# 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} }

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

#### 7 Citations

Characteristic polyhedra of singularities without completion: part II

- Mathematics
- 2014

Hironaka’s characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the… Expand

Generalized Loose Edge Factorization Theorems

- Mathematics
- 2018

We extend a factorization theorem by Gwo\'zdziewicz and Hejmej from the ring of formal power series to any complete regular local ring $ R $. More precisely, let $ f \in R $ and assume that its… Expand

Singularities and Homological Aspects of Commutative Algebra

- Mathematics
- 2020

Commutative algebra has recently witnessed a number of spectacular developments, resulting in the resolution of long-standing problems. The new techniques and perspectives, such as methods from the… Expand

About the algebraic closure of the field of power series in several variables in characteristic zero

- Mathematics
- 2013

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice… Expand

An irreducibility criterion for power series

- Mathematics
- 2016

We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous known results concerning quasi-ordinary polynomials.

Idealistic exponents: Tangent cone, ridge, characteristic polyhedra

- Mathematics
- 2014

Abstract We study Hironaka's idealistic exponents over Spec ( Z ) . We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

THE SEMIGROUP OF A QUASI-ORDINARY HYPERSURFACE

- Mathematics
- 2003

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 the… Expand

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 this… Expand

Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant

- Mathematics
- 1995

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 spaces… Expand

Toric embedded resolutions of quasi-ordinary hypersurface singularities

- Mathematics
- 2003

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 properties… Expand

Decomposition in bunches of the critical locus of a quasi-ordinary map

- Mathematics
- 2005

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 to… Expand

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 to… Expand

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 with… Expand

Jet schemes and generating sequences of divisorial valuations in dimension two

- Mathematics
- 2015

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 recover… Expand

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