Dimension theory for the asymptotic couple of the field of logarithmic transseries

TL;DR

This paper characterizes all dimension functions on models of the theory of the asymptotic couple of logarithmic transseries, establishing d-minimality and analyzing definable sets.

Contribution

It provides a complete classification of dimension functions and characterizes small definable sets in the theory of the asymptotic couple of logarithmic transseries.

Findings

All dimension functions are characterized for models of T_log.

T_log is shown to be d-minimal and does not eliminate imaginaries.

A new criterion for d-minimality is introduced with applications to valued fields.

Abstract

In this paper we completely characterize all dimension functions on all models of the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the "small" $1$-variable definable sets (Small Sets Theorem). As a byproduct, we show that $T_{\log}$ is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.