Summary: Sel. math., New ser. Online First
c 2005 Birkh¨auser Verlag Basel/Switzerland
Differentially algebraic gaps
Matthias Aschenbrenner, Lou van den Dries and
Joris van der Hoeven
Abstract. H-fields are ordered differential fields that capture some basic prop-
erties of Hardy fields and fields of transseries. Each H-field is equipped with
a convex valuation, and solving first-order linear differential equations in H-
field extensions is strongly affected by the presence of a "gap" in the value
group. We construct a real closed H-field that solves every first-order linear
differential equation, and that has a differentially algebraic H-field extension
with a gap. This answers a question raised in . The key is a combinatorial
fact about the support of transseries obtained from iterated logarithms by
algebraic operations, integration, and exponentiation.
Mathematics Subject Classification (2000). Primary 03C64, 16W60; Secondary
Keywords. H-fields, fields of transseries.