On the classification of quasihomogeneous singularities

TL;DR

This paper reviews combinatorial characterizations of quasihomogeneous singularities, establishes bounds related to Milnor numbers, and explores implications for prime Milnor numbers, aiding classification efforts.

Contribution

It introduces new bounds on the monodromy order based on Milnor numbers and analyzes the structure of weight systems for quasihomogeneous singularities.

Findings

Upper bound for monodromy order by Milnor number

Surprising consequences for prime Milnor numbers

Classification insights for quasihomogeneous singularities

Abstract

The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such weight systems for any number of variables. This leads to certain types and graphs of such weight systems. Using them, we prove an upper bound for the common denominator (and the order of the monodromy) by the Milnor number, and we show surprising consequences if the Milnor number is a prime number.