Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Let V, Ṽ be hypersurface germs in ℂᵐ , each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for V, V~V~ reduces to the linear equivalence problem for certain polynomials P, P~ arising from the moduli algebras of V, Ṽ. The polynomials P, P~ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for V, Ṽ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.The research is supported by the Australian Research Council. The fourth author is supported by the Russian Foundation for Basic Research and grant no. NSh-3476.2010.1 of the Leading Scientific Schools program