Distality for the asymptotic couple of the field of logarithmic transseries

TL;DR

This paper proves that the theory of the asymptotic couple of the field of logarithmic transseries is distal, providing new insights into its model-theoretic properties and its classification within NIP theories.

Contribution

It establishes the distality of the theory $T_{log}$, showing it is NIP but not strongly dependent, and clarifies its position in the hierarchy of model-theoretic properties.

Findings

$T_{log}$ is distal.

$T_{log}$ is NIP.

$T_{log}$ is not strongly dependent.

Abstract

We show that the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (= the non-independence property), this provides a new proof that $T_{\log}$ is NIP. Finally, we show that $T_{\log}$ is not strongly dependent, and in particular, it is not $\operatorname{dp}$-minimal and it does not have finite $\operatorname{dp}$-rank.