##
**Polynomial mappings.**
*(English)*
Zbl 0829.11002

Lecture Notes in Mathematics. 1600. Berlin: Springer-Verlag. vi, 130 p. (1995).

Let \(R\) be an integral domain and \(K\) its quotient field. The main purpose of the first part of the book is to study the ring \(\text{Int} (R) = \{f \in K [X]\); \(f(R) \subset R\}\). It is well known (G. Pólya, 1915) that, when \(R = \mathbb{Z}\), \(\text{Int} (R)\) is a free \(R\)-module generated by a family \(\{h_n\}_{n \in \mathbb{N}}\) of polynomials with \(\deg h_n = n\) (these are in fact the binomial polynomials). An open problem is the determination of all rings \(R\) with this property.

The author discusses this problem, in particular in the case when \(R = I_K\) is the ring of integers of a number field \(K\). Another interesting question is about the algebraic properties of \(\text{Int} (R)\): noetherianity, Skolem property, maximal and prime ideals, Krull dimension, Prüfer property, etc. This first part also deals with the values of the successive derivatives of polynomials or of rational functions.

The second part is devoted to the study of fully invariant subsets of a field by polynomial mappings: if \(f \in \mathbb{Q} [X]\) and \(S \subset \mathbb{Q}\) satisfy \(f(S) = S\), then either \(S\) is finite or \(\deg f = 1\). The aim of study is to determine the fields with this property or its analogue in the case of several variables. In particular, is this property stable by purely transcendental extension (yes) or by finite extension? The last chapter deals with polynomial cycles: by a theorem of I. N. Baker (1960) every polynomial of degree \(\geq 2\) in \(\mathbb{C} [X]\) has cycles of every order with at most one exception. The author considers this question in algebraic number fields.

This nice, short (130 pages) but dense book makes a sound review of the question. As often as possible, concise proofs are given. More technical results or related questions are described and references are given; the text is well supplemented by many exercises given at the end of each chapter. The appendix states a list of 21 open problems and the book contains 11 pages of bibliographical references (from 1895 to 1994). It is interesting to have such a synthesis on questions which are often studied but scattered in the literature.

The author discusses this problem, in particular in the case when \(R = I_K\) is the ring of integers of a number field \(K\). Another interesting question is about the algebraic properties of \(\text{Int} (R)\): noetherianity, Skolem property, maximal and prime ideals, Krull dimension, Prüfer property, etc. This first part also deals with the values of the successive derivatives of polynomials or of rational functions.

The second part is devoted to the study of fully invariant subsets of a field by polynomial mappings: if \(f \in \mathbb{Q} [X]\) and \(S \subset \mathbb{Q}\) satisfy \(f(S) = S\), then either \(S\) is finite or \(\deg f = 1\). The aim of study is to determine the fields with this property or its analogue in the case of several variables. In particular, is this property stable by purely transcendental extension (yes) or by finite extension? The last chapter deals with polynomial cycles: by a theorem of I. N. Baker (1960) every polynomial of degree \(\geq 2\) in \(\mathbb{C} [X]\) has cycles of every order with at most one exception. The author considers this question in algebraic number fields.

This nice, short (130 pages) but dense book makes a sound review of the question. As often as possible, concise proofs are given. More technical results or related questions are described and references are given; the text is well supplemented by many exercises given at the end of each chapter. The appendix states a list of 21 open problems and the book contains 11 pages of bibliographical references (from 1895 to 1994). It is interesting to have such a synthesis on questions which are often studied but scattered in the literature.

Reviewer: F.Gramain (Saint-Etienne)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11C08 | Polynomials in number theory |

12E05 | Polynomials in general fields (irreducibility, etc.) |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13B25 | Polynomials over commutative rings |

11T06 | Polynomials over finite fields |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

14E05 | Rational and birational maps |

11R09 | Polynomials (irreducibility, etc.) |