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2018-10-19 - Colloque/Abstract - Anglais - 33 page(s)

Point Françoise , "Topological large fields, their generic differential expansions and transfer results" in Kolchin seminar, New-York, Etats-Unis, 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) Title: Differential expansions of topological large fields and transfer results. Abstract. Given a theory T of large topological fields of characteristic $0$ admitting quantifier elimination (in some relational expansion L of the language of fields), we consider its (generic) expansion T_D to a theory of differential fields. Under some natural hypotheses, that we will detail, it is known that the class of existentially closed models of such expansions is axiomatizable and that its theory T_D^* admits quantifier elimination in L_D (the language L to which we add the derivation D). For instance if one starts with the class of real-closed fields, M. Singer showed that one obtains the class of closed ordered differential fields (CODF). We will first review a number of known transfer results between T and T_D^* and their consequences for the theory of dense pairs of models of T. Then we will concentrate on elimination of imaginaries, a property that allows one to associate with any definable set a code (for instance, the theory of differentially closed fields of characteristic zero has that property). Under the hypothesis that T_D^* has open core, namely any open L_D-definable set is already L-definable, we will show transfer of elimination of imaginaries between T and T_D^*, using a topological argument due to M. Tressl in the case of CODF. This is a joint work with Pablo Cubidès Kovacsics (Caen). There will be no prerequisites in model theory.

Notes :
  • (Anglais) https://cs.nyu.edu/~pogudin/ksda/