## Abstract

In [1] Casas-Alvero found decompositions of higher order polars of an irreducible plane complex analytic curve generalizing the results of Merle.
We improve his result obtaining a finer decomposition where we find out a kind of branches that we call *threshold semi-roots*. The existence of threshold semi-roots is a new phenomenon observed for the higher order polars. The topological type and the number of these branches is determined by the topological type of the original curve.

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