A polyhedral characterization of quasi-ordinary singularities

TL;DR

This paper introduces a new invariant based on a weighted polyhedron to identify quasi-ordinary hypersurface singularities and determine their embedded topology, providing a novel perspective on approximate roots.

Contribution

It constructs an invariant using a weighted Hironaka polyhedron to detect quasi-ordinary singularities and recover their semigroup and topology.

Findings

Invariant detects quasi-ordinary singularities.

Invariant determines the semigroup and embedded topology.

Provides a new approach to approximate roots.

Abstract

Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the singularity is quasi-ordinary with respect to the projection. The construction uses a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity in affine spaces of higher dimensions. When $ f $ is quasi-ordinary, our invariant determines the semigroup of the singularity and hence it encodes the embedded topology of the singularity $ \{ f = 0 \} $ in a neighbourhood of the origin when $ K = \mathbb{C};$ moreover, the construction yields the approximate roots, giving a new point of view on this subject.