2015/12/23 by Mourtada, Hussein, Schober, Bernd · 2 citations
#13F25 #14B05 #14E15 #32S05 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1512.07507
Given an irreducible hypersurface singularity of dimension d (defined by a polynomial f∈ K[[ \bf x ]][z]) and the projection to the affine space defined by K[[ \bf 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; moreover, the construction yields the approximate roots, giving a new point of view on this subject.