The tree model of a meromorphic plane curve

TL;DR

This paper introduces a tree model for meromorphic plane curves based on contact structure, providing new descriptions of derivatives, Jacobians, and characterizations of regularity and irregular values.

Contribution

It develops a novel tree model framework for meromorphic plane curves, linking contact structure to derivatives and regularity properties.

Findings

Tree model T(f) effectively encodes contact structure.

Descriptions of y-derivative and Jacobian in terms of T(f).

Characterization of regularity and bounds on irregular values.

Abstract

We associate with a plane meromorphic curve f a tree model T(f) based on its contact structure. Then we give a description of the y-derivative of f (resp. the Jacobien J(f,g)) in terms of T(f) (resp. T(fg)). We also characterize the regularity of f in terms of its tree and we give a bound for the number of its irregular values.