A Newtonian and Weierstrassian Approach to Local Resolution of Singularities in Characteristic Zero

TL;DR

This paper introduces an elementary algorithm for resolving singularities in characteristic zero using Newton polyhedra, monomial transformations, and Weierstrass preparation, with a focus on canonical reduction and variable management.

Contribution

It presents a novel, elementary algorithm for local resolution of singularities combining Newton polyhedra and Weierstrass techniques with new methods like canonical reduction.

Findings

Algorithm terminates after finite steps

Constructs a finite partition of unity near singularities

Uses singularity invariants based on primary variable orders

Abstract

This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial transformations and based on Weierstrass preparation theorem. This approach entails such new methods as canonical reduction and synthesis of monomial transformations as well as latency and revival of primary variables. The orders of primary variables serve as the decreasing singularity invariants for the algorithm albeit with some temporary increases. A finite partition of unity in a neighborhood of the singular point is constructed in an inductive way depending on the topological constraint imposed by Euler characteristic of the normal vector set of Newton polyhedron.