Introduction

One of the first tasks undertaken by Model Theory was to produce elimination results, for example methods of eliminating quantifiers in formulas of certain structures. In almost all cases those methods have been effective and thus provide algorithms for examining the truth of possible theorems. On the other hand, Gödel's Incompleteness Theorem and many subsequent results show that in certain structures, constructive elimination is impossible. The current article is a (very incomplete) effort to survey some results of each kind, with a focus on the decidability of existential theories, and ask some questions at the intersection of Logic and Number Theory. It has been written having in mind a mathematician without prior exposition to Model Theory. Our presentation will consist of four parts.

Part A deals with positive (decidability) results for analogues of Hilbert's tenth problem for substructures of the integers and for certain local rings.

Part B focuses on the ‘parametric problem’ and the relevance of Hilbert's tenth problem to conjectures of Lang.

Part C deals with the analogue of Hilbert's tenth problem for rings of Analytic and Meromorphic functions.

Part D is an informal discussion on the chances of proving a negative (or could it be positive?) answer to the analogue of Hilbert's tenth problem for the field of rational numbers.