The Semigroup of a Quasi-ordinary Hypersurface

Abstract

An analytically irreducible hypersurface germ (S, 0) ⊂ (Cd+1, 0) is quasi-ordinary if it canbe defined by the vanishing of the minimal polynomial f ∈ C{X}[Y ] of a fractional power series in the variables X = (X1, . . . , Xd) 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 quasiordinary polynomials and that, together with the coordinates X1, . . . , Xd, 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.