Bernstein polynomial of 2-Puiseux pairs irreducible plane curve singularities

TL;DR

This paper investigates the Bernstein polynomial of irreducible plane curve singularities with two Puiseux pairs, extending previous results and providing explicit root sets, bounds on invariants, and illustrative examples.

Contribution

It generalizes the understanding of Bernstein polynomials for singularities with two Puiseux pairs beyond cases with distinct monodromy eigenvalues.

Findings

Set of common roots of Bernstein polynomials determined

Bounds established for analytic invariants

Computational examples provided

Abstract

In 1982, Tamaki Yano proposed a conjecture predicting the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In \cite{ACLM-Yano2} we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with the hypotheses above. In particular the set of all common roots of those Bernstein polynomials is given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.