On the Classification of Quasihomogeneous Functions

TL;DR

This paper establishes criteria for the existence of non-degenerate quasihomogeneous polynomials within fixed weight configurations, relates these to Poincaré polynomial conditions, and provides an algorithm for specific singularity indices relevant to string theory.

Contribution

It introduces a criterion for quasihomogeneous polynomial existence, links it to Poincaré polynomial conditions, and offers an algorithm for configurations with singularity index 3.

Findings

Finiteness of configurations for a fixed singularity index

A criterion for the existence of non-degenerate quasihomogeneous polynomials

An algorithm for calculating configurations with index 3

Abstract

We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincar\'e polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.