A discriminant criterion of irreducibility

TL;DR

This paper introduces a new criterion for determining the irreducibility of complex power series in two variables using jacobian Newton diagrams, and explores the singularity at infinity of certain plane affine curves.

Contribution

It presents a novel irreducibility criterion based on jacobian Newton diagrams and analyzes singularities at infinity for specific plane affine curves.

Findings

Irreducibility criterion using jacobian Newton diagrams

Analysis of singularities at infinity for plane affine curves

Global implications of the irreducibility criterion

Abstract

In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Moreover we study the singularity at infinity of a plane affine curve with one point at infinity for which the global counterpart of our main result holds.