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Galois theory of linear differential equations. (English) Zbl 1036.12008
Grundlehren der Mathematischen Wissenschaften 328. Berlin: Springer (ISBN 3-540-44228-6/hbk). xvii, 438 p. (2003).
Let $$F$$ be a differential field with derivation $$D$$ and field of constants $$C$$, and let $$A$$ be an $$n \times n$$ matrix over $$F$$. This book is concerned with the fields generated by complete sets of solutions to the linear differential equation (in matrix form) $$Y^\prime =AY$$ (differential Galois extensions), their groups of differential automorphisms fixing $$F$$ (differential Galois groups), and the relation between their differential subfields and subgroups of the differential Galois group (differential Galois theory). This book, which is organized like a textbook with exercises, and intended by the authors to be accessible to a first-year graduate student, is a definitive account of the Galois theory of linear differential equations. In addition to its four chapters on algebraic theory and nine chapters on analytic theory, it has four appendices totaling nearly one hundred pages and a fourteen page bibliography of 306 items.
The first chapter, Picard-Vessiot theory, covers the basic construction of splitting fields, in characteristic zero and with algebraically closed field of constants, of linear differential equations and the algebraic group structure on their groups of differential automorphisms. Chapter 2 reformulates this in the language of differential modules and Tannakian categories. Chapter 3 considers the special case where $$F$$ is the field of formal power series over the complexes. Chapter 4 considers explicit determination of Galois groups in various contexts; this is the “direct problem” of “given the equation, determine the group”.
The analytic theory begins in the fifth chapter. Chapters 5 and 6 are devoted to the case where $$F$$ is the rational functions in one variable over the complexes, and with the potential Galois group given in various ways, to determine if an equation with that group exists, the so-called “inverse problem” of “given the group, find an equation realizing the group”. In the situation covered in these chapters, this is known as the Riemann-Hilbert problem. Chapters 7, 8, and 9 deal with the case that $$F$$ is the (quotient field of the ring of) convergent power series over the complexes. Chapter 10 calculates a differential Galois closure for the case where $$F$$ is either formal or convergent power series over the complexes. Chapter 11 looks at the inverse problem for arbitrary $$F$$. Chapter 12 is concerned with moduli spaces for (singular) differential equations over convergent power series, and in chapter 13 the Galois theory for the case where $$F$$ is of positive characteristic is presented. The four appendices cover algebraic geometry (including linear algebraic groups), Tannakian categories, sheaves and cohomology, and the extension to the case of partial differential equations (fields with a set of commuting derivations).
In sum, the book is a modern, comprehensive, and mostly self-contained account of the Galois theory of linear differential equations. It should be considered the standard reference in the field.

##### MSC:
 12H05 Differential algebra 12-02 Research exposition (monographs, survey articles) pertaining to field theory 34Mxx Ordinary differential equations in the complex domain 34A30 Linear ordinary differential equations and systems, general 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations