On the approximate jacobian Newton diagrams of an irreducible plane curve

TL;DR

This paper introduces approximate jacobian Newton diagrams for irreducible plane curves, proving they serve as complete topological invariants and generalize existing theorems on polar curve decomposition.

Contribution

It defines approximate jacobian Newton diagrams and demonstrates their role as complete topological invariants, extending prior results by Merle and Ephraim.

Findings

Approximate jacobian Newton diagrams are complete topological invariants.

Generalization of Merle and Ephraim's theorems on polar curve decomposition.

Provides a new tool for classifying irreducible plane curves.

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.