The Asymptotic Couple of the Field of Logarithmic Transseries

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