On the classification of quasihomogeneous singularities

Claus Hertling and Ralf Kurbel

Journal of Singularities
volume 4 (2012), 131-153

Received: 3 September 2010. Received in revised form: 11 July 2012.

DOI: 10.5427/jsing.2012.4h

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The motivations 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.


Quasihomogeneous polynomial, weight system, classification of quasihomogeneous singularities

Mathematical Subject Classification:

32S25, 14J17, 58K40

Author(s) information:

Claus Hertling Ralf Kurbel
Lehrstuhl für Mathematik VI Lehrstuhl für Mathematik VI
Universität Mannheim Universität Mannheim
Seminargebäude A 5, 6 Seminargebäude A 5, 6
68131 Mannheim, Germany 68131 Mannheim, Germany
email: hertling@math.uni-mannheim.de email: kurbel@math.uni-mannheim.de