Summary: Closed Asymptotic Couples
Matthias Aschenbrenner and Lou van den Dries*
University of Illinois at Urbana-Champaign, Department of Mathematics
Urbana, IL 61801, U.S.A.
E-mail: email@example.com; firstname.lastname@example.org
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 H-couples.
We define closed H-couples and show that every H-couple can be embedded into a closed
one. The key fact is that closed H-couples 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 H-couple of a maximal Hardy field is
closed, and this is also the case for the H-couple of the field of logarithmic-exponential
series over R. We analyse in detail finitely generated extensions of a given H-couple.
We describe here roughly the main result of the paper, and explain for non-experts
the role of model theory in its conception. Precise formulationsfollow in section 1,
and sections 24 contain the proof of the main result.