The Asymptotic Couple of the Field of Logarithmic Transseries

Abstract

The derivation on the differential-valued field $\mathbb{T}_{\log}$ 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 $T_{\log} = {\rm Th}(Γ_{\log},ψ)$ admits elimination of quantifiers in a natural first-order language. All models $(Γ,ψ)$ of $T_{\log}$ have an important discrete subset $Ψ:=ψ(Γ\setminus\{0\})$. We give explicit descriptions of all definable functions on $Ψ$ and prove that $Ψ$ is stably embedded in $Γ$.