A tale of two Liouville closures

TL;DR

This paper investigates the conditions determining whether an $H$-field has one or two Liouville closures, introducing the concept of $$-freeness and a novel yardstick technique involving growth rates of pseudoconvergence.

Contribution

It establishes that $$-freeness characterizes the dividing line for Liouville closures and introduces the yardstick argument for analyzing $H$-fields.

Findings

$$-freeness determines the number of Liouville closures.

$$-freeness is preserved under certain extensions.

The yardstick argument provides a new method for studying growth in $H$-fields.

Abstract

An $H$-field is a type of ordered valued differential field with a natural interaction between ordering, valuation, and derivation. The main examples are Hardy fields and fields of transseries. Aschenbrenner and van den Dries proved in~\cite{MZ} that every $H$-field $K$ has either exactly one or exactly two Liouville closures up to isomorphism over $K$, but the precise dividing line between these two cases was unknown. We prove here that this dividing line is determined by $\uplambda$-freeness, a property of $H$-fields that prevents certain deviant behavior. In particular, we show that under certain types of extensions related to adjoining integrals and exponential integrals, the property of $\uplambda$-freeness is preserved. In the proofs we introduce a new technique for studying $H$-fields, the \emph{yardstick argument} which involves the rate of growth of pseudoconvergence.