Exponents of an irreducible plane curve singularity

TL;DR

This paper provides an explicit formula for the exponents of irreducible plane curve singularities using Puiseux pairs and proves a conjecture relating the variance of these exponents to their range.

Contribution

It introduces a formula linking exponents to Puiseux pairs and confirms Hertling's conjecture on the variance bound for these exponents.

Findings

Explicit formula for exponents in terms of Puiseux pairs

Proof of Hertling's conjecture on variance bound

Bound on the variance of exponents

Abstract

We give an explicit formula for the exponents (i.e. the spectra up to the shift by one) of an irreducible plane curve singularity in terms of Puiseux pairs. As an application we prove in this case Hertling's conjecture that the variance (i.e. the square of the standard deviation) of the exponents is bounded by the difference between the maximal and minimal exponents divided by 12.