Irreducibility criterion for algebroid curves

TL;DR

This paper presents an algorithm to determine the irreducibility of reduced algebroid curves using a new local tropical variety concept and computes their value-semigroups.

Contribution

It introduces a novel local tropical variety notion and provides algorithms for irreducibility testing and value-semigroup computation of algebroid curves.

Findings

Algorithm successfully decides irreducibility of algebroid curves.

New local tropical variety concept extends previous tropism ideas.

Efficient computation of value-semigroups for irreducible curves.

Abstract

The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over an algebraically closed field of arbitrary characteristic. To do this, we introduce a new notion of local tropical variety which is a straightforward extension of tropism introduced by Maurer, and then give irreducibility criterion for algebroid curves in terms local tropical varieties. We also give an algorithm for computing the value-semigroups of irreducible algebroid curves. Combining the irreducibility criterion and the algorithm for computing the value-semigroups, we obtain an algorithm for deciding the irreducibility of algebroid curves.