Dimension in the realm of transseries

TL;DR

This paper investigates the properties of the dimension of definable sets within the differential field of transseries, linking topological and model-theoretic notions, especially for zero-dimensional sets.

Contribution

It establishes foundational properties of dimension in transseries, connecting topological, algebraic, and model-theoretic perspectives, and characterizes zero-dimensional sets.

Findings

Dimension relates to discreteness and co-analyzability.

Zero-dimensional sets are characterized topologically and model-theoretically.

The paper extends existing dimension theory to the context of transseries.

Abstract

Let $\mathbb T$ be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of ${\mathbb T}^n$, also in relation to its codimension in the ambient space ${\mathbb T}^n$. The case of dimension $0$ is of special interest, and can be characterized both in topological terms (discreteness) and in terms of the Herwig-Hrushovski-Macpherson notion of co-analyzability. The proofs use results by the authors from "Asymptotic Differential Algebra and Model Theory of Transseries", the axiomatic framework for "dimension" in [L. van den Dries, "Dimension of definable sets, algebraic boundedness and Henselian fields", Ann. Pure Appl. Logic 45 (1989), no. 2, 189-209], and facts about co-analyzability from [B. Herwig, E. Hrushovski, D. Macpherson, "Interpretable groups, stably embedded sets, and Vaughtian pairs", J. London Math. Soc. (2003) 68, no. 1, 1-11].