# A polyhedral characterization of quasi-ordinary singularities

@article{Mourtada2015APC,
title={A polyhedral characterization of quasi-ordinary singularities},
author={Hussein Mourtada and Bernd Schober},
journal={arXiv: Algebraic Geometry},
year={2015}
}
• Published 2015
• Mathematics
• arXiv: Algebraic Geometry
Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) 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… Expand
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#### References

SHOWING 1-10 OF 40 REFERENCES
THE SEMIGROUP OF A QUASI-ORDINARY HYPERSURFACE
An analytically irreducible hypersurface germ (S, 0) ⊂ (C d+1 , 0) is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial f ∈ C{X}(Y ) of a fractional power series in theExpand
Quasi-Ordinary Singularities: Tree Model, Discriminant, and Irreducibility
• Mathematics
• 2015
Let $f(Y)\in K[[X_1,\dots,X_d]][Y]$ be a quasi-ordinary Weierstrass polynomial with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. In thisExpand
Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
Summary.This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spacesExpand
Toric embedded resolutions of quasi-ordinary hypersurface singularities
A germ of a complex analytic variety is quasi-ordinary if there exists a finite projection to the complex affine space with discriminant locus contained in a normal crossing divisor. Some propertiesExpand
Decomposition in bunches of the critical locus of a quasi-ordinary map
A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P , with respect toExpand
Canonical Resolution of a Quasi-ordinary Surface Singularity
• Mathematics
• Canadian Journal of Mathematics
• 2000
Abstract We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $\left( V,\,p \right)$ which results from applying the canonical resolution of Bierstone-Milman toExpand
Quasi-ordinary singularities and Newton trees
• Mathematics
• 2012
In this paper we study some properties of the class of nu-quasi-ordinary hypersurface singularities. They are defined by a very mild condition on its (projected) Newton polygon. We associate withExpand
Jet schemes and generating sequences of divisorial valuations in dimension two
Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recoverExpand
The Abhyankar-Jung Theorem
• Mathematics
• 2011
We show that every quasi-ordinary Weierstrass polynomial $P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z]$, $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero $\K$,Expand
Characteristic polyhedra of singularities without completion
• Mathematics
• 2012
In Hironaka (J. Math. Kyoto Univ. 7(3):251–293, 1967) defines characteristic polyhedra of a singularity embedded in Spec(R), where R is a regular local ring. Unfortunately, to build these polyhedra,Expand