Invariant polynomials, gaps, and sparseness
John P. D'Angelo, Dusty E. Grundmeier, Daniel A. Lichtblau
TL;DR
The paper investigates invariant polynomials with non-negative coefficients under three families of cyclic groups acting on complex spaces, focusing on how many monomials such polynomials can contain (N(f)) while remaining invariant and equal to 1 on a line or plane. It establishes sharp gap results (via constructive and degree-based arguments) for the possible N(f) across the groups $Γ(m,1)$, $Γ(2r+1,2)$, and $Γ(7)$, revealing a tight connection between invariance, degree, and sparsity of solutions. The authors interpret these gaps through a sparsity lens in affine maps from invariant-parameter spaces to coefficient spaces, linking the algebraic structure of invariant polynomials to linear-algebraic “postage stamp” phenomena. They also discuss implications and extensions to equivariant settings and broader group families, highlighting rich combinatorial structures in invariant sphere-map problems.
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.