  Let $V$ be a projective hypersurface having only isolated singularities. We
show that these singularities are weighted homogeneous if and only if the
Koszul syzygies among the partial derivatives of an equation for $V$ are
exactly the syzygies with a generic first component vanishing on the singular
locus subscheme of $V$. This yields in particular a positive answer in this
setting to a question raised by Morihiko Saito and the first author. Finally we
explain how our result can be used to improve the listing of Jacobian syzygies
of a given degree by a computer algebra system such as Singular, CoCoA or
Macaulay2.
