On the polynomial vector fields on

Abstract

Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.