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 A_{μ} 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 Φ_{m},

(ii) and via its divisor (=sum of its zeros in the
group ring Z[S^1]), which is represented
as a sum of divisors Λ_{m} = (sum of m-th unit roots).

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

The tables are long, up to 17036 pages. Therefore for each of
the six cases (n=2,3,4 and Φ_{m} or Λ_{m}) 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)