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arXiv · 2208.06493

On real center singularities of complex vector fields on surfaces

V. León·B. Scárdua
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Abstract

One of the various versions of the classical Lyapunov-Poincar\'e center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincar\'e-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

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V. León, B. Scárdua. 2022-08-12. On real center singularities of complex vector fields on surfaces. https://arxiv.org/abs/2208.06493

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