Algorithmic local monomialization of a binomial: a comparison of different approaches

TL;DR

This paper compares various local algorithms for transforming binomials into monomials through blowing up, with implementations in Singular, focusing on local centers and their relation to p-adic integrals.

Contribution

It introduces explicit local algorithms for binomial monomialization and provides a comparative analysis with implementations in Singular.

Findings

Different approaches have comparable effectiveness on numerous examples.

Local monomialization techniques can be implemented explicitly in computer algebra systems.

Connections between monomialization and p-adic integral computations are established.

Abstract

We investigate different approaches to transform a given binomial into a monomial via blowing up appropriate centers. In particular, we develop explicit implementations in {\sc Singular}, which allow to make a comparison on the basis of numerous examples. We focus on a local variant, where centers are not required to be chosen globally. Moreover, we do not necessarily demand that centers are contained in the singular locus. Despite these restrictions, the techniques are connected to the computation of $ p $-adic integral whose data is given by finitely many binomials.