Truncation in Differential Hahn Fields

TL;DR

This paper investigates the properties of truncation in generalized series fields, especially in the context of derivations and extensions, with a focus on the logarithmic-exponential transseries field and its Liouville closures.

Contribution

It demonstrates the robustness of truncation-closedness under derivations and extensions, and establishes conditions under which Liouville closures preserve truncation-closedness in transseries fields.

Findings

Truncation-closedness is preserved under certain algebraic and transcendental extensions.

In generalized series fields with derivations, truncation-closedness remains robust.

Liouville closures of truncation closed differential subfields are also truncation closed under a splitting condition.

Abstract

Being closed under truncation for subsets of generalized series fields is a robust property in the sense that it is preserved under various algebraic and transcendental extension procedures. Nevertheless, in Chapter 4 of this dissertation, we show that generalized series fields with truncation as an extra primitive yields undecidability in several settings. Our main results, however, concern the robustness of being truncation closed in generalized series fields equipped with a derivation, and under extension procedures that involve this derivation. In the last chapter, we study this in the ambient field $\mathbb{T}$ of logarithmic-exponential transseries. It leads there to a theorem saying that under a natural `splitting' condition the Liouville closure of a truncation closed differential subfield of $\mathbb{T}$ is again truncation closed.