Topological types of quasi-ordinary singularities

Abstract

A germ ( X , x ) (X,x) of a complex analytic hypersurface in C d + 1 {\mathbb {C}^{d + 1}} is quasi-ordinary if it can be represented as the image of an open neighborhood of 0 0 in C d {\mathbb {C}^d} under the map ( s 1 , … , s d ) ↦ ( s 1 n , … , s d n , ζ ( s 1 , … , s d ) ) , n > 0 ({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0 , where ζ \zeta is a convergent power series. It is shown that the topological type of the singularity ( X , x ) ⊂ ( C d + 1 , 0 ) (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 ζ ( t 1 1 / n , … , t d 1 / n ) \zeta (t_1^{1/n}, \ldots ,t_d^{1/n}) .