 
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 ominimal structures.
We describe a common algebraic framework for these examples: the category
of Hfields. This unified setting leads to a better understanding of Hardy fields
and transseries from an algebraic and modeltheoretic perspective.
Contents
Introduction 1
1. Hardy Fields 4
2. The Field of LogarithmicExponential Series 14
3. HFields and Asymptotic Couples 21
4. Algebraic Differential Equations over HFields 28
References 35
Introduction
In taking asymptotic expansions `a la PoincarŽe we deliberately neglect transfinitely
small terms. For example, with f(x) := 1
1x1 + x log x
