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