A tale of two Liouville closures

Allen Gehret

Vol. 290 (2017), No. 1, 41–76

DOI: 10.2140/pjm.2017.290.41

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 (2002) proved 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 $\lambda$-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 $\lambda$-freeness is preserved. In the proofs we introduce a new technique for studying $H$-fields, the yardstick argument which involves the rate of growth of pseudoconvergence.

Keywords

$H$-fields, asymptotic fields, asymptotic couples, differential-valued fields, Liouville extensions, Liouville closures, $\lambda$-freeness

Mathematical Subject Classification 2010

Primary: 12H05, 12J10

Secondary: 06F20, 12J15, 26A12

Milestones

Received: 12 December 2016

Revised: 14 February 2017

Accepted: 14 February 2017

Published: 7 July 2017