Polynomial Automorphisms

1. Front Matter

   Pages N2-xviii

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2. Methods

   1. Front Matter

      Pages 1-1

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   2. Preliminaries

      • Arno van den Essen

      Pages 3-42

   3. Derivations and polynomial automorphisms

      • Arno van den Essen

      Pages 43-60

   4. Invertibility criteria and inversion formulae

      • Arno van den Essen

      Pages 61-75

   5. Injective morphisms

      • Arno van den Essen

      Pages 77-84

   6. The tame automorphism group of a polynomial ring

      • Arno van den Essen

      Pages 85-116

   7. Stabilization Methods

      • Arno van den Essen

      Pages 117-141

   8. Polynomial maps with nilpotent Jacobian

      • Arno van den Essen

      Pages 143-171

3. Applications

   1. Front Matter

      Pages 173-173

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   2. Applications of polynomial mappings to dynamical systems

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      Pages 175-201

   3. Group actions

      • Arno van den Essen

      Pages 203-237

   4. The Jacobian conjecture

      • Arno van den Essen

      Pages 239-273

4. Back Matter

   Pages 277-329

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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.

• Dimension
• Grad
• Gröbner basis
• algebra
• algebraic geometry
• algebraic group
• algebraic varieties
• automorphism
• commutative algebra
• field
• invariant theory
• matrices

"…This book is a valuable reference for the study of polynomial automorphisms, due to its breadth of coverage and clarity of exposition."

--Mathematical Reviews

• Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands

  Arno Essen

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