On the polynomial vector fields on

Source abstract

Let X be a polynomial vector field of degree n on M, M = ℝ m . The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝ m , 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.