A TALE OF TWO LIOUVILLE CLOSURES

作者：Gehret Allen^*

来源：Pacific Journal of Mathematics, 2017, 290(1): 41-76.

DOI：10.2140/pjm.2017.290.41

• H-fields
• asymptotic fields
• asymptotic couples
• differential-valued fields
• Liouville extensions
• Liouville closures
• lambda-freeness

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.

• 出版日期2017-9