Decompositions of the higher order polars of plane branches

TL;DR

This paper refines the decomposition of higher order polars of irreducible plane curves, identifying topological types and a new branch category called threshold semi-roots, building on Casas-Alvero's earlier work.

Contribution

It provides a more detailed decomposition of higher order polars, including the topological classification and the introduction of threshold semi-roots.

Findings

Finer decomposition of higher order polars

Determination of topological types of branches

Introduction of threshold semi-roots

Abstract

\noindent In \cite{Casas} Casas-Alvero found decompositions of higher order polars of an irreducible plane curve generalizing the results of Merle. We improve his result giving a finer decomposition where we determine the topological type and the number of a kind of branches that we call {\em threshold semi-roots}.