A polyhedral characterization of quasi-ordinary singularities

Let $ X $ be an irreducible hypersurface given by a polynomial $ f \in K[[x_1, \ldots, x_d]][z] $, where $ K $ denotes an algebraically closed field of characteristic zero. The variety $ X $ is called quasi-ordinary with respect to the projection to the affine space defined by $ K[[x_1, \ldots, x_d]] $ if the discriminant of $ f $ is a monomial times a unit. In my talk I am going to present the construction of an invariant that allows to detect whether a given polynomial $ f $ (with fixed projection) defines a quasi-ordinary singularity. This involves a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity in affine spaces of higher dimensions. Further, I will explain how the construction permits to view $ X $ as an "overweight deformation" of a toric variety which leads then to the proof of our characterization.