Paper detail

NIP for 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 $Γ_{\log}$ a certain map $ψ$. The structure $Γ= (Γ_{\log},ψ)$ is a divisible asymptotic couple. In~\cite{gehret} we began a study of the first-order theory of $(Γ_{\log},ψ)$ where, among other things, we proved that the theory $T_{\log} = \operatorname{Th}(Γ_{\log},ψ)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether $T_{\log}$ has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: $T_{\log}$ does have NIP. Our method of proof relies on a complete survey of the $1$-types of $T_{\log}$, which, in the presence of QE, is equivalent to a characterization of all simple extensions $Γ\langleα\rangle$ of $Γ$. We also show that $T_{\log}$ does not have the Steinitz exchange property and we weigh in on the relationship between models of $T_{\log}$ and the so-called \emph{precontraction groups} of~\cite{kuhlmann1}.