NIP for 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 $\Gamma = (\Gamma_{\log},\psi)$ is a divisible asymptotic couple. In~\cite{gehret} we began a study of the first-order theory of $(\Gamma_{\log},\psi)$ where, among other things, we proved that the theory $T_{\log} = \operatorname{Th}(\Gamma_{\log},\psi)$ 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 $\Gamma\langle\alpha\rangle$ of $\Gamma$. 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}.