On definable groups in real closed fields with a generic derivation, and related structures

Abstract

We study finite-dimensional groups definable in models of the theory RCF ∂ of real closed fields with a generic derivation (also known as CODF , the theory of closed ordered differential fields [28]). We prove that any such group Γ definably embeds in a “canonical” semialgebraic group G . We explain how our methods work in the general context of strongly model complete theories T of large “geometric” fields with a generic derivation, which includes the cases where T is the theory of pseudofinite fields and T = Th ( ℚ p ) . We also give a general theorem on recovering a definable group from generic data in the context of geometric theories. Finally we extend the methods to o -minimal theories with a generic derivation, due to Fornasiero and Kaplan, [10], and open theories of topological fields with a generic derivation, due to Cubides-Kovacsics and the third author, [7].