A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces

In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface $D=V(f)\subset \PP^n$ using the Jacobian syzygies of $f$. The characterization compares the ranks of the first syzygy matrices of the global Jacobian ideal $J_f$ and its quotient $J_f/(f)$. When $D$ has only isolated singularities, our characterization refines a recent result of Andrade-Beorchia-Dimca-Mir\'{o}-Roig. We also prove a generalization of this characterization to smooth projective toric varieties.