Notes on the DPRM Property for Listable Structures

Abstract

A celebrated result by Davis, Putnam, Robinson, and 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.

A Diophantine definition of the constants in $\mathbb{Q}(z)$

“…Thus, it follows from the results in Section 3 of [3] that Z is Diophantine in Q(z). From general principles (see for instance Section 4 in [10]) we know that listable subsets of Q are positive existentially definable over Q in the language {0, 1, +, ×, =, Z}. The result follows.…”

Universally defining finitely generated subrings of global fields

“…See the brief [Dem10, Subsection 1.1] for all this, using the adjective recursive instead of computable; see also the extensive survey [ST99, Sections 2 & 3] for more on computable and computably stable rings. The notion of listable sets of [Pas22] is the same as what we call computably enumerable here, and the relevant fact about F and O S is that they are uniquely listable in the sense introduced there, which is related (but not identical) to computable stability. Given the assumption that A is computably enumerable, B is also computably enumerable.…”

“…The question of whether Z is diophantine in Q is motivated by the search for an analogue of the Davis-Putnam-Robinson-Matiyasevich ('DPRM') Theorem [Mat70]: it is equivalent to the question of whether every computably enumerable subset of Q is diophantine. See [Pas22,§4] for an analysis of this property for very general structures, and in particular Proposition 4.15 there for the case of Q. Koenigsmann's universal definition of Z in Q yields the statement that every computably enumerable relation on Q can be defined using an existential-universal formula (see for instance [Daa24b, Corollary 6.2]); this is weaker than asserting that computably enumerable sets are diophantine. From our main results we obtain the following analogous statement (relying on [MS22,Theorem 7.1] instead of the DPRM Theorem):…”