Algebraic Theory of Locally Nilpotent Derivations

by Freudenburg, Gene

• ISBN13: 9783540295211
• ISBN10: 3540295216
• Format: Hardcover
• Copyright: 2006-10-30
• Publisher: Springer Verlag

Summary

This book explores the theory and application of locally nilpotent derivations, which is a subject of growing interest and importance not only among those in commutative algebra and algebraic geometry, but also in fields such as Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the entire theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane, right up to the most recent results, such as Makar-Limanov's Theorem for locally nilpotent derivations of polynomial rings. Topics of special interest include: progress in the dimension three case, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. The reader will also find a wealth of pertinent examples and open problems and an up-to-date resource for research.

Table of Contents

Introduction	1	(218)
1 First Principles	9	(26)
1.1 Basic Definitions for Derivations	9	(6)
1.2 Basic Facts about Derivations	15	(4)
1.3 Group Actions	19	(3)
1.4 First Principles for Locally Nilpotent Derivations	22	(9)
1.5 Ga-Actions	31	(4)
2 Further Properties of Locally Nilpotent Derivations	35	(14)
2.1 Irreducible Derivations	35	(2)
2.2 Minimal Local Slices	37	(2)
2.3 Three Lemmas about UFDs	39	(1)
2.4 The Defect of a Derivation	40	(4)
2.5 Exponential Automorphisms	44	(1)
2.6 Wronskians and Kernel Elements	45	(2)
2.7 The Star Operator	47	(2)
3 Polynomial Rings	49	(34)
3.1 Variables, Automorphisms, and Gradings	49	(1)
3.2 Derivations of Polynomial Rings	50	(11)
3.3 Group Actions on An	61	(2)
3.4 Locally Nilpotent Derivations of Polynomial Rings	63	(3)
3.5 Slices in Polynomial Rings	66	(1)
3.6 Triangular Derivations and Automoprhisms	66	(4)
3.7 Homogeneous Locally Nilpotent Derivations	70	(1)
3.8 Symmetric Locally Nilpotent Derivations	71	(1)
3.9 Some Important Early Examples	72	(4)
3.10 The Homogeneous Dependence Problem	76	(7)
4 Dimension Two	83	(24)
4.1 The Polynomial Ring in Two Variables over a Field	86	(5)
4.2 Locally Nilpotent R-Derivations of R[x,y]	91	(6)
4.3 Rank-Two Derivations of Polynomial Rings	97	(3)
4.4 Automorphisms Preserving Lattice Points	100	(1)
4.5 Newton Polygons	101	(2)
4.6 Appendix: Newton Polytopes	103	(4)
5 Dimension Three	107	(30)
5.1 Miyanishi's Theorem	108	(7)
5.2 Other Fundamental Theorems in Dimension Three	115	(4)
5.3 Questions of Triangularizability and Tameness	119	(2)
5.4 The Homogeneous (2, 5) Derivation	121	(1)
5.5 Local Slice Constructions	122	(5)
5.6 The Homogeneous Case	127	(4)
5.7 Graph of Kernels and Generalized Local Slice Constructions	131	(2)
5.8 G²a-Actions	133	(1)
5.9 Appendix: An Intersection Condition	134	(3)
6 Linear Actions of Unipotent Groups	137	(20)
6.1 The Finiteness Theorem	138	(1)
6.2 Linear Ga-Actions	139	(7)
6.3 Linear Counterexamples to the Fourteenth Problem	146	(5)
6.4 Linear G²a-Actions	151	(4)
6.5 Appendix: Finite Group Actions	155	(2)
7 Non-Finitely Generated Kernels	157	(24)
7.1 Roberts' Examples	157	(3)
7.2 Counterexample in Dimension Five	160	(9)
7.3 Proof for A'Campo-Neuen's Example	169	(1)
7.4 Quotient of a Ga-Module	170	(3)
7.5 Proof for the Linear Example in Dimension Eleven	173	(1)
7.6 Kuroda's Examples in Dimensions Three and Four	174	(1)
7.7 Locally Trivial Examples	175	(1)
7.8 Some Positive Results	176	(1)
7.9 Winkelmann's Theorem	177	(1)
7.10 Appendix: Van den Essen's Proof	178	(3)
8 Algorithms	181	(14)
8.1 Van den Essen's Algorithm	183	(2)
8.2 Image Membership Algorithm	185	(1)
8.3 Criteria for a Derivation to be Locally Nilpotent	186	(2)
8.4 Maubach's Algorithm	188	(2)
8.5 Extendibility Algorithm	190	(1)
8.6 Examples	190	(3)
8.7 Remarks	193	(2)
9 The Makar-Limanov and Derksen Invariants	195	(24)
9.1 Danielewski Surfaces	197	(2)
9.2 A Preliminary Result	199	(2)
9.3 The Threefold x + x²y + z² + t³ = 0	201	(3)
9.4 Characterizing k[x, y] by LNDs	204	(3)
9.5 Characterizing Danielewski Surfaces by LNDs	207	(2)
9.6 LNDs of Special Danielewski Surfaces	209	(3)
9.7 Further Properties of the ML Invariant	212	(3)
9.8 Further Results in the Classification of Surfaces	215	(4)
10 Slices, Embeddings and Cancellation	219	(16)
10.1 Some Positive Results	220	(3)
10.2 Torus Action Formula	223	(2)
10.3 Asanuma's Torus Actions	225	(5)
10.4 Vénéreau Polynomials	230	(3)
10.5 Open Questions	233	(2)
11 Epilogue	235	(8)
11.1 Rigidity of Kernels for Polynomial Rings	235	(1)
11.2 The Extension Property	236	(1)
11.3 Nilpotency Criterion	236	(1)
11.4 Calculating the Makar-Limanov Invariant	236	(1)
11.5 Relative Invariants	237	(1)
11.6 Structure of LND(B)	237	(1)
11.7 Maximal Subalgebras	238	(1)
11.8 Invariants of a Sum	238	(1)
11.9 Finiteness Problem for Extensions	239	(1)
11.10 Geometric Viewpoint	239	(1)
11.11 Paragonic Varieties	240	(1)
11.12 Stably Triangular Ga-Actions	241	(1)
11.13 Extending Ga-Actions to Larger Group Actions	241	(1)
11.14 Variable Criterion	241	(1)
11.15 Bass's Question on Rational Triangularization	242	(1)
11.16 Popov's Questions	242	(1)
11.17 Miyanishi's Question	242	(1)
References	243	(14)
Index	257