Quasi-ordinary singularities and Newton trees

TL;DR

This paper investigates nu-quasi-ordinary hypersurface singularities, introduces Newton trees for them, and provides a formula for discriminant computation, enhancing algorithmic resolution methods.

Contribution

It characterizes quasi-ordinary singularities via Newton trees and offers a new discriminant formula that avoids determinants, aiding in computational applications.

Findings

Newton trees characterize quasi-ordinary singularities.

Discriminant formula uses Newton tree decorations.

Transversal section Newton trees do not always match the original.

Abstract

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 them a Newton tree and characterize quasi-ordinary hypersurface singularities among nu-quasi-ordinary hypersurface singularities in terms of their Newton tree. A formula to compute the discriminant of a quasi-ordinary Weierstrass polynomial in terms of the decorations of its Newton tree is given. This allows to compute the discriminant avoiding the use of determinants and even for non Weierstrass prepared polynomials. This is important for applications like algorithmic resolutions. We compare the Newton tree of a quasi-ordinary singularity and those of its curve transversal sections. We show that the Newton trees of the transversal sections do not give the tree of the quasi-ordinary singularity in general. It does if we know that the Newton tree of the quasi-ordinary singularity has only one arrow.