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Transactions of the American Mathematical Society

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

Authors: Olivier Le Gal, Fernando Sanz and Patrick Speissegger
Journal: Trans. Amer. Math. Soc. 370 (2018), 2211-2229
MSC (2010): Primary 34C08, 03C64, 34M30
Published electronically: November 1, 2017
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Abstract: Let $ \xi $ be an analytic vector field at $ (\mathbb{R}^3,0)$ and $ \mathcal {I}$ be an analytically non-oscillatory integral pencil of $ \xi $; i.e., $ \mathcal {I}$ is a maximal family of analytically non-oscillatory trajectories of $ \xi $ at 0 all sharing the same iterated tangents. We prove that if $ \mathcal {I}$ is interlaced, then for any trajectory $ \Gamma \in \mathcal {I}$, the expansion $ \mathbb{R}_{\textup {an},\Gamma }$ of the structure $ \mathbb{R}_{\textup {an}}$ by $ \Gamma $ is model-complete, o-minimal and polynomially bounded.

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Additional Information

Olivier Le Gal
Affiliation: Université de Savoie, Laboratoire de Mathématiques, Bâtiment Chablais, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France

Fernando Sanz
Affiliation: Universidad de Valladolid, Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Campus Miguel Delibes, Paseo de Belén, 7, E-47011 Valladolid, Spain

Patrick Speissegger
Affiliation: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

Keywords: Ordinary differential equations, o-minimal structures, multisummable series, Stokes phenomena
Received by editor(s): October 9, 2013
Received by editor(s) in revised form: January 16, 2017
Published electronically: November 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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