Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Abstract

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.