Polynomial Automorphisms: And the Jacobian ConjectureSpringer Science & Business Media, 2000 - 329 páginas 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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Contenido
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 |
321 | |
325 | |
Otras ediciones - Ver todas
Polynomial Automorphisms: And the Jacobian Conjecture Arno van den Essen Sin vista previa disponible - 2000 |
Polynomial Automorphisms: And the Jacobian Conjecture Arno van den Essen Sin vista previa disponible - 2000 |
Términos y frases comunes
A-module Aff(R affine varieties algebraic set algebraically closed algebraically independent algorithm Applied Algebra assume Chapter characteristic zero coefficients commutative ring coordinate Corollary counterexample cubic homogeneous D₁ deduce defined deg F deg h deg Q degree denote dimension domain easily verifies eigenvalues elements embedding equivalent example Exercise exists F is invertible field of characteristic finitely generated k-algebra flat follows Furthermore Ga-action given gives Gröbner basis group actions H₁ hence Hubbers hypothesis ideal implies induction integer inverse function theorem irreducible isomorphism Jacobian Conjecture JF(x JH is nilpotent k-algebra Lemma Let F linear linearizable linearly locally finite locally nilpotent derivation map F Math matrix monomial morphism Newton polygons Noetherian non-zero observe polynomial automorphism polynomial map polynomial ring problem Proposition Pure and Applied resp result ring homomorphism satisfies semisimple Show strongly nilpotent suffices to prove surjective tends to infinity variables whence
Referencias a este libro
Combinatorial and Computational Algebra: International Conference on ... Kai-Yuen Chan Sin vista previa disponible - 2000 |
Combinatorial and Computational Algebra: International Conference on ... Kai-Yuen Chan Sin vista previa disponible - 2000 |