Revisiting closed asymptotic couples

TL;DR

This paper investigates the structure of closed asymptotic couples, showing that discrete definable subsets are contained within finite-dimensional linear subspaces, revealing new structural insights about the differential-valued field of transseries.

Contribution

It demonstrates that the value group of the transseries field has more definable structure than previously understood, especially in relation to its asymptotic couple and scalar multiplication.

Findings

Discrete definable subsets are contained in finite-dimensional linear subspaces.

The transseries field's value group has richer structure than its asymptotic couple.

Scalar multiplication by real numbers influences the structure of the asymptotic couple.

Abstract

Every discrete definable subset of a closed asymptotic couple with ordered scalar field $\boldsymbol k$ is shown to be contained in a finite-dimensional $\boldsymbol k$-linear subspace of that couple. It follows that the differential-valued field $\mathbb T$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb R$ as the underlying set for the second sort.