http://dx.doi.org/10.1090/surv/048 Selected Titles in This Series 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Preese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1993 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors, The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 (Continued in the back of this publication) This page intentionally left blank Integer-Valued Polynomials This page intentionally left blank Mathematical Surveys and Monographs Volume 48 Integer-Valued Polynomials Paul-Jean Cahen Jean-Luc Chabert American Mathematical Society Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair Howard A. Masur Michael Renardy The first author was supported in part by the CNRS URA 225, and the second author by the LaMfa-Amiens. 1991 Mathematics Subject Classification. Primary 13A15, 13G05; Secondary 11R04, 13B25, 13C20, 13E99, 13F05, 13F20. Library of Congress Cataloging-in-Publication Data Cahen, Paul-Jean, 1946- Integer-valued polynomials / Paul-Jean Cahen, Jean-Luc Chabert. p. cm.—(Mathematical surveys and monographs, ISSN 0076-5376 ; v. 48) Includes bibliographical references and index. ISBN 0-8218-0388-3 (alk. paper) 1. Polynomials. 2. Ideals (Algebra) 3. Rings of integers. I. Chabert, Jean-Luc. II. Title. III. Series: Mathematical surveys and monographs ; no. 48. QA161.P59C34 1996 512'.4—dc20 96-35954 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97 Contents Foreword xi Historical Introduction xiii Mathematical Introduction xvii Conventions and Notation xix Chapter I. Coefficients and Values 1 1.1. Various rings of integer-valued polynomials 2 1.2. Localization 6 1.3. Trivial cases 9 1.4. Polynomial torsion theory 14 Exercises 19 Chapter II. Additive Structure 25 II. 1. Characteristic ideals 26 11.2. Discrete valuation domains 29 11.3. Dedekind domains 34 11.4. Polya fields 40 Exercises 43 Chapter III. Stone-Weierstrass 51 III.l. a-adic topology 52 III.2. Continuity 53 III.3. Mahler series 56 III.4. Reduction to one-dimensional local domains 60 III.5. Analytically irreducible domains 63 Exercises 66 viii CONTENTS Chapter IV. Integer-Valued Polynomials on a Subset 73 IV. 1. Polynomial closure 74 IV.2. Globalization 79 IV.3. Dedekind domains as polynomially dense subrings 82 IV.4. Integral closure 86 Exercises 90 Chapter V. Prime Ideals 99 V.l. Fibers and Krull dimension 100 V.2. Discrete valuation domains 103 V.3. Unibranched and non-unibranched domains 108 V.4. Large fibers 111 Exercises 114 Chapter VI. Multiplicative Properties 123 VI. 1. Priifer domains and valuation overrings 124 VI.2. Various transfer properties 129 VI.3. Factorization properties 134 VI.4. Priifer or not Priifer 140 Exercises 151 Chapter VII. Skolem Properties 159 VII. 1. Skolem properties of rings of algebraic integers 160 VII.2. Unitary and non-unitary ideals 165 VII.3. Strong Skolem properties 170 VII.4. Almost Skolem rings 175 VII.5. Skolem rings and the Nullstellensatz property 178 Exercises 183 Chapter VIII. Invertible Ideals and the Picard Group 193 VIII.l. The short exact sequence 194 VIII.2. Invertible unitary ideals and value functions 198 VIII.3. Generalization to analytically irreducible domains 203 VIII.4. The strong two-generator property 208 VIII.5. Prime ideals and the Picard group 213 Exercises 220 Chapter IX. Integer-Valued Derivatives and Finite Differences 227 IX. 1. More rings of integer-valued polynomials 228 IX.2. Prime ideals 233 IX.3. Bases for finite differences 241 IX.4. Bases for derivatives 247 Exercises 252 CONTENTS ix Chapter X. Integer-Valued Rational Functions 257 X.l. From polynomials to rational functions 258 X.2. Continuity and prime ideals 261 X.3. Multiplicative properties 265 X.4. Valuation domains revisited 270 Exercises 277 Chapter XL Integer-Valued Polynomials in Several Indeterminates 285 XI. 1. From one to several indeterminates 286 XL 2. Prime ideals and related topics 291 XI.3. The Nullstellensatz property 295 XI.4. Miscellaneous 301 Exercises 303 References 307 List of Symbols 317 Index 319