On the approximate Jacobian Newton diagrams of an irreducible plane curve

Abstract

We introduce the notion of an approximate Jacobian Newton diagram which is the Jacobian Newton diagram of the morphism $(f^{(k)},f)$, where $f$ is a branch and $f^{(k)}$ is a characteristic approximate root of $f$. We prove that the set of all approximate Jacobian Newton diagrams is a complete topological invariant. This generalizes theorems of Merle and Ephraim about the decomposition of the polar curve of a branch.