Polynomial automorphisms and the Jacobian conjecture
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 derivations connected with polynomial mappings satisfying the Jacobian condition and to conditions equivalent to the Jacobian conjecture expressed in terms of derivations. Chapter 3 is about formulas for the inverse of a polynomial mapping by using Gröbner basis and the elimination theory. Chapter 4 concerns the interesting topic of criteria of injectivity of regular mappings between affine varieties. In chapter 5 the author describes the difficult and interesting problem of the structure of the group of polynomial automorphisms of affine spaces (and topics connected with the problem of embeddings of affine spaces). Chapter 6 is connected with the previous one and studies the structure of polynomial automorphisms by extending them trivially to automorphisms of affine spaces of greater dimensions (so called stabilization problems). The second part, ``Applications'', gives applications of methods and results of the first part to other branches of affine geometry which are also connected to the Jacobian conjecture. The chapter 8 gives applications to dynamical systems, precisely to conditions equivalent to Jacobian conjecture in terms of asymptotic stability of vector fields generated by a given polynomial mapping (in \({\mathbb R^n}\)). In chapter 9 we find applications to group actions on affine varieties. In particular the author describes the famous linearization and cancellation problem. Chapter 10 describes selected methods (and the results obtained) of attacks on the Jacobian conjecture (conceived by many authors). In particular the Newton polygon methods and Pinchuk's counterexample to the real Jacobian conjecture. In the third part, ``Appendices'', the author collects some basic definitions and results (used in the text) from commutative algebra, algebraic geometry, and theory of the Gröbner bases. The last interesting ``Appendix F'' delivers many special examples and counterexamples connected with the problems considered in the main text, which the author (an expert in this topics) encountered in his mathematical life. Summing up, the book is very useful to all interested in affine geometry and especially in the Jacobian conjecture. So far, it is the only book on this subject. Although it contains many results without proofs it may serve as a reference book. It can also be used in graduate courses (since the methods used are algebraic and rather elementary and almost each section ends with exercises). The text is well arranged and clear. One can feel from the text the personal connection and engagement of the author to the problems described. Evidences of this are personal reminiscences in the text (e.g. in ``Introduction on K. Adjamagbo'' and in chapter 8 on a solution of the Markus-Yamabe problem), although in the reviewer's opinion this is needless in monographs. So, I may recommend this book to any interested in algebraic geometry who deals with affine geometry and especially with the Jacobian conjecture.
scientific article; zbMATH DE number 1518743