# The precontraction group of the field of logarithmic transseries $\mathbb{T}_{\log}$

## TL;DR

This paper investigates the precontraction group structure induced by the logarithm in the field of logarithmic transseries, establishing its model completeness and characterizing definable subsets.

## Contribution

It provides the first detailed analysis of the theory of the precontraction group associated with $ ext{T}_{ ext{log}}$, including model completeness and definability results.

## Key findings

- The theory of $(
abla_{ ext{log}}, ext{chi})$ is model complete.
- The theory of $(
abla_{ ext{log}}, ext{chi})$ is complete.
- All definable subsets of $ ext{chi}(
abla_{ ext{log}})$ are characterized.

## Abstract

As a first step to understand the theory of the structure $\mathbb{T}_{\log}$ of logarithmic transseries as an ordered valued logarithmic field, we focus on the map $\chi$ induced by the logarithm of $\mathbb{T}_{\log}$ in its value group $\Gamma_{\log}$ and study the theory of the precontraction group $(\Gamma_{\log},\chi)$. Particularly, we show that this theory is model complete and complete, and we characterize all definable subsets of the discrete set $\chi(\Gamma_{\log})$.