On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov\u2019s Theorem

Martin Klimeš,Christiane Rousseau

doi:10.1007/s12346-020-00416-y

Martin Klimeš, Christiane Rousseau

Open Access

https://doi.org/10.1007/s12346-020-00416-y

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Abstract

In this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the {mathcal {C}}^infty case, where we show that only versality is possible.

Highlights

Let us consider a singular point of a germ of analytic vector field X on (C, 0)
The first normal form is more frequent in the older works of the Russian school
When the singular point is parabolic, Kostov proved that the following standard deformation of (1.1) is versal [7]: X1(x, y) =

Summary

Introduction

Let us consider a singular point of a germ of analytic vector field X on (C, 0). If the singular point is simple, the germ of vector field is analytically linearizable.

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