A Modified Coefficient Ideal for Use with the Strict Transform

TL;DR

This paper proposes a hybrid algorithm for resolution of singularities that utilizes the strict transform, reducing complexity compared to traditional methods that rely on the Hilbert-Samuel stratification, offering a potential complementary approach.

Contribution

A novel hybrid algorithmic approach enabling the use of the strict transform without full Hilbert-Samuel stratification complexity in resolution processes.

Findings

Reduces complexity of the resolution process

Allows integration of strict transform in existing algorithms

Potential for improved efficiency in specific cases

Abstract

Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main difference is the use of different notions of transforms during the resolution process and the use of a stratification by the Hilbert-Samuel function in the one using the strict transform. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert-Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the implemented one which uses the weak transform, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic.