  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.
