Invariant polynomials, gaps, and sparseness

Abstract

We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on the appropriate line or hyperplane. Our result provides crucial information about gaps in the possible target dimensions for certain invariant polynomial sphere maps. We interpret our results in terms of sparseness for solutions of certain linear systems.