Newton vector fields on the plane and on the torus

Abstract

In this article, we show that complex vector fields on the punctured Riemann sphere are Newton vector fields and as a consequence all meromorphic vector fields and all elliptic vector fields are Newton vector fields. Moreover, for a large class of vector fields, which includes all elliptic vector fields, the proof is constructive in the sense that one can explicitly construct the function characterizing the Newton vector field.