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 = $\mathbb{C}$;$ moreover, the construction yields the approximate roots, giving a new point of view on this subject.