Definable groups in topological fields with a generic derivation
We study a class of tame L-theories T of topological fields and their extensions by a generic derivation δ. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We axiomatize the class of the existentially closed Lδ-expansions.
We show that T∗δ 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 T∗δ.
\par Then letting K be a model of T∗δ and M a |K|+-saturated elementary extension of K, we first associate with an Lδ(K)-definable group G in M, a pro-L-definable set G∗∗∞ in which the differential prolongations G∇∞ of elements of G are dense, using the L-open core property of T∗δ. Following the same ideas as in the group configuration theorem in o-minimal structures as developed by K. Peterzil, we construct a type L-definable topological group H∞⊂G∗∗∞, acting on a K-infinitesimal neighbourhood of a generic element of G∗∗∞ in a faithful, continuous and transitive way. Further H∞∩G∇∞ is dense in H∞ and the action of H∞∩G∇∞ coincides with the one induced by the initial Lδ-group action.
\par The first part of this work is joint with Pablo Cubid\`es Kovacsics.