 
Summary: Closed Asymptotic Couples
Matthias Aschenbrenner and Lou van den Dries*
University of Illinois at UrbanaChampaign, Department of Mathematics
Urbana, IL 61801, U.S.A.
Email: maschenb@math.uiuc.edu; vddries@math.uiuc.edu
Communicated by Leonard Lipshitz
The derivation of a Hardy field induces on its value group a certain function . If a
Hardy field extends the real field and is closed under powers, then its value group is also a
vector space over R. Such "ordered vector spaces with function" are called Hcouples.
We define closed Hcouples and show that every Hcouple can be embedded into a closed
one. The key fact is that closed Hcouples have an elimination theory: solvability of an
arbitrary system of equations and inequalities (built up from vector space operations, the
function , parameters, and the unknowns to be solved for) is equivalent to an effective
condition on the parameters of the system. The Hcouple of a maximal Hardy field is
closed, and this is also the case for the Hcouple of the field of logarithmicexponential
series over R. We analyse in detail finitely generated extensions of a given Hcouple.
INTRODUCTION
We describe here roughly the main result of the paper, and explain for nonexperts
the role of model theory in its conception. Precise formulationsfollow in section 1,
and sections 24 contain the proof of the main result.
