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.

Keywords:

• meromorphic and elliptic Newton flows
• pullback vector fields
• branched covering maps
• Mittag–Leffler expansion
• Weirstrass ℘-function

AMS Subject Classification::

• 37F75
• 32M25
• 37F10
• 33E05
• 30D30
• 33E20

Acknowledgements

The first author would like to thank Professor Jesús Muciño-Raymundo for introducing him to the study of complex vector fields and for many interesting, insightful and productive discussions.

The other two authors wish to acknowledge and thank the first author for the opportunity to work, under his guidance, on their undergraduate thesis which led to some of the results of this article.

This work was made possible in part by UABC grant 0196.

Notes

Notes

1. In fact, Benzinger studies the solutions to the differential equation ż = R(z), where R is any rational function, but this is equivalent to studying the rational vector fields R(z).

2. The choice of z ₀ ∈ U is determined by the function f (z) in order to satisfy (3).

3. If 0 is not a pole then G ₀(z) = 0.