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

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