Summary: ASYMPTOTIC DIFFERENTIAL ALGEBRA
MATTHIAS ASCHENBRENNER AND LOU VAN DEN DRIES
Abstract. We believe there is room for a subject named as in the title of
this paper. Motivating examples are Hardy fields and fields of transseries.
Assuming no previous knowledge of these notions, we introduce both, state
some of their basic properties, and explain connections to o-minimal structures.
We describe a common algebraic framework for these examples: the category
of H-fields. This unified setting leads to a better understanding of Hardy fields
and transseries from an algebraic and model-theoretic perspective.
1. Hardy Fields 4
2. The Field of Logarithmic-Exponential Series 14
3. H-Fields and Asymptotic Couples 21
4. Algebraic Differential Equations over H-Fields 28
In taking asymptotic expansions `a la PoincarŽe we deliberately neglect transfinitely
small terms. For example, with f(x) := 1
1-x-1 + x- log x