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On non-Diophantine sets in rings of functions

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On non-Diophantine sets in rings of functions

For a ring R, a subset of a cartesian power of R is said to be Diophantine if it is positive existentially definable over R with parameters from R. In general, Diophantine sets over rings are not well-understood even in very natural situations; for instance, we do not know if the ring of integers Z is Diophantine in the field of rational numbers. To show that a set is Diophantine requires to produce a particular existential formula that defines it. However, to show that a set is not Diophantine is a more subtle task; in lack of a good description of Diophantine sets it requires to find at least a property shared by all of them. I will give an outline of some recent joint work with Garcia-Fritz and Pheidas on showing that several sets and relations over rings of polynomials and rational functions that are not Diophantine.

- Séminaire Géométrie et théorie des modèles

Détails :

Orateur / Oratrice : Hector Pasten
Date : 25 novembre 2022
Horaire : 16h00 - 17h30
Lieu : IHP salle 01 et zoom