76 research outputs found
Towards a Model Theory for Transseries
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.Comment: Notre Dame J. Form. Log., to appear; 33 p
Dimension in the realm of transseries
Let be the differential field of transseries. We establish some
basic properties of the dimension of a definable subset of ,
also in relation to its codimension in the ambient space . The
case of dimension is of special interest, and can be characterized both in
topological terms (discreteness) and in terms of the
Herwig-Hrushovski-Macpherson notion of co-analyzability. The proofs use results
by the authors from "Asymptotic Differential Algebra and Model Theory of
Transseries", the axiomatic framework for "dimension" in [L. van den Dries,
"Dimension of definable sets, algebraic boundedness and Henselian fields", Ann.
Pure Appl. Logic 45 (1989), no. 2, 189-209], and facts about co-analyzability
from [B. Herwig, E. Hrushovski, D. Macpherson, "Interpretable groups, stably
embedded sets, and Vaughtian pairs", J. London Math. Soc. (2003) 68, no. 1,
1-11].Comment: 16 pp; version 2, taking into account comments by the refere
On the Pila-Wilkie theorem
This expository paper gives an account of the Pila-Wilkie counting theorem
and some of its extensions and generalizations. We use semialgebraic cell
decomposition to simplify part of the original proof. Included are complete
treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-
Gromov theorem that are used in this proof.Comment: 44 page
- …