Let${\mathbb K}$be an algebraically bounded structure, and letTbe its theory. IfTis model complete, then the theory of${\mathbb K}$endowed with a derivation, denoted by$T^{\delta }$, has a model completion. Additionally, we prove that if the theoryTis stable/NIP then the model completion of$T^{\delta }$is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove (...) a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.