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NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES

Part of: Computability and recursion theory Arithmetic algebraic geometry Diophantine approximation, transcendental number theory Number theory: Connections with logic

Published online by Cambridge University Press:  23 November 2021

HECTOR PASTEN
HECTOR PASTEN*
Affiliation:
DEPARTAMENTO DE MATEMÁTICAS FACULTAD DE MATEMÁTICAS PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE4860 AVENUE VICUÑA MACKENNA MACUL, SANTIAGO, CHILEE-mail:hpasten@gmail.com
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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.

Keywords

Diophantine setlistable structureglobal fields

MSC classification

Primary: 11U09: Model theory
Secondary: 03D45: Theory of numerations, effectively presented structures 11G35: Varieties over global fields 11J68: Approximation to algebraic numbers
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 273 - 312
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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Footnotes

This research was supported by ANID (ex CONICYT) FONDECYT Regular grant 1190442 from Chile.

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