On two methods for reconstructing homogeneous hypersurface singularities from their Milnor algebras

By the well-known Mather-Yau theorem, a complex hypersurface germ VV with isolated singularity is fully determined by its moduli algebra A(V)A(V). The proof of this theorem does not provide an explicit procedure for recovering VV from A(V)A(V), and finding such a procedure is a long-standing open problem. In the present paper we survey and compare two recently proposed methods for reconstructing VV from A(V)A(V) up to biholomorphic equivalence under the assumption that the singularity of VV is homogeneous (in which case A(V)A(V) coincides with the Milnor algebra of VV). As part of our discussion of one of the methods, we give a characterization of the algebras arising from finite polynomial maps with homogeneous components of equal degrees