Summary: Towards a Model Theory for Transseries
Matthias Aschenbrenner, Lou van den Dries,
and Joris van der Hoeven
For Anand Pillay, on his 60th birthday.
Abstract The differential field of transseries extends the field of real
Laurent series, and occurs in various context: asymptotic expansions,
analytic vector fields, o-minimal structures, to name a few. We give an
overview of the algebraic and model-theoretic aspects of this differential
field, and report on our efforts to understand its first-order theory.
We shall describe a fascinating mathematical object, the differential field T
of transseries. It is an ordered field extension of R and is a kind of universal
domain for asymptotic real differential algebra. In the context of this paper,
a transseries is what is called a logarithmic-exponential series or LE-series
in . Here is the main problem that we have been pursuing, intermittently,
for more than 15 years.
Conjecture. The theory of the ordered differential field T is model complete,
and is the model companion of the theory of H-fields with small derivation.
With slow progress during many years, our understanding of the situation
has recently increased at a faster rate, and this is what we want to report