Polynomial Automorphisms: and the Jacobian Conjecture

Arno van den Essen (auth.)9783764363505, 3764363509, 0817663509

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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Table of contents :
Front Matter….Pages N2-xviii
Front Matter….Pages 1-1
Preliminaries….Pages 3-42
Derivations and polynomial automorphisms….Pages 43-60
Invertibility criteria and inversion formulae….Pages 61-75
Injective morphisms….Pages 77-84
The tame automorphism group of a polynomial ring….Pages 85-116
Stabilization Methods….Pages 117-141
Polynomial maps with nilpotent Jacobian….Pages 143-171
Front Matter….Pages 173-173
Applications of polynomial mappings to dynamical systems….Pages 175-201
Group actions….Pages 203-237
The Jacobian conjecture….Pages 239-273
Back Matter….Pages 277-329