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

On the syzygies and Alexander polynomials of nodal hypersurfaces