Functions holomorphic along holomorphic vector fields

TL;DR

This paper generalizes Forelli's theorem by showing that functions holomorphic along certain integral curves of a linearizable holomorphic vector field with positive real eigenvalue ratios are actually holomorphic near the singularity.

Contribution

It extends Forelli's theorem to a broader class of vector fields with specific eigenvalue ratio conditions, including a necessary example.

Findings

Functions with asymptotic Taylor expansions are holomorphic near the singularity.

The eigenvalue ratio condition is necessary for the generalization.

Provides an example illustrating the necessity of the eigenvalue ratio condition.

Abstract

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $\phi$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.