Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal Journal Articles

Let ξ \xi be an analytic vector field at ( R 3 , 0 ) (\mathbb {R}^3,0) and I \mathcal {I} be an analytically non-oscillatory integral pencil of ξ \xi ; i.e., I \mathcal {I} is a maximal family of analytically non-oscillatory trajectories of ξ \xi at 0 all sharing the same iterated tangents. We prove that if I \mathcal {I} is interlaced, then for any trajectory Γ ∈ I \Gamma \in \mathcal {I} , the expansion R a n , Γ \mathbb {R}_{\mathrm {an},\Gamma } of the structure R a n \mathbb {R}_{\mathrm {an}} by Γ \Gamma is model-complete, o-minimal and polynomially bounded.