We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued field is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn (...) field $$\mathbb {R}((\mathbb {Z}))$$ are definable. Similarly, all types over the quotient field of the Witt ring $$W(\mathbb {F}_p^{\text {alg}})$$ are definable. This extends a work of Cubides and Delon and of Cubides and Ye. (shrink)

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

We exhibit a theory where definable types lack the amalgamation property.