Classification of complex polynomial vector fields in one complex variable

Abstract

This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur ℂ. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields in the same combinatorial class have flows that are quasi-conformally equivalent.