Involution (eBook)

W.M. Seiler is professor for computational mathematics (algorithmic algebra) at Kassel University. His research fields include differential equations, commutative algebra and mechanics. He is particularly interested in combining geometric and algebraic approaches. For many years, he has been an external developer for the computer algebra system MuPAD.

Involution 
5 
1 Introduction 20
2 Formal Geometry of Differential Equations 28
2.1 A Pedestrian Approach to Jet Bundles 29
2.2 An Intrinsic Approach to Jet Bundles 37
Addendum: The Contact Structure à la Gardner–Shadwick 46
2.3 Differential Equations 48
2.4 Some Examples 67
2.5 Notes 77
3 Involution I: Algebraic Theory 81
3.1 Involutive Divisions 82
Addendum: Some Algorithmic Considerations 90
3.2 Polynomial Algebras of Solvable Type 94
3.3 Hilbert's Basis Theorem and Gröbner Bases 104
Iterated Polynomial Algebras of Solvable Type 105
Polynomial Algebras with Centred Commutation Relations 107
Filtered Algebras 109
Polynomial Algebras over Fields 110
3.4 Involutive Bases 112
3.5 Notes 118
4 Completion to Involution 123
4.1 Constructive Divisions 124
4.2 Computation of Involutive Bases 128
Addendum: Right and Two-Sided Ideals 136
4.3 Pommaret Bases and -Regularity 140
4.4 Construction of Minimal Bases and Optimisations 150
4.5 Semigroup Orders 159
4.6 Involutive Bases over Rings 174
4.7 Notes 179
5 Structure Analysis of Polynomial Modules 184
5.1 Combinatorial Decompositions 185
5.2 Dimension and Depth 192
5.3 Noether Normalisation and Primary Decomposition 199
Addendum: Standard Pairs 207
5.4 Syzygies and Free Resolutions 210
Addendum: Iterated Polynomial Algebras of Solvable Type 224
5.5 Minimal Resolutions and Castelnuovo–Mumford Regularity 227
5.6 Notes 245
6 Involution II: Homological Theory 252
6.1 Spencer Cohomology and Koszul Homology 253
6.2 Cartan's Test 263
6.3 Pommaret Bases and Homology 271
6.4 Notes 277
7 Involution III: Differential Theory 280
7.1 (Geometric) Symbol and Principal Symbol 281
7.2 Involutive Differential Equations 298
7.3 Completion of Ordinary Differential Equations 313
Addendum: Constrained Hamiltonian Systems 319
7.4 Cartan–Kuranishi Completion 322
7.5 The Principal Symbol Revisited 327
7.6 -Regularity and Extended Principal Symbols 334
7.7 Notes 339
8 The Size of the Formal Solution Space 345
8.1 General Solutions 346
8.2 Cartan Characters and Hilbert Function 350
8.3 Differential Relations and Gauge Symmetries 359
Addendum: Einstein's Strength 368
8.4 Notes 369
9 Existence and Uniqueness of Solutions 372
9.1 Ordinary Differential Equations 373
9.2 The Cauchy–Kovalevskaya Theorem 385
9.3 Formally Well-Posed Initial Value Problems 389
9.4 The Cartan–Kähler Theorem 399
9.5 The Vessiot Distribution 407
Addendum: Generalised Prolongations 419
Addendum: Symmetry Theory and the Method of Characteristics 422
9.6 Flat Vessiot Connections 427
9.7 Notes 439
10 Linear Differential Equations 445
10.1 Elementary Geometric Theory 446
10.2 The Holmgren Theorem 450
10.3 Elliptic Equations 454
10.4 Hyperbolic Equations 463
10.5 Basic Algebraic Analysis 472
10.6 The Inverse Syzygy Problem 480
Addendum: Computing Extension Groups 487
Addendum: Algebraic Systems Theory 489
10.7 Completion to Involution 494
10.8 Linear Systems of Finite Type with Constant Coefficients 508
10.9 Notes 518
A Miscellaneous 522
A.1 Multi Indices and Orders 522
Addendum: Computing Derivative Trees 528
A.2 Real-Analytic Functions 530
A.3 Elementary Transformations of Differential Equations 532
Reduction to first order 532
Quasi-Linearisation 534
Transformation to one dependent variable 535
A.4 Modified Stirling Numbers 538
B Algebra 542
B.1 Some Basic Algebraic Structures 543
B.2 Homological Algebra 557
B.3 Coalgebras and Comodules 572
B.4 Gröbner Bases for Polynomial Ideals and Modules 580
C Differential Geometry 598
C.1 Manifolds 598
C.2 Vector Fields and Differential Forms 605
C.3 Distributions and the Frobenius Theorem 613
C.4 Connections 617
C.5 Lie Groups and Algebras 621
C.6 Symplectic Geometry and Generalisations 623
References 630
Glossary 650
Index 652