On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem

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 𝒞^∞ case, where we show that only versality is possible.