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＜電子ブック＞
Galois Theory of Linear Differential Equations / by Marius van der Put, Michael F. Singer
(Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics. ISSN:21969701 ; 328)

版	1st ed. 2003.
出版者	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
出版年	2003
本文言語	英語
大きさ	XVII, 438 p : online resource
著者標目	*van der Put, Marius author Singer, Michael F author SpringerLink (Online service)
件　名	LCSH:Mathematical analysis LCSH:Number theory LCSH:Commutative algebra LCSH:Commutative rings LCSH:Algebraic geometry LCSH:Differential equations FREE:Analysis FREE:Number Theory FREE:Commutative Rings and Algebras FREE:Algebraic Geometry FREE:Differential Equations
一般注記	Algebraic Theory -- 1 Picard-Vessiot Theory -- 2 Differential Operators and Differential Modules -- 3 Formal Local Theory -- 4 Algorithmic Considerations -- Analytic Theory -- 5 Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group -- 6 Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem -- 7 Exact Asymptotics -- 8 Stokes Phenomenon and Differential Galois Groups -- 9 Stokes Matrices and Meromorphic Classification -- 10 Universal Picard-Vessiot Rings and Galois Groups -- 11 Inverse Problems -- 12 Moduli for Singular Differential Equations -- 13 Positive Characteristic -- 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 Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students HTTP:URL=https://doi.org/10.1007/978-3-642-55750-7

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