中文名: 線性微分方程的伽羅瓦理論
原名: Galois Theory of Linear Differential Equations
作者: (荷)Marius van der Put, Michael F.Singer
資源格式: PDF
版本: 清晰版
出版社: Springer
書號: 3540442286
發行時間: 2003年
地區: 美國
語言: 英文
簡介:

內容簡介：
本書專門論述線性微分方程的伽羅瓦理論，涉及諸多方面：代數理論(尤其足微分伽羅瓦理論)、形式理論、分類、有限項可解性判定算法、單值性、希爾伯特21問題、漸近性和可求和性、反問題以及具正特征值的線性微分方程。附錄是本書所用到的代數幾何、線性代數群、層及Tannakian范疇中的一些概念。.
本書將成為該領域所有數學家和研究生的標准參考書。
內容截圖：

目錄:
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 g) = k[a] 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
3.2 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
4.3 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
5.2 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
6.4 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,A01/k
7.5 The Equation (8 - q)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(S1, STS)
9.4 Explicit l-cocycles for H](Sl, STS)
9.5 H1(S1, STS) as an Algebraic Variety
10 Universal Picard-Vessiot Rings and Galois Groups
10.1 Introduction
10.2 Regular Singular Differential Equations
10.3 Formal Differential Equations
10.4 Meromorphic Differential Equations
11 Inverse Problems
11.1 Introduction
11.2 The Inverse Problem for C((z))
11.3 Some Topics on Linear Algebraic Groups
11.4 The Local Theorem
11.5 The Global Theorem
11.6 More on Abhyankar's Conjecture
11.7 The Constructive Inverse Problem
12 Moduli for Singular Differential Equations
12.1 Introduction
12.2 The Moduli Functor
12.3 An Example
12.4 Unramified Irregular Singularities
12.5 The Ramified Case
12.6 The Meromorphic Classification
13 Positive Characteristic
13.1 Classification of Differential Modules
13.2 Algorithmic Aspects
13.3 Iterative Differential Modules
Appendices
A Algebraic Geometry
A.1 Affine Varieties
A.2 Linear Algebraic Groups
B Tannakian Categories
B.1 Galois Categories
B.2 Affine Group Schemes
B.3 Tannakian Categories
C Sheaves and Cohomology
C.1 Sheaves: Definition and Examples
C.2 Cohomology of Sheaves
D Partial Differential Equations
D.1 The Ring of Partial Differential Operators
D.2 Picard-Vessiot Theory and Some Remarks
Bibliography
List of Notation
Index