Quasi-analytic solutions of analytic ordinary 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 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.

Quantifier elimination and rectilinearization theorem for generalized quasianalytic algebras

“…• R an,H , where F is a collection of functions containing all real analytic functions restricted to the unit cube and a solution H of a first order analytic differential equation which is singular at the origin (see [RSS07]). …”

“…(4) The structure R an,H , where F is a collection of functions containing all real analytic functions restricted to the unit cube and a solution H of a first-order analytic differential equation which is singular at the origin (see [20]). (5) The structure R Q , in which certain Dulac transition maps are definable (see [13] for a complete survey on Dulac's problem and [14] for the proof of model-completeness and ominimality).…”

Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes

“…-The Main Theorem applies to all the classes mentioned in the introduction, where the morphism T is the Taylor expansion at zero in cases a), b) and d), the identity in case c) and the asymptotic expansion f → T (f ) in case e). In fact, quasianalyticity is tautological in case c), it is proven in [16] in case b) and it follows by classical theorems in cases a), d) and e) (see [19,20,10]). Moreover, the closure and compatibility conditions in 2.…”

“…A function f , defined on an open set U ⊆ R m , is said to be A H -analytic if for every a ∈ U there exists a germ ϕ a (x) ∈ A H such that the germ of f (x) at a is equal to the germ ϕ a (x − a). It is proven in [16] that the collection of all A H -analytic functions forms a quasianalytic class of C ∞ functions. [6]).…”

“…Suppose furthermore that H admits an asymptotic expansion for x → 0 + as in [16, 2.2]. As in [16,Section 3], we let A H be the smallest collections of real…”