ON TWO METHODS FOR RECONSTRUCTING HOMOGENEOUS HYPERSURFACE SINGULARITIES FROM THEIR MILNOR ALGEBRAS

By the well-known Mather-Yau theorem, a complex hypersurface germ V with isolated singularity is fully determined by its moduli algebra A(V). The proof of this theorem does not provide an explicit procedure for recovering V from 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 V from A(V) up to biholomorphic equivalence under the assumption that the singularity of V is homogeneous (in which case A(V) coincides with the Milnor algebra of V). 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.