Completeness is determined by any non-algebraic trajectory

TL;DR

This paper proves that for polynomial vector fields in two complex variables, completeness on any non-algebraic trajectory implies overall completeness of the vector field.

Contribution

It establishes a new criterion linking the completeness of polynomial vector fields to their behavior on non-algebraic trajectories.

Findings

Completeness on a non-algebraic trajectory implies overall completeness.

Polynomial vector fields in two complex variables are characterized by their trajectories.

The result bridges local trajectory properties with global vector field completeness.

Abstract

It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.