Two-Dimensional Homogeneous Polynomial Vector Fields with Common Factors

Invariant characterizations are obtained for the existence of one or more common factors in two-dimensional homogeneous polynomial vector fields of arbitrary degree, r. The presence of a common factor is related to the existence of nonisolated critical points of the vector fields. The particular case where the highest common factor is of maximal degree, r, is studied further, from the invariant point of view. The linear (r = 1) and quadratic (r = 2) cases are then examined in the context of the general theory, and the results are contrasted with those which are obtained using more conventional approaches. The relevance of the investigation to certain inhomogeneous polynomial vector fields is briefly discussed. This brings to light an invariant characterization of the classical Jacobi differential equation.