# Moscow Mathematical Journal

Volume 18, Issue 4, October–December 2018 pp. 755–785.

A Polyhedral Characterization of Quasi-Ordinary Singularities

**Authors**:
Hussein Mourtada (1) and Bernd Schober (2)

**Author institution:**(1) Institut Mathématique de Jussieu-Paris Rive Gauche, Université Paris 7, B&acaron;timent Sophie Germain, case 7012, 75205 Paris Cedex 13, France

(2) Johannes Gutenberg-Universität Mainz, Fachbereich 08, Staudingerweg 9, 55099 Mainz, Germany

**Summary: **

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

2010 Math. Subj. Class. 14B05, 32S05, 13F25, 14E15.

**Keywords:**Quasi-ordinary singularities, characteristic polyhedron, overweight deformations.

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