Polynomial Automorphisms: And the Jacobian Conjecture

כריכה קדמית
Springer Science & Business Media, 2000 - 329 עמודים
Motivated by some notorious open problems, such as the Jacobian conjecture and the tame generators problem, the subject of polynomial automorphisms has become a rapidly growing field of interest. This book, the first in the field, collects many of the results scattered throughout the literature. It introduces the reader to a fascinating subject and brings him to the forefront of research in this area. Some of the topics treated are invertibility criteria, face polynomials, the tame generators problem, the cancellation problem, exotic spaces, DNA for polynomial automorphisms, the Abhyankar-Moh theorem, stabilization methods, dynamical systems, the Markus-Yamabe conjecture, group actions, Hilbert's 14th problem, various linearization problems and the Jacobian conjecture. The work is essentially self-contained and aimed at the level of beginning graduate students. Exercises are included at the end of each section. At the end of the book there are appendices to cover used material from algebra, algebraic geometry, D-modules and Gröbner basis theory. A long list of ''strong'' examples and an extensive bibliography conclude the book.
 

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תוכן

1 Preliminaries
3
12 Derivations
14
13 Locally finite derivations
22
14 Algorithms for locally nilpotent derivations
37
2 Derivations and polynomial automorphisms
43
22 Derivations and the Jacobian Condition
50
23 The degree of the inverse of a polynomial automorphism
55
formulae
61
of the Markus Yamabe Conjecture
185
84 Meisters cubiclinear linearization conjecture and the MYC revisited
195
9 Group actions
203
92 Hilberts finitencss theorem
208
Derksens algorithm to compute the invariants for reductive groups
211
94 A linearization conjecture for reductive group actions
215
95 Ga actions
219
96 Gnactions and Hilberts fourteenth problem
228

32 An invertibility algorithm for morphisms between finitely generated kalgebras
63
33 A resultant criterion and formula for the inversion of a polynomial map in two variables
70
4 Injective morphisms
77
42 Injective endomorphisms of affine algebraic sets are automorphisms
79
43 A short proof of Theorem
82
44 Injective morphisms between irreducible affine varieties of the same dimension
83
5 The tame automorphism group of a polynomial ring
85
51 The tame automorphism group of RX Y
86
52 The tame automorphism group in dimension 3
95
53 Embeddings of affine algebraic varieties and tame automorphisms
99
54 The AbhyankarMoh theorem
106
6 Stabilization Methods
117
62 Stable equivalence
119
63 Applications to the Jacobian conjecture
125
64 GorniZampieri pairing
131
7 Polynomial maps with nilpotent Jacobian
143
72 The class n A
148
73 Hn A Dn A and stable tameness
155
74 Strongly nilpotent Jacobian matrices
164
Part II Applications
173
8 Applications of polynomial mappings to dynamical systems
175
and the LaSalle problem in dimension two
180
10 The Jacobian conjecture
239
102 The twodimensional Jacobian conjecture
245
103 Polynomial maps with integer coefficients and the Jacobian Conjecture in positive characteristic
257
104 Smodules and the Jacobian conjecture
263
105 Endomorphisms sending coordinates to coordinates
269
Part III Appendices
275
Appendix A Some commutative algebra
277
A3 Localization
278
A4 Completions
279
A6 The universal coefficients method
280
Appendix B Some basic results from algebraic geometry
283
B2 Morphisms of irreducible affine algebraic varieties
284
Appendix C Some results from Grobner basis theory
287
Appendix D Flatness
291
D2 Flat morphisms between affine algebraic varieties
292
Appendix E smodules
295
E2 Direct and inverse images
297
Appendix F Special examples and counterexamples
299
Bibliography
307
Authors Index
321
Index
325
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