2005/05/04 by Jean-Philippe Rolin, Rolin, J. -P., Fernando Sanz +3
Mathematics · #03C64 #30D60 #34M30 #34M40 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.math/0505073
openalex publication_date 2005/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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