Topological fields with a generic derivation

TL;DR

This paper investigates the extension of tame topological field theories by a generic derivation, establishing key model-theoretic properties and applications to dense pairs, including distal expansions of certain field theories.

Contribution

It proves that the expansion by a generic derivation preserves tameness properties like open core, cell decomposition, and elimination of imaginaries, and applies these results to dense pairs.

Findings

Expansion by a generic derivation has $\\mathcal{L}$-open core.

Cell decomposition and elimination of imaginaries transfer to the extended theory.

Theories of pairs of real closed fields and related fields admit distal expansions.

Abstract

We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_\delta$-extension $T_{\delta}^*$ by a generic derivation $\delta$. 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 $\mathcal{L}$-open core (i.e., every $\mathcal{L}_\delta$-definable open set is $\mathcal{L}$-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 $T_\delta^*$. 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.