Multiplier ideals of analytically irreducible plane curves

TL;DR

The paper introduces a new method to construct generators of multiplier ideals for analytically irreducible plane curves using special polynomials related to Puiseux series, unifying and extending previous results.

Contribution

It provides a novel construction of polynomials that generate multiplier ideals for irreducible plane curves, connecting maximal contact elements and approximate roots.

Findings

New set of polynomials for multiplier ideal generation

Unified approach recovers previous results

Explicit generators for multiplier ideals in terms of Puiseux series

Abstract

Let $S$ be a Puiseux series of the germ of an analytically irreducible plane curve $Z$. We provide a new perspective to construct a set of polynomials $F=\{F_1,\ldots, F_{g-1}\}$ associated to $S$, which is a special choice of maximal contact elements constructed in [AMB17] and approximate roots defined in [Dur18], [AM73a], [AM73b]. Using these polynomials as building blocks, we describe a set of generators of multiplier ideals of the form $\mathfrak{I}(\alpha Z)$ with $0<\alpha<1$ a rational number, which recovers the results about irreducible plane curves in [AMB17], [Dur18].