Polynomial Mappings
The book deals with certain algebraic and arithmetical questions concerning polynomial mappings in one or several variables. Algebraic properties of the ring Int(R) of polynomials mapping a given ring R into itself are presented in the first part, startin
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
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Wladyslaw Narkiewicz
Polynomial Mappings
Springer
Author Wladyslaw Narkiewicz Institute of Mathematics Wroclaw University Plac Grunwaldzki 2/4 PL-50-384-Wroclaw, Poland E-mail: [email protected]
Mathematics Subject Classification (1991): lIC08, lIR09, lIT06, 12E05, 13B25, 13F20, 14E05
ISBN 3-540-59435-3 Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready output by the author SPIN: 10130271 46/3142-543210 - Printed on acid-free paper
Preface
1. Our aim is to give a survey of results dealing with certain algebraic and arithmetic questions concerning polynomial mappings in one or several variables. The first part will be devoted to algebraic properties of the ring Int(R) of polynomials which map a given ring R into itself. In the case R = Z the first result goes back to G.P61ya who in 1915 determined the structure of Int(Z) and later considered the case when R is the ring of integers in an algebraic number field. The rings Int(R) have many remarkable algebraic properties and are a source of examples and counterexamples in commutative algebra. E.g. the ring Int(Z) is not Noetherian and not a Bezout ring but it is a Priifer domain and a Skolem ring. We shall present classical results in this topic due to G.P6Iya, A.Ostrowski and T.Skolem as well as modern development.
2. In the second part we shall deal with fully invariant sets for polynomial X. In the mappings in one or several variables, i.e. sets X satisfying case of complex polynomials this notion is closely related to Julia sets and the modern theory of fractals, however we shall concentrate on much more modest questions and consider polynomial maps in fields which are rather far from being algebraically closed. Our starting point will be the observation that if f is a polynomial with rational coefficients and X is a subset of the rationals satisfying f(X) = X, then either X is finite or f is linear. It turns out that the same assertion holds for certain other fields in place of the rationals and also for a certain class of polynomial mappings in several variables. We shall survey the development of these question and finally we shall deal with cyclic points of a polynomial mapping,
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