On the intersection multiplicity of plane branches

TL;DR

This paper establishes a new intersection formula for plane branches using semigroups and key polynomials, enhances Bayer's theorem on intersection numbers, and applies it to analyze the logarithmic distance between branches.

Contribution

It introduces a novel intersection formula for plane branches and strengthens Bayer's theorem, with applications to the logarithmic distance in branch space.

Findings

Derived a new intersection formula for plane branches.

Provided a stronger version of Bayer's theorem.

Applied results to the logarithmic distance between branches.

Abstract

We prove an intersection formula for two plane branches in terms of their semigroups and key polynomials. Then we provide a strong version of Bayer's theorem on the set of intersection numbers of two branches and apply it to the logarithmic distance in the space of branches.