Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal
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- by Olivier Le Gal, Fernando Sanz and Patrick Speissegger PDF
- Trans. Amer. Math. Soc. 370 (2018), 2211-2229 Request permission
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 $\mathbb {R}_{\mathrm {an},\Gamma }$ of the structure $\mathbb {R}_{\mathrm {an}}$ by $\Gamma$ is model-complete, o-minimal and polynomially bounded.References
- Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
- Patrick Bonckaert and Freddy Dumortier, Smooth invariant curves for germs of vector fields in $\textbf {R}^3$ whose linear part generates a rotation, J. Differential Equations 62 (1986), no. 1, 95–116. MR 830049, DOI 10.1016/0022-0396(86)90107-5
- Felipe Cano, Robert Moussu, and Fernando Sanz, Pinceaux de courbes intégrales d’un champ de vecteurs analytique, Astérisque 297 (2004), 1–34 (French, with English and French summaries). Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. MR 2135673
- F. Cano, R. Moussu, and F. Sanz, Oscillation, spiralement, tourbillonnement, Comment. Math. Helv. 75 (2000), no. 2, 284–318 (French, with English and French summaries). MR 1774707, DOI 10.1007/s000140050127
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- M. Coste, An introduction to o-minimal structures, Dipartimento di Matematica, Università di Pisa, Int. Ed. e Poligr. Int. (2000).
- Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Jean-Marie Lion and Jean-Philippe Rolin, Volumes, feuilles de Rolle de feuilletages analytiques et théorème de Wilkie, Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 1, 93–112 (French, with English and French summaries). MR 1658452
- O. Le Gal, F. Sanz, and P. Speissegger, Non-interlaced solutions of 2-dimensional systems of linear ordinary differential equations, Proc. Amer. Math. Soc. 141 (2013), no. 7, 2429–2438. MR 3043024, DOI 10.1090/S0002-9939-2013-11614-X
- B. Malgrange and J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 353–368 (French, with English summary). MR 1162566
- J.-P. Ramis and Y. Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 3, 811–848 (English, with English and French summaries). MR 1303885
- J.-P. Rolin, F. Sanz, and R. Schäfke, Quasi-analytic solutions of analytic ordinary differential equations and o-minimal structures, Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 413–442. MR 2352566, DOI 10.1112/plms/pdm016
- F. Sanz, Course on non-oscillatory trajectories, Fields Institute Communications, vol. 62, Amer. Math. Soc., Providence, RI, 2012, pp. 111–177.
- Patrick Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211. MR 1676876, DOI 10.1515/crll.1999.026
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
- Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0203188
Additional Information
- Olivier Le Gal
- Affiliation: Université de Savoie, Laboratoire de Mathématiques, Bâtiment Chablais, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France
- MR Author ID: 831839
- Email: Olivier.Le-Gal@univ-savoie.fr
- 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
- MR Author ID: 623470
- Email: fsanz@agt.uva.es
- Patrick Speissegger
- Affiliation: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada
- MR Author ID: 361060
- Email: speisseg@math.mcmaster.ca
- Received by editor(s): October 9, 2013
- Received by editor(s) in revised form: January 16, 2017
- Published electronically: November 1, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2211-2229
- MSC (2010): Primary 34C08, 03C64, 34M30
- DOI: https://doi.org/10.1090/tran/7205
- MathSciNet review: 3739207