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 к together with a derivation д : к —» к, that is, an additive map that satisfies d(ab) = d(a)b + ad(b) for all a,b E к (we will usually denote da for a E к as a'). Except for Chapter 13, all differential fields will be of characteristic zero. A linear differential equation is an equation of the form dY = AY where A is an n x n matrix with entries in к although sometimes we shall also consider scalar linear differential equations L(y) = dny + an-\dn~1y + • • • + a^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
