CONTEMPORARY MATHEMATICS

270

Hilbert/s Tenth Problem: Relations with Arithmetic and

AlgebraiC Geometry Workshop on Hilbert's Tenth Problem:

Relations with Arithmetic and Algebraic Geometry November 2-5, 1999

Ghent University, Belgium

Jan Denef Leonard Lipshitz

Thanases Pheidas Jan Van Geel

Editors

Selected Titles in This Series

270 Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000

269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in geometry, topology and dynamics, 2001

268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000

267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000

266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000

265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000

264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000

263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul lgodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors,

Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability

on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in

analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjaar Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999

251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000

250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer

Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum

affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and David Royal Larson, Editors, The functional and

harmonic analysis of wavelets and frames, 1999 (Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/270

Hilbert's Tenth Problem: Relations with Arithmetic and

Algebraic Geometry

CoNTEMPORARY MATHEMATICS

270

Hilbert's Tenth Problem: Relations with Arithmetic and

Algebraic Geometry Workshop on Hilbert's Tenth Problem:

Relations with Arithmetic and Algebraic Geometry November 2-5, 1999

Ghent University, Belgium

Jan Denef Leonard Lipshitz

Thanases Pheidas Jan Van Geel

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume is the outcome of a workshop on relations between Hilbert's Tenth Problem and arithmetic and algebraic geometry held at Ghent University, Belgium, November 2-5, 1999, with support from the Fund for Scientific Research-Flanders (FWO-Vlaandaren), the FWO Research Network (WOG) in Mathematics, the Department for Pure Mathe-matics and Computer Algebra of Ghent University and the Department of Mathematics of the Katholiel,{e Universiteit Leuven.

2000 Mathematics Subject Classification. Primary OOB25, 03B25, 03D20, 03D35, 11U05, 14Gxx, 65Y20, 68Q15.

Library of Congress Cataloging-in-Publication Data Hilbert's tenth problem : relations with arithmetic and algebraic geometry/ Jan Denef ... [et al.], editors.

p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 270) Includes bibliographical references. ISBN 0-8218-2622-0 (alk. paper) 1. Hilbert's tenth. problem. 2. Arithmetical algebraic geometry. 3. Geometry, Algebraic.

I. Denef, Jan, 1951- II. Contemporary mathematics (American Mathematical Society) ; v. 270. QA242 .H55 2000 512'.7-dc21 00-052575

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissioniDams. org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 2000 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.

Visit the AMS home page at URL: http: I /www. ams. org/

10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00

Contents

Introduction 1x

List of Participants x1

Hilbert's tenth problem: What was done and what is to be done YURI MATIYASEVICH 1

Undecidability of existential theories of rings and fields: A survey THANASES PHEIDAS AND KARIM ZAHIDI 49

Hilbert's tenth problem over number fields, a survey ALEXANDRA SHLAPENTOKH 107

Defining constant polynomials MIHAl PRUNESCU 139

Decidability and local-global principles L. DARNIERE 145

Applications of local-global principles to arithmetic and geometry LAURENT MORET-BAILLY 169

Regularly T-closed fields JOACHIM SCHMID 187

Skolem density problems over large Galois extensions of global fields MOSHE JARDEN AND AHARON RAZON WITH AN APPENDIX BY WULF-DIETER GEYER 213

An effort to prove that the existential theory of Q is undecidable THANASES PHEIDAS 237

Topology of diophantine sets: Remarks on Mazur's conjectures GUNTHER CORNELISSEN AND KARIM ZAHIDI 253

Diagonal quadratic forms and Hilbert's tenth problem PAUL VOJTA 261

Algebraic geometry over four rings and the frontier to tractability J. MAURICE ROJAS 275

Some model theory of compact complex spaces ANAND PILLAY 323

vii

viii CONTENTS

Double coset decompositions for algebraic groups over K[t] K. H. KIM AND F. W. ROUSH 339

Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers CURTIS D. BENNETT, LISAK. ELDERBROCK, AND ANDREW M. W. GLASS 357

Introduction

This volume is the outcome of a meeting on relations between Hilbert's tenth problem, and arithmetic and algebraic geometry, that took place in the histori-cal surroundings of the former Dominican Abbey "Het Pand" in Ghent, Belgium in November 1999. Interrelations between these different fields have had a grow-ing presence in the literature ever since Julia Robinson's landmark papers on the elementary theory of the rational numbers (using the local-global structure of qua-dratic forms), and later on through the use of unit groups of quadratic fields (dis-guised as Fibonacci numbers) in Matijasevich's solution to Hilbert's tenth problem for the rational integers and the use of the arithmetic of elliptic curves in the works of Denef and others. More recently, we have seen connections to the real topological structure of sets of rational points on varieties through the work of Mazur, linking diophantine sets more tightly to local-global principles; the work of Hrushovski, ap-plying model theory to conjectures of Manin-Mumford and Mordell-Lang (of which excellent surveys already exist); and the work of Rojas making connections to com-plexity theory in computational geometry and paving the way for applications .in theoretical computer science. Moreover Rumely's local-global principle, implying the decidability of Hilbert's tenth problem for the ring of all algebraic integers, has been generalized to very powerful local-global principles for "big" rings, having applications to geometry and number theory.

The original optimism of Hilbert, expressed in his tombstone engraving Wir muss en wissen, wir werden wissen 1 , was originally dampened by undecidability results of various kinds, starting with Godel's famous work in the thirties. The negative solution to the tenth problem seems to have come (in the early seventies) as a confirmation of the scepticism in the power of algorithms and computers, lead-ing to Matijasevich's theorem being best known in the literature via a formulation starting "there is no algorithm to decide ... " , whereas the complete content of the theorem actually is the equivalence between diophantine sets (i.e., images of pro-jections from sets of integral points on affine varieties) and recursively enumerable sets. The above formulation is an immediate corollary of this and the existence of a recursively enumerable, non-recursive set. It is the latter (full) formulation of the theorem that we would like to become pre-eminent, with its arithmetical and geometrical content made visible for application and reflection.

During the meeting, some instructional sets of lectures were given, some of which were historical surveys summing up the current status of the field (e.g., Hilbert's tenth problem for various rings and fields), whereas others treated novel aspects (like Rojas's theory). Of most of these lectures, this volume contains a

1 We must know, we shall know ix

X INTRODUCTION

detailed write-up. One exception is the notes from a lecture series by Colliot-Thel(me on the Hasse principle, the text of which has been published in another volume of this series (The Hasse principle in a pencil of algebraic varieties. Number theory (Tiruchirapalli, 1996), 19-39, Contemp. Math. 210, 1998). There were also several research presentations, and a selection of research papers is contained in this volume. Some invited speakers who were not able to attend the meeting did agree to contribute papers to these proceedings. We want to thank A. Glass, K. Kim and F. Roush, J. Schmid and P. Vojta for doing so.

Here is an outline of how the papers in the volume are grouped. First comes a paper of Matijasevich presenting a survey of the history of the problem. It is followed by papers on Hilbert's tenth problem for various rings and fields. Then we group papers on local-global principles. The next three papers treat the relation between Hilbert's tenth problem and conjectures in arithmetic geometry relating to the structure of the set of rational points on a variety (as opposed to the set of integral points in the original problem of Hilbert), via elliptic curves, Mazur's conjecture and Biichi's problem. This is followed by a paper of Rojas summing up his Ghent lectures concerning complexity results for computing with complex algebraic sets, and novel relations between the decidability of specific predicates like 3\i:J and height bounds for integral points on curves. Next come papers on model theory of algebraic groups and analytic geometry. We conclude with a paper on good zero bounds applied to the explicit solution of diophantine equations in few variables. This result "on the positive side" leads us back to the original problem, which could be paraphrased by saying: what do we know, and-getting back to Hilbert-what can we know, about the general structure of the set of solutions of polynomial equations (in various structures)? We hope this volume allows the reader to view this question from several angles, including arithmetical, geometrical, topological, model-theoretic and computational. We also hope it will stimulate further research and interaction between number theorists and logicians, much as it was the case at the Ghent meeting.

It is our pleasant duty to thank the following institutions for financial sup-port: the Fund for Scientific Research-Flanders (FWO-Vlaanderen), the FWO Research Network (WOG) in Mathematics, the Department for Pure Mathematics and Computer Algebra of Ghent University and the Department of Mathematics of the Katholieke Universiteit Leuven. We thank the AMS for publishing this vol-ume. We also want to express our warmest gratitude to the staff of "Het Pand" for their flexibility and help on the practical side, to Gunther Cornelissen and Karim Zahidi of Ghent University for much help in many ways, and Christine Thivierge and Barbara Beeton of the AMS for their guidance in editing this volume.

September 2000,

The editors

Belair Luc Bes Alexis Boffa Maurice Chatzidakis Zoe Cherlin Gregory Colliot- Thelene Jean-Louis Coppens Marc Cornelissen Gunther Cluckers Raj Darniere Luck Davis Martin Delon Fran9oise Denef Jan Derakhshan Jamshid De Volder Cindy Donnely Steve Duret Jean-Louis Ershov Yuri Gardeyn Francis Herremans Adriaan H oornaert Kathleen Jarden Moshe Lipshitz Leonard Loeser Fran9ois Matiyasevich Yuri

Michaux Christian Moosa Rahim Moret-Bailly Laurent Moroz Boris Pheidas Thanases Pierce David Pillay A nand Point Fran9oise Poonen Bjorn Prestel Alexander Prunescu Mihai

List of Participants

Universite de Quebec Universite Paris 7 Universite de Mons-Hainaut Universite Paris 7 Rutgers University Universite de Paris-Sud Katholiek Hogeschool Kempen MPIM Bonn and Ghent University Katholieke Universiteit Leuven Universite d'Angers Courant Institute, New York Universite Paris 7 Katholieke Universiteit Leuven Einstein Institute of Mathematics Ghent University University of Georgia Universite d'Angers Novosibirsk State University Ghent University Katholieke Universiteit Leuven Katholieke Universiteit Leuven Tel Aviv University Purdue University Universite de Paris 7 Steklov Institute of Mathematics,

St.-Petersburg Universite de Mons-Hainaut Universite de Paris 7 Universite de Rennes 1 MPIM Bonn University of Crete McMaster University University of Illinois at Urbana-Champaign Universite de Mons-Hainaut University of California, Berkeley Universitiit Konstanz Universitiit Greifswald

xi

xii

Razon Aharon Ribenboim Paolo Rojas Maurice Segers Dirk Shlapentokh Alexandra Sundaram Narasimhan Tschinkel Yuri Van Geel Jan Van Lierde Veronique Veys Wim Vidaux Xavier Videla Carlos Vsemirnov Maxim

Zahidi Karim

LIST OF PARTICIPANTS

Bar-Ilan University Queen's University City University Hong Kong Katholieke Universiteit Leuven East Carolina University State University of New York, Albany University of Illinois, Chicago Ghent University Katholieke Universiteit Leuven Katholieke Universiteit Leuven Universite d'Angers Instituto Politecnico N acional 2508, Mexico Steklov Institute of Mathematics,

St.-Petersburg Ghent University

Selected Titles in This Series (Continued from the front of this publication)

246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in dynamics, 1999

245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias Lopez, and Joseph Lipman, Studies in

duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and

education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric

aspects of partial differential equations, 1999 241 Piotr Pragacz, Michal Szurek, and Jaroslaw Wisniewski, Editors, Algebraic

geometry: Hirzebruch 70, 1999 240 Angel Carocca, Victor Gonzalez-Aguilera, and Rubi E. Rodriguez, Editors,

Complex geometry of groups, 1999 239 Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors, Homotopy

invariant algebraic structures, 1999 238 Gui-Qiang Chen and Emmanuele DiBenedetto, Editors, Nonlinear partial

differential equations, 1999 237 Thomas Branson, Editor, Spectral problems in geometry and arithmetic, 1999 236 Bruce C. Berndt and Fritz Gesztesy, Editors, Continued fractions: From analytic

number theory to constructive approximation, 1999 235 Walter A. Carnielli and !tala M. L. D'Ottaviano, Editors, Advances in

contemporary logic and computer science, 1999 234 Theodore P. Hill and Christian Houdre, Editors, Advances in stochastic

inequalities, 1999 233 Hanna Nencka, Editor, Low dimensional topology, 1999 232 Krzysztof Jarosz, Editor, Function spaces, 1999 231 Michael Farber, Wolfgang Liick, and Shmuel Weinberger, Editors, Tel Aviv

topology conference: Rothenberg Festschrift, 1999 230 Ezra Getzler and Mikhail Kapranov, Editors, Higher category theory, 1998 229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the

representation theory of finite dimensional algebras, 1998 228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang,

Editors, Operator algebras and operator theory, 1999 227 John McCleary, Editor, Higher homotopy structures in topology and mathematical

physics, 1999 226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation:

Applications to geometry and optimization, 1999 225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications,

and algorithms, 1999 224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent

progress in algebra, 1999 223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances

in discrete and computational geometry, 1999 222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in

Pohang, 1999

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstoref.

ISBN D-8218-2622-0

9 780821 826225 CONM/270