Claus Hertling and Ralf Kurbel

Tables of weight systems of quasihomogeneous singularities

A quasihomogeneous polynomial is a polynomial in n variables with complex coefficients such that with respect to a chosen system of (rational or natural) weights all monomials with nonzero coefficients have the same weighted degree. It has at 0 an isolated singularity if the first derivatives vanish together only at 0, else it has at 0 a non-isolated singularity (see e.g. the paper below for definitions and references).

The existence of quasihomogeneous polynomials with an isolated singularity at 0 imposes combinatorial conditions on a given weight system. These are reviewed in chapter 2 (Lemma 2.1 and Theorem 2.2) of the following paper.

On the classification of quasihomogeneous singularities by C. Hertling and R. Kurbel ( Journal of Singularities, vol. 4 (2012), 131-153).

The conditions had been found in different forms and different generality independently by many people (see the Remarks 2.3 of this paper). The first complete characterization is due to Kouchnirenko (1976).

Associated to a quasihomogeneous polynomial with an isolated singularity at 0 (short: "quasihomogeneous singularity") is its Milnor number μ. It can be computed via the weights.

We (Claus Hertling and Ralf Kurbel) used Theorem 2.2 in the paper and a PC to compute for n=2,3,4 (n=1 is known, only the Aμ singularities exist) essentially all weight systems which allow quasihomogeneous singularities up to Milnor number 9000 (for n=2 or 3) and 2000 (for n=4).

We restricted to rational weights < 1/2 such that the polynomial is of weighted degree 1. This is justified by the Splitting Lemma and a result of K. Saito (1971) (see e.g. Theorem 3.7 in the paper above).

We also used a formula of Milnor and Orlik (1970) (cited on page 27 in the paper above) to compute the characteristic polynomial of the monodromy of the quasihomogeneous singularity. It depends only on the weight system.

The characteristic polynomial is presented in two forms:
(i) as a product of cyclotomic polynomials Φm,
(ii) and via its divisor (=sum of its zeros in the group ring Z[S^1]), which is represented as a sum of divisors Λm = (sum of m-th unit roots).

In the following tables the weight systems (w1,...,wn) are ordered first by growing Milnor number, then by decreasing wn, then by decreasing wn-1, etc. The number d is the lcm of the denominators of the (rational) weights w1,...,wn.

The tables are long, up to 17036 pages. Therefore for each of the six cases (n=2,3,4 and Φm or Λm) the first 20 pages are cut out to give a first idea.

n = 2 μ ≤ 9000 Λ-polynomials 3130 pages 16 MB table ws 2 9000 lambda table ws 2 9000 lambda 20 pages
n = 2 μ ≤ 9000 Φ-polynomials 3405 pages 21 MB table ws 2 9000 phi table ws 2 9000 phi 20 pages
n = 3 μ ≤ 9000 Λ-polynomials 15305 pages 93 MB table ws 3 9000 lambda table ws 3 9000 lambda 20 pages
n = 3 μ ≤ 9000 Φ-polynomials 17036 pages 117 MB table ws 3 9000 phi table ws 3 9000 phi 20 pages
n = 4 μ ≤ 2000 Λ-polynomials 4305 pages 31 MB table ws 4 2000 lambda table ws 4 2000 lambda 20 pages
n = 4 μ ≤ 2000 Φ-polynomials 4681 pages 34 MB table ws 4 2000 phi table ws 4 2000 phi 20 pages

We hope that these tables will be useful. Looking at them we had conjectured Theorem 6.1 in the paper, which makes precise that in the case Milnor number = prime number the weight systems and the characteristic polynomials are rather special.

People interested in the programs with which the tables are calculated may contact us. (15.08.2011)