Open Access
2014 Briot-Bouquet's theorem in high dimension
S. A. Carrillo, F. Sanz
Publ. Mat. 58(S1): 135-152 (2014).


Let $X$ be a germ of holomorphic vector field at $0\in\mathbb{C}^n$ and let $E$ be a linear subspace of $\mathbb C^n$ which is invariant for the linear part of $X$ at $0$. We give a sufficient condition that imply the existence of a non-singular invariant manifold tangent to $E$ at $0$. It generalizes to higher dimensions the conditions in the classical Briot--Bouquet's Theorem: roughly speaking, we impose that the convex hull of the eigenvalues $\mu_i$ corresponding to $E$ does not contain $0$ and there are no resonances between the $\mu_i$ and the complementary eigenvalues. As an application, we propose an elementary proof of the analyticity of the local stable and unstable manifolds of a real analytic vector field at a singular point.


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S. A. Carrillo. F. Sanz. "Briot-Bouquet's theorem in high dimension." Publ. Mat. 58 (S1) 135 - 152, 2014.


Published: 2014
First available in Project Euclid: 19 May 2014

zbMATH: 1365.37029
MathSciNet: MR3211830

Primary: 34C45
Secondary: 32S65 , 34A20 , 37D10 , 37F75

Keywords: Briot-Bouquet's theorem , formal and convergent invariant manifolds , Holomorphic vector fields , non-resonance , stable analytic manifold

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. S1 • 2014
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