An approach to plane algebroid branches

TL;DR

This paper presents a new method to analyze plane algebroid branches using logarithmic distance, avoiding Hamburger-Noether expansions, and aims to reprove fundamental results in the theory of plane algebraic curve branches.

Contribution

It introduces a novel approach based on logarithmic distance to study branches, providing a direct calculation method without Hamburger-Noether expansions.

Findings

Reproves basic results of plane algebraic curve branches

Uses logarithmic distance to simplify calculations

Applicable over fields of arbitrary characteristic

Abstract

Our aim is to reprove the basic results of the theory of branches of plane algebraic curves over algebraically closed fields of arbitrary characteristic. We do not use the Hamburger-Noether expansions. Our basic tool is the logarithmic distance on the set of branches satisfying the strong triangle inequality which permits to make calculations directly on the equations of branches.