Topological Types of Quasi-Ordinary Singularities

Abstract

A germ $(X,x)$ of a complex analytic hypersurface in ${\mathbb {C}^{d + 1}}$ is quasi-ordinary if it can be represented as the image of an open neighborhood of $0$ in ${\mathbb {C}^d}$ under the map $({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0$, where $\zeta$ is a convergent power series. It is shown that the topological type of the singularity $(X,x) \subset ({\mathbb {C}^{d + 1}},0)$ is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series $\zeta (t_1^{1/n}, \ldots ,t_d^{1/n})$.