  We give sharp lower bounds for the degree of the syzygies involving the
partial derivatives of a homogeneous polynomial defining a nodal hypersurface.
The result gives information on the position of the singularities of a nodal
hypersurface expressed in terms of defects or superabundances.
  The case of Chebyshev hypersurfaces is considered as a test for this result
and leads to a potentially infinite family of nodal hypersurfaces having
nontrivial Alexander polynomials.
