《線性微分方程的伽羅瓦理論 | 影印版》是科學出版社出版的圖書，作者是(荷)Marius van der Put, Michael F.Singer。
基本介紹
- 中文名：線性微分方程的伽羅瓦理論 | 影印版
- 作者：(荷)Marius van der Put, Michael F.Singer
- 出版社：科學出版社
- ISBN：9787030183019
內容簡介,圖書目錄,
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(尤其是微分伽羅瓦理論)、形式理論、分類、有限項可解性判定算法、單值性、希爾伯特211問題、漸近性和可求和性、反問題以及具正特徵值的線性微分方程。附錄是本書所用到的代數幾何、線性代數群、層及Tannakian範疇中的一些概念。
本書將成為該領域所有數學家和研究生的標準參考書。
圖書目錄
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 D=k[e] 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, A01/k
7.5 The Equation (δ - 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 HI(s1, STS)
9.4 Explicit 1-cocycles for HI(s1, STS)
9.4.1 One Level k
9.4.2 Two Levels k19.4.3 The General Case
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
12.3.1 Construction of the Moduli Space
12.3.2 Comparison with the Meromorphic Classification
12.3.3 Invariant Line Bundles
12.3.4 The Differential Galois Group
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
13.2.1 The Equation b(p-1) q- 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 Sheavesand Cohomology
C.1 Sheaves: Definition and Examples
C.I.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.1.5 Complexes and Exact Sequences
C.2 Cohomology of Sheaves
C.2.1 The Idea and the Formalism
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
Bibliography
List of Notation
Index
