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2018-08-05 - Colloque/Abstract - Anglais - 29 page(s)

Point Françoise , "Topological large fields, their generic differential expansions and transfer results" in Model-Theoretic Methods in Number Theory and Algebraic Differential Equations, Manchester, Angleterre, 2018

  • Codes CREF : Algèbre - théorie des anneaux - théorie des corps (DI1147), Logique mathématique (DI1170)
  • Unités de recherche UMONS : Logique mathématique (S838)
  • Instituts UMONS : Institut de Recherche sur les Systèmes Complexes (Complexys)
  • Centres UMONS : Algèbre, Géométrie et Interactions fondamentales (AGIF)

Abstract(s) :

(Anglais) TOPOLOGICAL LARGE FIELDS, THEIR GENERIC DIFFERENTIAL EXPANSIONS AND TRANSFER RESULTS Abstract. We start with a theory T of topological fields admitting quantifier elimination (in a relational expansion L of the theory of fields). Under natural hypotheses and in particular that topological fields satisfy a property of largeness, and assuming they are endowed with a definable topology, it is known that the class of existentially closed expansions to differential fields are models of a theory TD∗ and that TD∗ admits quantifier elimination in Lδ (the language L to which we add the derivation δ). Note that in any model of TD∗ , we get a dense pair of models of T. For instance if one starts with the class of real-closed fields, one obtains the class of closed ordered differential fields; an axiomatization known as CODF was given by M. Singer. We will first review a number of known transfer results between T and TD∗ and their consequences for the theory of dense pairs of models of T. Then we will concentrate on elimination of imaginaries. In the case of CODF, there are now several proofs, for instance one using the description of definable types. We will show transfer of elimination of imaginaries between T and TD∗ , using a topological argument due to M. Tressl in the case of CODF. This is a joint work with Pablo Cubides Kovacsics (Caen).