The $GL(2,\mathbb R)$-orbits of the homogeneous polynomial differential systems

In this work, we study the generic homogeneous polynomial differential system $\dot{x}_1= P_k(x_1, x_2)$, $\dot{x}_2=Q_k(x_1,x_2)$ under the action of the center-affine group of transformations of the phase space, $GL(2,\mathbb R)$. We show that if the dimension of the $GL(2,\mathbb R)$-orbits of this system is smaller than four, then $deg(GCD(P_k,Q_k))\geq k-1$.