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 $\Gamma{\log}$ a certain map $\psi$. The structure $(\Gamma_{\log},\psi)$ is a divisible asymptotic couple. We prove that the theory $T_{\log} = {\rm Th}(\Gamma_{\log},\psi)$ admits elimination of quantifiers in a natural first-order language. All models $(\Gamma,\psi)$ of $T_{\log}$ have an important discrete subset $\Psi:=\psi(\Gamma\setminus{0})$. We give explicit descriptions of all definable functions on $\Psi$ and prove that $\Psi$ is stably embedded in $\Gamma$.