Rotation number for a homogeneous vector field in ${\mathbf{C}}^2$

In this paper, we use affine surfaces to describe completely the real-time trajectories of a homogeneous vector field in ${\mathbf{C}}^2$. We prove the existence of a continuous “rotation number” in ${\mathbf{C}}^2$ which is constant along real-time trajectories. A description of the trajectories depending on their rotation number is given.