**2018-10-03 - Colloque/Abstract**- Anglais - 32 page(s)

Cubides Kovacsics Pablo, Point Françoise , "Topological large fields, their generic differential expansions and transfer results" in Workshop on Tame Expansions of O-minimal Structures, Konstanz, Allemagne, 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: Topological large fields, their generic differential expansions and transfer results. Pablo Cubides Kovacsics (Caen) and Francoise Point (FNRS-FRS, UMons) Abstract: We consider a dp-minimal (not strongly minimal) theory $T$, expanding the theory of fields of characteristic 0. We assume it admits quantifier elimination (in some relational expansion $\cL$ of the language of fields), we consider its expansion $T_D$ to a theory of differential fields. Under some natural hypotheses, it is known that the class of existentially closed models of such expansions is elementary and that its theory $T_{D}^*$ admits quantifier elimination in $L_{\delta}$ (the language $\cL$ to which we add the derivation $\delta$). In particular, in any model of $T^*_D$, we get a dense pair of models of $T$. 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 (e.i.). In the case of CODF, there are now several proofs (of e.i.), for instance one using the description of definable types. 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 the theory of closed ordered fields.

(Anglais) Title: Topological large fields, their generic differential expansions and transfer results. Pablo Cubides Kovacsics (Caen) and Francoise Point (FNRS-FRS, UMons) Abstract: We consider a dp-minimal (not strongly minimal) theory $T$, expanding the theory of fields of characteristic 0. We assume it admits quantifier elimination (in some relational expansion $\cL$ of the language of fields), we consider its expansion $T_D$ to a theory of differential fields. Under some natural hypotheses, it is known that the class of existentially closed models of such expansions is elementary and that its theory $T_{D}^*$ admits quantifier elimination in $L_{\delta}$ (the language $\cL$ to which we add the derivation $\delta$). In particular, in any model of $T^*_D$, we get a dense pair of models of $T$. 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 (e.i.). In the case of CODF, there are now several proofs (of e.i.), for instance one using the description of definable types. 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 the theory of closed ordered fields.

(Anglais) Title: Topological large fields, their generic differential expansions and transfer results. Pablo Cubides Kovacsics (Caen) and Francoise Point (FNRS-FRS, UMons) Abstract: We consider a dp-minimal (not strongly minimal) theory $T$, expanding the theory of fields of characteristic 0. We assume it admits quantifier elimination (in some relational expansion $\cL$ of the language of fields), we consider its expansion $T_D$ to a theory of differential fields. Under some natural hypotheses, it is known that the class of existentially closed models of such expansions is elementary and that its theory $T_{D}^*$ admits quantifier elimination in $L_{\delta}$ (the language $\cL$ to which we add the derivation $\delta$). In particular, in any model of $T^*_D$, we get a dense pair of models of $T$. 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 (e.i.). In the case of CODF, there are now several proofs (of e.i.), for instance one using the description of definable types. 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 the theory of closed ordered fields.