
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.
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Taxonomy
TopicsTopological and Geometric Data Analysis
