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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.

Allan Adler

adler@hera.wku.edu

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