Polynomial and Inverse Forms

In a recent paper, this author studied invariant ideals in abelian group algebras under the action of certain infinite, locally finite, quasi-simple groups. While the main result was reasonably definitive, there are nevertheless certain natural extensions that should be considered. One approach to a proof of such extensions is to improve the basic tools that were used in the original work. However, we show here that the two obvious improvements to the polynomial form results fail in general. Specifically, we prove that the final value of a polynomial form f : A → S is not necessarily a subgroup of S. We also show that polynomial forms cannot be extended to “rational function forms” without losing the key properties we require.