On the classification of 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.

30 pages. In the 2nd version a misprint at the end of the proof of lemma 2.1 is corrected, the statements of lemma 2.4 and lemma 3.5 are corrected, and the table in section 5 is extended. The 2nd version coincides except for the new correction of lemma 3.5 with the published version