A tale of two Liouville closures

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.

arXiv comment25 pp; updated according to suggestions by the reviewer; to appear in Pacific J. Math

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