Topological fields with a generic derivation

Pablo Cubides Kovacsics, Françoise Point

Published 2019-12-17Version 1

We study a class of tame theories $T$ of topological fields and their extension $T_{\delta}^*$ by a generic derivation. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. For most examples, we show that the associated expansion by a generic derivation has the open core property (i.e., there are no new open definable sets). In addition, we show various transfer results between tame properties of $T$ and $T_\delta^*$, including relative elimination of field sort quantifiers, NIP, distality and elimination of imaginaries, among others. 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.