# 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 fK[[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.