Topological fields with a generic derivation

Abstract

We study a class of tame L-theories T of topological fields and their Lδ-extension Tδ⁎ by a generic derivation δ. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has L-open core (i.e., every Lδ-definable open set is L-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of T such as relative elimination of field sort quantifiers, NIP and distality also transfer to Tδ⁎. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of p-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.