Solvability of Planar Complex Vector Fields with Homogeneous Singularities

TL;DR

This paper investigates the solvability of planar complex vector fields with homogeneous singularities, linking solution properties to number theoretic aspects of associated complex numbers.

Contribution

It introduces a novel analysis connecting the solvability of the equation $Lu=f$ to the number theoretic properties of complex parameters of the vector field.

Findings

Solution properties depend on number theoretic characteristics of complex parameters

Established criteria for solvability based on these properties

Enhanced understanding of singularities in complex vector fields

Abstract

In this paper we study the equation $Lu=f$, where $L$ is a $\C$-valued vector field in $\R^2$ with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers attached to the vector field.