The log-canonical threshold of a plane curve

TL;DR

This paper provides an explicit formula for calculating the log-canonical threshold of a plane curve germ, based on its branches' contact values and intersection multiplicities, improving previous formulas.

Contribution

It introduces a new explicit formula for the log-canonical threshold of plane curve germs, simplifying calculations using only initial contact values and intersection data.

Findings

Explicit formula for log-canonical threshold derived

Formula depends on first two contact values and intersection multiplicities

Improves upon previous two-branch formulas by Kuwata

Abstract

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in a paper by Kuwata in Amer. J. Math. 121.