A characterization of quasi-homogeneous singularities of free and nearly free plane curves

The goal of this paper is to establish a new and efficient characterization of quasi-homogeneous isolated singularities of free curves and nearly free curves $C$ in $\mathbb{P}^2$. The criterion will be in terms of the first syzygy matrix associated with the Jacobian ideal of $f$ where $f=0$ is the equation of the plane curve $C$.
Comment: 20 pages, 4 figures