Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility

TL;DR

This paper characterizes jacobian Newton polygons of plane branches, providing combinatorial criteria for irreducibility of complex series and necessary conditions for a curve to be a discriminant of a plane branch.

Contribution

It introduces two new characterizations of jacobian Newton polygons and derives novel criteria for irreducibility and discriminant conditions in complex plane branches.

Findings

Provides combinatorial criteria for irreducibility

Establishes necessary conditions for discriminants of plane branches

Characterizes jacobian Newton polygons in two distinct ways

Abstract

In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.