On the Zil’ber Principle for Structures Definable in O-minimal Models

S. Starchenko

Let M be a fixed o-minimal model, and N - a structure definable in M (i.e. the universe of N is a definable subset of M and all basic relations and functions of N are definable in M).

A general question one can ask is whether the Zil’ber principle holds for such N. In this talk we will discuss mainly the following part of the Zil’ber principle (so called “purity”):

If M is an expansion of a field then every M-definable set is definable in the pure field language.

This part of the Zil’ber principle, does not hold in general. For example, ℝ_{an}, the expansion of the field of reals by restricted analytic functions, is a proper expansion of the field of reals.

However we will show that “purity” holds under extra assumption that M is stable. From this one can derive another proof of Chow’s theorem and also the “purity” principle for the category of

compact analytic complex manifolds.

Address: Department of Mathematics, University of Notre Dame, USA

E-mail: starchenko.1@nd.edu