# MAA

## Seminar

### Quasi-analytic solutions of differential equations and o-minimal structures

Seminar Room 1, Newton Institute

#### Abstract

We present the results of a joint work with F. Sanz and R. Schaefke. Consider a non-oscillating trajectory of real analytic vector field. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in that is not definable in any o-minimal extension of the reals.