We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

Abstract. The derivation on the differential-valued field T log of logarithmic transseries induces on its value group Γ log a certain map ψ. The structure (Γ log , ψ) is a divisible asymptotic couple. We prove that the theory T log = Th(Γ log , ψ) admits elimination of quantifiers in a natural first-order language. All models (Γ, ψ) of T log have an important discrete subset Ψ := ψ(Γ\{0}). We give explicit descriptions of all definable functions on Ψ and prove that Ψ is stably embedded in Γ.

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

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