Algebraic Theory of Locally Nilpotent Derivations

Contents......Page 7
Introduction......Page 10
1.1 Basic Definitions for Derivations......Page 17
1.2 Basic Facts about Derivations......Page 23
1.3 Group Actions......Page 27
1.4 First Principles for Locally Nilpotent Derivations......Page 30
1.5 G[sub(a)]-Actions......Page 39
2.1 Irreducible Derivations......Page 42
2.2 Minimal Local Slices......Page 44
2.3 Three Lemmas about UFDs......Page 46
2.4 The Defect of a Derivation......Page 47
2.5 Exponential Automorphisms......Page 51
2.6 Wronskians and Kernel Elements......Page 52
2.7 The Star Operator......Page 54
3.1 Variables, Automorphisms, and Gradings......Page 55
3.2 Derivations of Polynomial Rings......Page 56
3.3 Group Actions on A[sup(n)]......Page 67
3.4 Locally Nilpotent Derivations of Polynomial Rings......Page 69
3.6 Triangular Derivations and Automoprhisms......Page 72
3.7 Homogeneous Locally Nilpotent Derivations......Page 76
3.8 Symmetric Locally Nilpotent Derivations......Page 77
3.9 Some Important Early Examples......Page 78
3.10 The Homogeneous Dependence Problem......Page 82
4. Dimension Two......Page 89
4.1 The Polynomial Ring in Two Variables over a Field......Page 92
4.2 Locally Nilpotent R-Derivations of R[x, y]......Page 97
4.3 Rank-Two Derivations of Polynomial Rings......Page 103
4.4 Automorphisms Preserving Lattice Points......Page 106
4.5 Newton Polygons......Page 107
4.6 Appendix: Newton Polytopes......Page 109
5. Dimension Three......Page 113
5.1 Miyanishi's Theorem......Page 114
5.2 Other Fundamental Theorems in Dimension Three......Page 121
5.3 Questions of Triangularizability and Tameness......Page 125
5.4 The Homogeneous (2, 5) Derivation......Page 127
5.5 Local Slice Constructions......Page 128
5.6 The Homogeneous Case......Page 133
5.7 Graph of Kernels and Generalized Local Slice Constructions......Page 137
5.8 G[sup(2)][sub(a)]-Actions......Page 139
5.9 Appendix: An Intersection Condition......Page 140
6. Linear Actions of Unipotent Groups......Page 143
6.1 The Finiteness Theorem......Page 144
6.2 Linear G[sub(a)]-Actions......Page 145
6.3 Linear Counterexamples to the Fourteenth Problem......Page 152
6.4 Linear G[sup(2)][sub(a)]-Actions......Page 157
6.5 Appendix: Finite Group Actions......Page 161
7.1 Roberts' Examples......Page 163
7.2 Counterexample in Dimension Five......Page 166
7.3 Proof for A'Campo-Neuen's Example......Page 175
7.4 Quotient of a G[sub(a)]-Module......Page 176
7.5 Proof for the Linear Example in Dimension Eleven......Page 179
7.6 Kuroda's Examples in Dimensions Three and Four......Page 180
7.7 Locally Trivial Examples......Page 181
7.8 Some Positive Results......Page 182
7.9 Winkelmann's Theorem......Page 183
7.10 Appendix: Van den Essen's Proof......Page 184
8. Algorithms......Page 187
8.1 Van den Essen's Algorithm......Page 189
8.2 Image Membership Algorithm......Page 191
8.3 Criteria for a Derivation to be Locally Nilpotent......Page 192
8.4 Maubach's Algorithm......Page 194
8.6 Examples......Page 196
8.7 Remarks......Page 199
9. The Makar-Limanov and Derksen Invariants......Page 201
9.1 Danielewski Surfaces......Page 203
9.2 A Preliminary Result......Page 205
9.3 The Threefold x + x[sup(2)]y + z[sup(2)] + t[sup(3)] = 0......Page 207
9.4 Characterizing k[x, y] by LNDs......Page 210
9.5 Characterizing Danielewski Surfaces by LNDs......Page 213
9.6 LNDs of Special Danielewski Surfaces......Page 215
9.7 Further Properties of the ML Invariant......Page 218
9.8 Further Results in the Classification of Surfaces......Page 221
10. Slices, Embeddings and Cancellation......Page 224
10.1 Some Positive Results......Page 225
10.2 Torus Action Formula......Page 228
10.3 Asanuma's Torus Actions......Page 230
10.4 Vénéreau Polynomials......Page 235
10.5 Open Questions......Page 238
11.1 Rigidity of Kernels for Polynomial Rings......Page 240
11.4 Calculating the Makar-Limanov Invariant......Page 241
11.6 Structure of LND(B)......Page 242
11.8 Invariants of a Sum......Page 243
11.10 Geometric Viewpoint......Page 244
11.11 Paragonic Varieties......Page 245
11.14 Variable Criterion......Page 246
11.17 Miyanishi's Question......Page 247
References......Page 248
D......Page 262
K......Page 263
R......Page 264
W......Page 265
Z......Page 266