1. Quasi-analytic solutions of analytic ordinary differential equations and o-minimal structures.
- Author
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J.-P. Rolin, F. Sanz, and R. Schäfke
- Subjects
QUASIANALYTIC functions ,FOLIATIONS (Mathematics) ,VECTOR fields ,DIFFERENTIAL equations ,ASYMPTOTIC theory of algebraic ideals ,MINIMAL surfaces ,NUMERICAL solutions to differential equations - 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 of whether 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 ℝ5 that is not definable in any o-minimal extension of ℝ. [ABSTRACT FROM AUTHOR]
- Published
- 2007
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