Polynomial Automorphisms

by Van Den Essen, Arno

• ISBN13: 9783764363505
• ISBN10: 3764363509
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface
Introduction
Preliminaries	p. 3
Derivations and polynomial automorphisms	p. 43
Invertibility criteria and inversion formulae	p. 61
Injective morphisms	p. 77
The tame automorphism group of a polynomial ring	p. 83
Stabilization Methods	p. 117
Polynomial maps with nilpotent Jacobian	p. 143
Applications of polynomial mappings to dynamical systems	p. 175
Group actions	p. 203
The Jacobian Conjecture	p. 239
Some commutative algebra	p. 277
Some basic results from algebraic geometry	p. 283
Some results from Grobner basis theory	p. 287
Flatness	p. 291
D-Modules	p. 295
App. F: Special examples and counterexamples	p. 299
Bibliography	p. 307
Authors Index	p. 321
Index	p. 325
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