Strong Density of Definable Types and Closed Ordered Differential Fields

Abstract

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X ⊆ M n , there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

Generic derivations on o-minimal structures

“…Moreover, K is endowed with a dimension function dim, associating to every set X definable with parameters some natural number, satisfying the axioms in [vdD89]. This function dim is invariant under automorphisms of the ambient structure: equivalently, dim is "codedefinable" in the sense of [BCKP19].…”

“…• RCF with m commuting generic derivations: see [FK20,CKP23] for a proof based on M. Tressl's idea, see also [BCKP19,Poi11] for different proofs; • DCF 0,m,nc (see [MS14]). We have seen that the above conjecture holds for certain topological structures (see §11).…”

“…Using the known techniques, it is quite plausible that the above conjecture could be also proved when T is simple (see [MS14]). For the general case, we think new ideas are needed (but see [BCKP19]). 15.2.…”