Normal forms for singularities of one dimensional holomorphic vector fields

Abstract

Abstract: We study the normal form of the ordinary differential equation $dot z=f(z)$, $zinmathbb{C}$, in a neighbourhood of a point $pinmathbb{C}$, where $f$ is a one-dimensional holomorphic function in a punctured neighbourhood of $p$. Our results include all cases except when $p$ is an essential singularity. We treat all the other situations, namely when $p$ is a regular point, a pole or a zero of order $n$. Our approach is based on a formula that uses the flow associated with the differential equation to search for the change of variables that gives the normal form.

Keywords

• Meromorphic vector field
• holomorphic vector field
• normal form.