Extensions of Hilbert's tenth problem

Abstract

The present article is an attempt to bridge the gap between the researchers that work in the areas adjacent to Hilbert's Tenth Problem (for short, HTP), mainly, number theory and mathematical logic. It presents the main results that have been obtained and asks some of the open questions in the area, leading to the main unanswered question (at least in the opinion of the author)—the analogue of HTP for the rational numbers. In this respect, the article is a successor of [Da] where the reader can find more information on the solution of the initial problem. Most of the results that are presented are not new, but some of the proofs are. As the purpose of a survey article is the “unification” of the methods involved, we present a proof of the analogue of HTP in the case of polynomial rings which is more geometric in nature than the initial proof of Denef (although it uses essentially the same tools). This allows, in our opinion, an easy introduction to the utilisation of the concepts used in the more recent approaches, like that of elliptic curves. We must say that the approach to the subject that we suggest through the questions we ask (mostly consisting of generalisations of HTP to various algebraic and analytic domains) is not the only one.