Galois Theory of Linear Differential Equations

by Marius van der Put, Michael F. Singer

Overview

From the reviews: "This is a great book, which will hopefully become a classic in the subject of differential Galois theory. […] the specialist, as well as the novice, have long been missing an introductory book covering also specific and advanced research topics. This gap is filled by the volume under review, and more than satisfactorily." Mathematical Reviews

Product Details

ISBN-13:	9783642629167
Publisher:	Springer Berlin Heidelberg
Publication date:	10/23/2012
Series:	Grundlehren der mathematischen Wissenschaften , #328
Edition description:	2003
Pages:	438
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Algebraic Theory.- 1 Picard-Vessiot Theory.- 1.1 Differential Rings and Fields.- 1.2 Linear Differential Equations.- 1.3 Picard-Vessiot Extensions.- 1.4 The Differential Galois Group.- 1.5 Liouvillian Extensions.- 2 Differential Operators and Differential Modules.- 2.1 The Ring D= k[?] of Differential Operators.- 2.2 Constructions with Differential Modules.- 2.3 Constructions with Differential Operators.- 2.4 Differential Modules and Representations.- 3 Formal Local Theory.- 3.1 Formal Classification of Differential Equations.- 3.1.1 Regular Singular Equations.- 3.1.2 Irregular Singular Equations.- 3.2 The Universal Picard-Vessiot Ring of K.- 3.3 Newton Polygons.- 4 Algorithmic Considerations.- 4.1 Rational and Exponential Solutions.- 4.2 Factoring Linear Operators.- 4.2.1 Beke’s Algorithm.- 4.2.2 Eigenring and Factorizations.- 4.3 Liouvillian Solutions.- 4.3.1 Group Theory.- 4.3.2 Liouvillian Solutions for a Differential Module.- 4.3.3 Liouvillian Solutions for a Differential Operator.- 4.3.4 Second Order Equations.- 4.3.5 Third Order Equations.- 4.4 Finite Differential Galois groups.- 4.4.1 Generalities on Scalar Fuchsian Equations.- 4.4.2 Restrictions on the Exponents.- 4.4.3 Representations of Finite Groups.- 4.4.4 A Calculation of the Accessory Parameter.- 4.4.5 Examples.- Analytic Theory.- 5 Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group.- 5.1 Monodromy of a Differential Equation.- 5.1.1 Local Theory of Regular Singular Equations.- 5.1.2 Regular Singular Equations on P1.- 5.2 A Solution of the Inverse Problem.- 5.3 The Riemann-Hilbert Problem.- 6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem.- 6.1 Differentials and Connections.- 6.2 Vector Bundles and Connections.- 6.3 Fuchsian Equations.- 6.3.1 From Scalar Fuchsian to Matrix Fuchsian.- 6.3.2 A Criterion for a Scalar Fuchsian Equation.- 6.4 The Riemann-Hilbert Problem, Weak Form.- 6.5 Irreducible Connections.- 6.6 Counting Fuchsian Equations.- 7 Exact Asymptotics.- 7.1 Introduction and Notation.- 7.2 The Main Asymptotic Existence Theorem.- 7.3 The Inhomogeneous Equation of Order One.- 7.4 The Sheaves A, A0, A1/k, $$ A_{{1/k}}sub0 $$.- 7.5 The Equation $$ (\delta - q)\hat{f} = g $$ Revisited.- 7.6 The Laplace and Borel Transforms.- 7.7 The k-Summation Theorem.- 7.8 The Multisummation Theorem.- 8 Stokes Phenomenon and Differential Galois Groups.- 8.1 Introduction.- 8.2 The Additive Stokes Phenomenon.- 8.3 Construction of the Stokes Matrices.- 9 Stokes Matrices and Meromorphic Classification.- 9.1 Introduction.- 9.2 The Category Gr2.- 9.3 The Cohomology Set H1(S1STS).- 9.4 Explicit 1-cocycles for H1(S1, STS).- 9.4.1 One Level k.- 9.4.2 Two Levels k1 Meromorphic Classification.- 13 Positive Characteristic.- 13.1 Classification of Differential Modules.- 13.2 Algorithmic Aspects.- 13.2.1 The Equation b(p-1)+bp = a.- 13.2.2 The p-Curvature and Its Minimal Polynomial.- 13.2.3 Example: Operators of Order Two.- 13.3 Iterative Differential Modules.- 13.3.1 Picard-Vessiot Theory and Some Examples.- 13.3.2 Global Iterative Differential Equations.- 13.3.3 p-Adic Differential Equations.- Appendices.- A Algebraic Geometry.- A.1 Affine Varieties.- A. 1.1 Basic Definitions and Results.- A. 1.2 Products of Affine Varieties over k.- A. 1.3 Dimension of an Affine Variety.- A. 1.4 Tangent Spaces, Smooth Points, and Singular Points.- A.2 Linear Algebraic Groups.- A.2.1 Basic Definitions and Results.- A.2.2 The Lie Algebra of a Linear Algebraic Group.- A.2.3 Torsors.- B Tannakian Categories.- B.1 Galois Categories.- B.2 Affine Group Schemes.- B.3 Tannakian Categories.- C Sheaves and Cohomology.- C.l Sheaves: Definition and Examples.- C.1.1 Germs and Stalks.- C.1.2 Sheaves of Groups and Rings.- C. 1.3 From Presheaf to Sheaf.- C. 1.4 Moving Sheaves.- C.l.5 Complexes and Exact Sequences.- C.2 Cohomology of Sheaves.- C.2.1 The Idea and the Formahsm.- C.2.2 Construction of the Cohomology Groups.- C.2.3 More Results and Examples.- D Partial Differential Equations.- D. 1 The Ring of Partial Differential Operators.- D.2 Picard-Vessiot Theory and Some Remarks.- List of Notation.

Editorial Reviews

From the reviews:

"This book offers a detailed and thorough introduction to Galois Theory for differential fields and its applications to linear differential equations." (A. Cap, Monatshefte für Mathematik, Vol. 145 (4), 2005)

"At last, a thorough exposition, including most of the facets it presents nowadays, of this beautiful analogue of the Galois theory of field extensions … . The book is in fact already becoming a standard reference, not only for differential Galois theory proper, but also for the many areas which have accompanied its recent growth … . Any … student working in these areas will benefit from this book, which clearly belongs to all mathematical libraries." (D.Bertrand, Jahresberichte der Deutschen Mathematiker Vereinigung, Vol. 106 (4), 2004)

"This book, which is organized like a textbook with exercises … is a definitive account of the Galois theory of linear differential equations. … 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." (Andy R. Magid, Zentralblatt MATH, Vol. 1036 (11), 2004)

"This is a great book, which will hopefully become a classic in the subject of differential Galois theory. … The book is carefully written: the authors have made a great effort to state the results in a language as common as possible, making use of specialized terminology only when strictly necessary. The material is introduced step by step and with a clear distinction of what is ‘common knowledge’ … and what is ‘specifically required’… ." (Pedro Fortuny Ayuso, Mathematical Reviews, 2004 c)

"This book is an introduction to the algebraic, algorithmic and analytic aspects of Galois theory of homogenous linear differential equations. … This book presents many of the recent results and approaches to this classical field. … This book is comprehensively written and thorough … ." (Ernie Kalnin, New Zealand Mathematical Society Newsletter, Issue 90, April, 2004)

"The book is aimed to be an introduction into the theory and the authors attempted to make the subject accessible to anyone with a background in algebra and analysis … . It contains many examples and exercises. Without any doubt, the book is devoted to a very interesting and useful topic and can be highly recommended." (J.Synnatzschke, Zeitschrift für Analysis und ihre Anwendungen – ZAA, Vol. 22 (3), 2003)

From the Publisher