Galois theory of linear differential equations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2003]
|
| Series: | Die Grundlehren der mathematischen Wissenschaften
328 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 438 Seiten Illustrationen |
| ISBN: | 9783540442288 9783642629167 3540442286 |
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MARC
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| 100 | 1 | |a Put, Marius van der |d 1941- |0 (DE-588)115533249 |4 aut | |
| 245 | 1 | 0 | |a Galois theory of linear differential equations |c Marius van der Put ; Michael F. Singer |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2003] | |
| 264 | 4 | |c © 2003 | |
| 300 | |a xvii, 438 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Die Grundlehren der mathematischen Wissenschaften |v 328 | |
| 650 | 7 | |a Differentiaalvergelijkingen |2 gtt | |
| 650 | 4 | |a Galois, Théorie de | |
| 650 | 7 | |a Galois-theorie |2 gtt | |
| 650 | 7 | |a Théorie de Galois |2 rasuqam | |
| 650 | 7 | |a Théorie différentielle de Galois |2 rasuqam | |
| 650 | 7 | |a Équation différentielle linéaire |2 rasuqam | |
| 650 | 4 | |a Équations différentielles linéaires | |
| 650 | 4 | |a Differential equations, Linear | |
| 650 | 4 | |a Galois theory | |
| 650 | 0 | 7 | |a Galois-Theorie |0 (DE-588)4155901-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Differentialgleichung |0 (DE-588)4206889-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lineare Differentialgleichung |0 (DE-588)4206889-7 |D s |
| 689 | 0 | 1 | |a Galois-Theorie |0 (DE-588)4155901-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Singer, Michael F. |d 1950- |0 (DE-588)115533265 |4 aut | |
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Record in the Search Index
| _version_ | 1804129649953865728 |
|---|---|
| adam_text | Table of Contents
Algebraic Theory 1
1 Picard Vessiot Theory 3
1.1 Differential Rings and Fields 3
1.2 Linear Differential Equations 6
1.3 Picard Vessiot Extensions 12
1.4 The Differential Galois Group 18
1.5 Liouvillian Extensions 33
2 Differential Operators and Differential Modules 37
2.1 The Ring D = k[d] of Differential Operators 37
2.2 Constructions with Differential Modules 42
2.3 Constructions with Differential Operators 47
2.4 Differential Modules and Representations 53
3 Formal Local Theory 59
3.1 Formal Classification of Differential Equations 59
3.1.1 Regular Singular Equations 63
3.1.2 Irregular Singular Equations 68
3.2 The Universal Picard Vessiot Ring of K 71
3.3 Newton Polygons 86
4 Algorithmic Considerations 99
4.1 Rational and Exponential Solutions 100
4.2 Factoring Linear Operators 110
4.2.1 Beke s Algorithm Ill
4.2.2 Eigenring and Factorizations 114
XIV Table of Contents
4.3 Liouvillian Solutions 116
4.3.1 Group Theory 117
4.3.2 Liouvillian Solutions for a Differential Module 118
4.3.3 Liouvillian Solutions for a Differential Operator 121
4.3.4 Second Order Equations 125
4.3.5 Third Order Equations 128
4.4 Finite Differential Galois groups 130
4.4.1 Generalities on Scalar Fuchsian Equations 131
4.4.2 Restrictions on the Exponents 133
4.4.3 Representations of Finite Groups 134
4.4.4 A Calculation of the Accessory Parameter 135
4.4.5 Examples 136
Analytic Theory 141
5 Monodromy, the Riemann Hilbert Problem,
and the Differential Galois Group 143
5.1 Monodromy of a Differential Equation 143
5.1.1 Local Theory of Regular Singular Equations 144
5.1.2 Regular Singular Equations on P1 147
5.2 A Solution of the Inverse Problem 150
5.3 The Riemann Hilbert Problem 153
6 Differential Equations on the Complex Sphere
and the Riemann Hilbert Problem 157
6.1 Differentials and Connections 157
6.2 Vector Bundles and Connections 160
6.3 Fuchsian Equations 168
6.3.1 From Scalar Fuchsian to Matrix Fuchsian 168
6.3.2 A Criterion for a Scalar Fuchsian Equation 172
6.4 The Riemann Hilbert Problem, Weak Form 174
6.5 Irreducible Connections 175
6.6 Counting Fuchsian Equations 181
Table of Contents XV
7 Exact Asymptotics 187
7.1 Introduction and Notation 187
7.2 The Main Asymptotic Existence Theorem 193
7.3 The Inhomogeneous Equation of Order One 199
7.4 The Sheaves A, A0, Ai/k, A°yk 203
7.5 The Equation (8 q)f~ g Revisited 208
7.6 The Laplace and Borel Transforms 209
7.7 The fc Summation Theorem 212
7.8 The Multisummation Theorem 218
8 Stokes Phenomenon and Differential Galois Groups 229
8.1 Introduction 229
8.2 The Additive Stokes Phenomenon 230
8.3 Construction of the Stokes Matrices 234
9 Stokes Matrices and Meromorphic Classification 245
9.1 Introduction 245
9.2 The Category Gr2 246
9.3 The Cohomology Set // (S1, STS) 248
9.4 Explicit 1 cocycles for Hl(Sl, STS) 252
9.4.1 One Level k 254
9.4.2 Two Levels kl k2 256
9.4.3 The General Case 257
9.5 // (S1, STS) as an Algebraic Variety 259
10 Universal Picard Vessiot Rings and Galois Groups 261
10.1 Introduction 261
10.2 Regular Singular Differential Equations 262
10.3 Formal Differential Equations 263
10.4 Meromorphic Differential Equations 264
11 Inverse Problems 271
11.1 Introduction 271
11.2 The Inverse Problem for C((z)) 273
XVI Table of Contents
11.3 Some Topics on Linear Algebraic Groups 274
11.4 The Local Theorem 278
11.5 The Global Theorem 282
11.6 More on Abhyankar s Conjecture 284
11.7 The Constructive Inverse Problem 286
12 Moduli for Singular Differential Equations 295
12.1 Introduction 295
12.2 The Moduli Functor 297
12.3 An Example 299
12.3.1 Construction of the Moduli Space 299
12.3.2 Comparison with the Meromorphic Classification 300
12.3.3 Invariant Line Bundles 303
12.3.4 The Differential Galois Group 305
12.4 Unramified Irregular Singularities 306
12.5 The Ramified Case 310
12.6 The Meromorphic Classification 313
13 Positive Characteristic 317
13.1 Classification of Differential Modules 317
13.2 Algorithmic Aspects 322
13.2.1 The Equation b(P~X) + bP = a 323
13.2.2 The p Curvature and Its Minimal Polynomial 324
13.2.3 Example: Operators of Order Two 326
13.3 Iterative Differential Modules 327
13.3.1 Picard Vessiot Theory and Some Examples 328
13.3.2 Global Iterative Differential Equations 331
13.3.3 p Adic Differential Equations 333
Appendices 337
A Algebraic Geometry 339
A.I Affine Varieties 342
A.I .1 Basic Definitions and Results 342
Table of Contents XVII
A. 1.2 Products of Affine Varieties over k 351
A. 1.3 Dimension of an Affine Variety 355
A. 1.4 Tangent Spaces, Smooth Points, and Singular Points 357
A.2 Linear Algebraic Groups 360
A.2.1 Basic Definitions and Results 360
A.2.2 The Lie Algebra of a Linear Algebraic Group 368
A.2.3 Torsors 370
B Tannakian Categories 373
B.I Galois Categories 373
B.2 Affine Group Schemes 377
B.3 Tannakian Categories 383
C Sheaves and Cohomology 391
C.I Sheaves: Definition and Examples 391
C.I.I Germs and Stalks 393
C. 1.2 Sheaves of Groups and Rings 394
C.1.3 From Presheaf to Sheaf 395
C.1.4 Moving Sheaves 397
C.1.5 Complexes and Exact Sequences 398
C.2 Cohomology of Sheaves 401
C.2.1 The Idea and the Formalism 401
C.2.2 Construction of the Cohomology Groups 404
C.2.3 More Results and Examples 408
D Partial Differential Equations 409
D. 1 The Ring of Partial Differential Operators 409
D.2 Picard Vessiot Theory and Some Remarks 414
Bibliography 419
List of Notation 433
Index 435
|
| any_adam_object | 1 |
| author | Put, Marius van der 1941- Singer, Michael F. 1950- |
| author_GND | (DE-588)115533249 (DE-588)115533265 |
| author_facet | Put, Marius van der 1941- Singer, Michael F. 1950- |
| author_role | aut aut |
| author_sort | Put, Marius van der 1941- |
| author_variant | m v d p mvd mvdp m f s mf mfs |
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| callnumber-sort | QA 3174.2 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 SK 500 |
| classification_tum | MAT 110f MAT 340f |
| ctrlnum | (OCoLC)51003831 (DE-599)BVBBV014892786 |
| dewey-full | 515/.354 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.354 |
| dewey-search | 515/.354 |
| dewey-sort | 3515 3354 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014892786 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:08:30Z |
| institution | BVB |
| isbn | 9783540442288 9783642629167 3540442286 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010069843 |
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| physical | xvii, 438 Seiten Illustrationen |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Die Grundlehren der mathematischen Wissenschaften |
| series2 | Die Grundlehren der mathematischen Wissenschaften |
| spelling | Put, Marius van der 1941- (DE-588)115533249 aut Galois theory of linear differential equations Marius van der Put ; Michael F. Singer Berlin ; Heidelberg Springer [2003] © 2003 xvii, 438 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften 328 Differentiaalvergelijkingen gtt Galois, Théorie de Galois-theorie gtt Théorie de Galois rasuqam Théorie différentielle de Galois rasuqam Équation différentielle linéaire rasuqam Équations différentielles linéaires Differential equations, Linear Galois theory Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Lineare Differentialgleichung (DE-588)4206889-7 gnd rswk-swf Lineare Differentialgleichung (DE-588)4206889-7 s Galois-Theorie (DE-588)4155901-0 s DE-604 Singer, Michael F. 1950- (DE-588)115533265 aut Erscheint auch als Online-Ausgabe 978-3-642-55750-7 Die Grundlehren der mathematischen Wissenschaften 328 (DE-604)BV000000395 328 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010069843&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Put, Marius van der 1941- Singer, Michael F. 1950- Galois theory of linear differential equations Die Grundlehren der mathematischen Wissenschaften Differentiaalvergelijkingen gtt Galois, Théorie de Galois-theorie gtt Théorie de Galois rasuqam Théorie différentielle de Galois rasuqam Équation différentielle linéaire rasuqam Équations différentielles linéaires Differential equations, Linear Galois theory Galois-Theorie (DE-588)4155901-0 gnd Lineare Differentialgleichung (DE-588)4206889-7 gnd |
| subject_GND | (DE-588)4155901-0 (DE-588)4206889-7 |
| title | Galois theory of linear differential equations |
| title_auth | Galois theory of linear differential equations |
| title_exact_search | Galois theory of linear differential equations |
| title_full | Galois theory of linear differential equations Marius van der Put ; Michael F. Singer |
| title_fullStr | Galois theory of linear differential equations Marius van der Put ; Michael F. Singer |
| title_full_unstemmed | Galois theory of linear differential equations Marius van der Put ; Michael F. Singer |
| title_short | Galois theory of linear differential equations |
| title_sort | galois theory of linear differential equations |
| topic | Differentiaalvergelijkingen gtt Galois, Théorie de Galois-theorie gtt Théorie de Galois rasuqam Théorie différentielle de Galois rasuqam Équation différentielle linéaire rasuqam Équations différentielles linéaires Differential equations, Linear Galois theory Galois-Theorie (DE-588)4155901-0 gnd Lineare Differentialgleichung (DE-588)4206889-7 gnd |
| topic_facet | Differentiaalvergelijkingen Galois, Théorie de Galois-theorie Théorie de Galois Théorie différentielle de Galois Équation différentielle linéaire Équations différentielles linéaires Differential equations, Linear Galois theory Galois-Theorie Lineare Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010069843&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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