| Résume | A celebrated result by Davis, Putnam, Robinson and Matiyasevich shows that over the integers, listable sets are the same as Diophantine
sets. There is the question of whether other interesting rings satisfy the same DPRM property and the most common setting where this problem
has been considered is that of recursive rings. However, recursive rings do not seem to be the most appropriate framework: one would like
to allow more general structures (not just rings) and it seems natural to allow the signature of the structure to be expanded by
positive-existentially definable relations which, in general, might fail to be recursive. In this talk we will discuss the DPRM property
on structures endowed with a listable presentation (rather than a recursive one) and we will present several results addressing
foundational material around this notion such as uniqueness of the listable presentation, transference of the DPRM structure under
interpretation, and characterization of the DPRM property in terms of p.e. bi-interpretability. As a consequence, we will obtain proofs of
several folklore "facts" repeatedly claimed as results elsewhere in the literature but whose proofs are absent. Another application of the
theory is that it will allow us to link various Diophantine conjectures to the question of whether or not the DPRM property holds
for the field of rational numbers and for k(t) with k a finite field. |
