THE SEMIGROUP OF A QUASI-ORDINARY HYPERSURFACE

Abstract

An analytically irreducible hypersurface germ $(S,0)\subset(\bm{C}^{d+1},0)$ is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial $f\in\bm{C}\{X\}[Y]$ of a fractional power series in the variables $X=(X_1,\dots,X_d)$ which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case ($d=1$). We prove that the set of vertices of Newton polyhedra of resultants of $f$ and $h$ with respect to the indeterminate $Y$, for those polynomials $h$ which are not divisible by $f$, is a semigroup of rank $d$, generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial $f$ are irreducible quasi-ordinary polynomials and that, together with the coordinates $X_1,\dots,X_d$, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of $(S,0)$ are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ $(S,0)$ as characterized by the work of Gau and Lipman.

AMS 2000 Mathematics subject classification: Primary 14M25; 32S25