessary for the Picard-Vessiot theory.
k .
For any differential equation ∂Y = AY over k one can define a corresponding
k [ ∂ ]-module in much the same way that one can associate an F [ X ]-module to
any linear transformation of a vector space over a field F .If ∂Y = A 1 Y and
∂Y = A 2 Y are differential equations over k and M 1 and M 2 are their asso-
ciated k [ ∂ ]-modules, then M 1
∈
M 2 as k [ ∂ ]-modules if and only if here is an
invertible matrix Z with entries in k such that Z − 1 ( ∂
−
A 1 ) Z = ∂
−
A 2 ,that
Z − 1 Z . We say two equations are equivalent over k if such
a relation holds. We show equivalent equations have the same Galois groups
and so can define the Galois group of a k [ ∂ ]-module. This chapter is devoted
to further studying the elementary properties of modules over k [ ∂ ]andtheir
relationship to linear differential equations. Further the Tannakian equivalence
between differential modules and representations of the differential Galois group
is presented.
−
dz . The main result is to classify k [ ∂ ]-
modules over this ring or, equivalently, show that any differential equation ∂Y =
AY can be put in a normal form over an algebraic extension of k (an analogue of
the Jordan Normal Form of complex matrices). In particular, we show that any
equation ∂Y = AY is equivalent (over a field of the form C (( t )) ,t m = z for some
integer m> 0) to an equation ∂Y = BY where B is a block diagonal matrix
where each block B i is of the form B i = q i I + C i where where q i ∈
d
t − 1 C [ t − 1 ]and
C i is a constant matrix. We give a proof (and formal meaning) of the classical
fact that any such equation has a solution matrix of the form Z = Hz L e Q ,
where H is an invertible matrix with entries in C (( t )), L is a constant matrix
(i.e. with coecients in C ), where z L means e log( z ) L , Q is a diagonal matrix
whose entries are polynomials in t − 1 without constant term. A differential
equation of this type is called quasi-split (because of its block form over a finite
extension of C (( z )) ). Using this, we are able to explicitly give a universal
Picard-Vessiot extension containing solutions for all such equations. We also
show that the Galois group of the above equation ∂Y = AY over C (( z )) is
In Chapter 2, we introduce the ring k [ ∂ ] of differential operators over a differ-
ential field k , that is, the (in general, noncommutative) ring of polynomials in
the symbol ∂ where multiplication is defined by ∂a = a + a∂ for all a
is A 2 = Z − 1 A 1 Z
In Chapter 3, we study differential equations over the field of fractions k =
C (( z )) of the ring of formal power series C [[ z ]] over the field of complex numbers,
provided with the usual differentiation
134565442.001.png
v
the smallest linear algebraic group containing a certain commutative group of
diagonalizable matrices ( the exponential torus ) and one more element ( the formal
monodromy ) and these can be explicitly calculated from its normal form. In this
chapter we also begin the study of differential equations over C (
{
z
}
), the field
of fractions of the ring of convergent power series C
{
z
}
.f A has entries in
), we show that the equation ∂Y = AY is equivalent over C (( z )) to a
unique (up to equivalence over C (
{
z
}
)that
is quasi-split . This latter fact is key to understanding the analytic behavior of
solutions of these equations and will be used repeatedly in succeeding chapters.
In Chapter 2 and 3, we also use the language of Tannakian categories to describe
some of these results. This theory is explained in Appendix B. This appendix
also contains a proof of the general result that the category of k [ ∂ ]-modules
for a differential field k forms a Tannakian category and how one can deduce
from this the fact that the Galois groups of the associated equations are linear
algebraic groups. In general, we shall use Tannakian categories throughout the
book to deduce facts about categories of special k [ ∂ ]-modules, i.e., deduce facts
about the Galois groups of restricted classes of differential equations.
{
z
}
)) equation with entries in C (
{
z
}
u of L (over k or over an algebraic extension
of k ) corresponds to a special solution f of L ( f ) = 0, which can be rational,
exponential or liouvillian. Some of the ideas involved here are already present
in Beke’s classical work on factoring differential equations.
The “inverse” problem, namely to construct a differential equation over k with
a prescribed differential Galois group G and action of G on the solution space
is treated for a connected linear algebraic group in Chapter 11. In the opposite
case that G is a finite group (and with base field Q ( z )) an effective algorithm
is presented together with examples for equations of order 2 and 3. We note
that some of the algorithms presented in this chapter are ecient and others
are only the theoretical basis for an ecient algorithm.
−
Starting with Chapter 5, we turn to questions that are, in general, of a more
analytic nature. Let ∂Y = AY be a differential equation where A has en-
tries in C ( z ), where C is the field of complex numbers and ∂z =1. Apoint
c
, standard existence theorems imply that there exists
an invertible matrix Z of functions, analytic in a neighbourhood of p ,such
that ∂Z = AZ . Furthermore, one can analytically continue such a matrix of
the singular points
}
C (
In Chapter 4, we consider the “direct” problem, which is to calculate explicitly
for a given differential equation or differential module its Picard-Vessiot ring
and its differential Galois group. A complete answer for a given differential
equation should, in principal, provide all the algebraic information about the
differential equation. Of course this can only be achieved for special base fields
k ,suchas Q ( z ) ,∂z =1(where Q is the algebraic closure of the field of rational
numbers). The direct problem requires factoring many differential operators L
over k . A right hand factor ∂
C is said to be a singular point of the equation ∂Y = AY if some en-
try of A is not analytic at c (this notion can be extended to the point at
infinity on the Riemann sphere P as well). At any point p on the manifold
P
∈
\{
134565442.002.png 134565442.003.png

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