Quasianalytic solutions of differential equations and o-minimal structures

Details Quasianalytic solutions of differential equations and o-minimal structures Quasianalytic solutions of differential equations and o-minimal structures

2005-05-04

arxiv.org/abs/math/0505073v2

Mathematics

Classical Analysis and ODEs

33 pages

Scientific paper

It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real analytic vector fields. 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 dimension 5 that is not definable in any o-minimal extension of the reals

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