Truncation In Unions of Hahn Fields with a Derivation

TL;DR

This paper investigates conditions under which truncation properties in unions of Hahn fields are preserved when extending to differential rings and Liouville closures, with applications to logarithmic-exponential transseries.

Contribution

It introduces IL-closedness as a key condition ensuring truncation preservation in unions of Hahn fields during extensions.

Findings

IL-closedness guarantees truncation preservation in differential extensions.

Truncation is preserved in Liouville closures under IL-closedness.

Applicable to fields of logarithmic-exponential transseries.

Abstract

Truncation in Generalized Series fields is a robust notion, in the sense that it is preserved under various algebraic and some transcendental extensions. In this paper, we study conditions that ensure that a truncation closed set extends naturally to a truncation closed differential ring, and a truncation closed differential field has a truncation closed Liouville closure. In particular, we introduce the Notion of IL-closedness in Unions of Hahn fields in order to determine that this condition is sufficient to preserve truncation in those two settings for constructions such as the field of logarithmic-exponential transseries.