Polynomial automorphisms and the Jacobian conjecture
type: Item
instance of: scholarly article
title: Polynomial automorphisms and the Jacobian conjecture
zbMATH Open document ID: 0962.14037
published in: Progress in Mathematics
publication date: 2000-10-17T00:00:00Z
review text: In the Mathematical Subject Classification 2000 a new section ``Affine geometry'' was added and in it new subsections ``Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)'' and ``Jacobian problem''. So far there has not appeared any monograph devoted to this subjects. The book under review is the first one. It presents precisely the topics mentioned above. It summaries the actual knowledge of the main part of the theory of polynomial automorphisms. The topics selected reflect rather the own research interests of the author and the same can be said about the methods -- almost entirely algebraic. So, out of this book there are other methods used in the theory of polynomial automorphisms and the jacobian conjecture as for instance topological (Lê Dũng Tráng, W. Neumann -- study of fibers of polynomials), geometrical (Z. Jelonek -- identity and determining sets for polynomial automorphisms) and analytical (J.Chądzyński and the reviewer -- the Łojasiewicz exponent at infinity and components of polynomial automorphisms). As it was said the presentation method is algebraic and the book is written from the perspective of the Jacobian conjecture. Because of the wide spectrum of results described, some of them are given without proofs. Nevertheless they are well presented and described (with good references). The main feature of results is that they are given in the form as general as possible (for instance ground fields are replaced by k-algebras). So, many proofs are new and some of them simpler than the known ones. Let us now outline the contents of the book. It consists of three parts: 1. Methods, 2. Applications, 3. Appendices. The first part, ``Methods'', presents algebraic tools and methods used in the attack of the Jacobian conjecture. In chapter 1 we have the formal inverse function theorem and the basic properties of derivations of rings (this last notion is a key notion for almost all results in the book). Chapter 2 is devoted to der…
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 1518743
zbMATH Keywords: polynomial automorphism
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: derivation
zbMATH Keywords: affine space
zbMATH Keywords: tame automorphism
zbMATH Keywords: Markus-Yamabe conjecture
zbMATH Keywords: Gröbner basis
zbMATH Keywords: injective endomorphism
zbMATH Keywords: affine variety
zbMATH Keywords: cancellation problem
zbMATH Keywords: locally nilpotent derivation
zbMATH Keywords: face polynomial
zbMATH Keywords: triangular automorphism
author: Arno van den Essen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1518743