Diophantine Maps

TL;DR

This paper introduces Diophantine maps and equivalences to formalize the process of transferring solutions of Hilbert's tenth problem between rings, enhancing understanding of its decidability across different algebraic structures.

Contribution

It formalizes Diophantine maps and equivalences, providing a framework to transfer Hilbert's tenth problem solutions between rings.

Findings

Formalization of Diophantine maps and equivalences

Framework to transfer positive or negative solutions

Comparison between Diophantine and recursive cases

Abstract

To prove that Hilbert's tenth problem over a ring R has a negative answer, usually the integers or another ring for which Hilbert's tenth problem has a negative solution is modelled inside the ring of interest. In this paper, we formalize this practice by introducing the notions of a Diophantine map and a Diophantine equivalence map. We compare the Diophantine case to the recursive case. We formalize a general version of Hilbert's tenth problem and show that we can transfer a positive or negative answer to Hilbert's tenth problem using effective Diophantine maps.