Ecuaciones Diferenciales - Springer-Verlag - Galois Theory of Linear Differential Equations.pdf

Galois Theory of Linear
Differential Equations
Marius van der Put
Department of Mathematics
University of Groningen
800
9700 AV Groningen
herlands

Department of Mathematics
North Carolina State University
Box 8205
Raleigh, . 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 19th Century and was put on a firm footing
by Kolchin in the middle of the 20th 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)=
∂(a)b + a∂(b) for all a, b ∈ k (we will usually denote ∂a for a ∈ 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 × n
matrix with entries in k although sometimes we shall also consider scalar li