Invariant polynomials, gaps, and sparseness

TL;DR

This paper investigates invariant polynomials related to cyclic group representations, analyzing their term counts and sparseness, which informs the possible dimensions of invariant polynomial sphere maps.

Contribution

It introduces new bounds on the number of terms in invariant polynomials and links these to gaps in target dimensions for sphere maps.

Findings

Bounds on the number of terms in invariant polynomials.

Identification of gaps in target dimensions for sphere maps.

Connection between sparseness and solutions of linear systems.

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.