Algebraic and topological classification of homogeneous quartic vector fields in the plane (Q2118005)

The authors consider the problem of the algebraic and topological classification of the following planar homogeneous quartic system: \[ \dot{x}=P(x, y),\quad \dot{y}=Q(x, y), \] where \(P\) and \(Q\) are homogeneous polynomials of degree four. They first provide all of the possible canonical forms of homogeneous polynomials of degree five. Then they present all the possible phase portraits in the Poincaré disk of the above quartic system. They prove that the system has exactly 23 different topological phase portraits.