Matrice magique associ\'ee \`a un germe de courbe plane et division par l'id\'eal Jacobien

TL;DR

This paper studies solutions to a specific functional equation in the ring of holomorphic functions at the origin of C^2, introducing intersection multiplicities and constructing an explicit functional equation related to a plane curve germ.

Contribution

It introduces a new matrix associated with a plane curve germ and uses intersection multiplicities to analyze solutions of a functional equation, providing explicit formulas.

Findings

Defined intersection multiplicities relative to w and f'_y

Constructed an explicit functional equation for f

Analyzed solutions (u,v) using valuations

Abstract

In the ring of holomorphic functions at the origin of C^2, we consider the equation uf'_x+vf'_y=wf where f and w are given. We introduce intersection multiplicities relative to w and f'_y along the branches of f, and we study the solutions (u,v) using these valuations. As an application, we construct an explicit functional equation satisfied by f.