Explicit reconstruction of homogeneous isolated hypersurface singularities from their milnor algebras

By the Mather-Yau theorem, a complex hypersurface germ V with isolated singularity is completely determined by its moduli algebra A(V). The proof of the 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 this paper, we present an explicit way 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.