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

Abstract : 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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Contributor : Reinhard Schäfke <>
Submitted on : Tuesday, January 16, 2007 - 3:43:05 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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Jean-Philippe Rolin, Fernando Sanz, Reinhard Schäfke. Quasi-analytic solutions of differential equations and o-minimal structures. Proceedings of the London Mathematical Society, London Mathematical Society, 2007, 95 (2), pp.413-442. ⟨10.1112/plms/pdm016⟩. ⟨hal-00124875⟩

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