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