Definable groups in topological differential fields

Abstract

For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $\mathcal{L}\cup \{D\}$-definable sets and their $\mathcal{L}$-reducts, where $\mathcal{L}$ is a relational expansion of the field language and $D$ a symbol for a derivation. This enables us to associate with an $\mathcal{L}\cup \{D\}$-definable group in models of such theories, a local $\mathcal{L}$-definable group. As a byproduct, we show that in closed ordered differential fields, one has the descending chain condition on centralisers.