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Quasi‐analytic solutions of analytic ordinary differential equations and o‐minimal structures
Author(s) -
Rolin J.P.,
Sanz F.,
Schäfke R.
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm016
Subject(s) - mathematics , ordinary differential equation , mathematical analysis , analytic function , pure mathematics , differential equation
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 ℝ 5 that is not definable in any o‐minimal extension of ℝ.