A polyhedral characterization of quasi-ordinary singularities

Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ \mathbf{x} ]][z]$) and the projection to the affine space defined by $K[[ \mathbf{x} ]]$, 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}$ and $ f $ is complex analytic; moreover, we explain the relation between the construction and the approximate roots.