Seven combinatorial problems around quasihomogeneous singularities

TL;DR

This paper introduces seven combinatorial problems related to the formulas for characteristic polynomials and spectral numbers of quasihomogeneous singularities, including a new conjecture that revises Orlik's old conjecture.

Contribution

It presents a new conjecture on the characteristic polynomial and formulates seven combinatorial problems inspired by monodromy and spectral data of quasihomogeneous singularities.

Findings

Proposes a new conjecture on characteristic polynomial.

Formulates seven combinatorial problems.

Amends Orlik's old conjecture.

Abstract

This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of a quasihomogeneous singularity. One of them is a new conjecture on the characteristic polynomial. It is an amendment to an old conjecture of Orlik on the integral monodromy of a quasihomogeneous singularity. The search for a combinatorial proof of the new conjecture led us to the seven purely combinatorial problems.