Notes on the DPRM property for listable structures

TL;DR

This paper explores the DPRM property in listable structures, establishing foundational results and linking it to Diophantine conjectures and definability complexities in number theory.

Contribution

It provides new insights into the DPRM property, including uniqueness, transferability, and its relation to positive existential bi-interpretability, with applications to Diophantine problems.

Findings

Proved foundational results on the DPRM property in listable structures.

Linked the DPRM property to Diophantine conjectures in global fields.

Analyzed the complexity of defining Diophantine sets with existential quantifiers.

Abstract

A celebrated result by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.