The Asymptotic Couple of the Field of Logarithmic Transseries

TL;DR

This paper studies the structure of the value group associated with logarithmic transseries, proving quantifier elimination and describing definable functions, thereby advancing the understanding of asymptotic couples in differential-valued fields.

Contribution

It establishes quantifier elimination for the theory of the asymptotic couple of logarithmic transseries and characterizes definable functions on a key discrete subset.

Findings

Quantifier elimination in the theory $T_{log}$.

Explicit descriptions of definable functions on $ ext{ extbackslash Psi}$.

Stable embedding of $ ext{ extbackslash Psi}$ in $ ext{ extbackslash Gamma}$.

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 $(\Gamma_{\log},\psi)$ is a divisible asymptotic couple. We prove that the theory $T_{\log} = {\rm Th}(\Gamma_{\log},\psi)$ admits elimination of quantifiers in a natural first-order language. All models $(\Gamma,\psi)$ of $T_{\log}$ have an important discrete subset $\Psi:=\psi(\Gamma\setminus\{0\})$. We give explicit descriptions of all definable functions on $\Psi$ and prove that $\Psi$ is stably embedded in $\Gamma$.