Thomas Breuer wrote:
The GAP implementation uses --besides the usual polynomials
and rational functions-- a special handling of rational functions
whose denominators are polynomials of the form (1-z^r)^k;
representing a Molien series as a sum of such rational functions
is useful for computing specific coefficients of the series
using a Taylor series for each summand.
I would like to know more about this handling of such rational
functions. Are any details available or does one have to read
the source code?
There are a lot of different expressions for the Molien series as
a rational function whose denominator is a product of n polynomials of
the form (1-z^k)^r, where n is the degree of the representation, and where
the numerator has coefficients that are positive integers. Maybe this
algorithm finds a particular one but it is also of interest to be able
to describe, in some concise way, all of the possible denominators.
To simplify the discussion, let me simply repeat factors 1-z^k instead
of writing a product of r of them as (1-z^k)^r. The denominator is
then described by an n-tuple (k1,...,kn) where the k's are monotone.
One can replace any k by a multiple of itself and, after reordering,
obtain another acceptable denominator. This generates a partial order
on the n-tuples and the problem is how to list the minimal elements
of this partial order.
This is useful to do since one would like to know in general when a
denominator of this type, giving a numerator with nonnegative integer
coefficients, actually reflects the structure of the ring of invariants.
If this holds for the minimal elements of the partial order, then it
is true for every such denominator. If not, then one has to look further
into the partial order for the minimal admissible denominators.