Forms of Nice Functions

TL;DR

The paper studies functional equations arising from finite group actions on the domain or image of an unknown function, showing that a single linear equation can determine a unique solution when the associated determinant is nonzero. It identifies this determinant with the classical group determinant from the regular representation and explains its factorization into the forms $d(\phi)$ corresponding to irreducible representations over $\mathbb{C}$. A collection of explicit examples for groups such as $C_2$, $C_3$, and $S_3$ demonstrates solvability and yields concrete values like $f(2024)$ and $f(3)$ under various symmetry constraints, including actions on the plane and unit disk. The paper also discusses integral-form phenomena, multiplicativity of determinant values $d(S^G)$, and suggests educational applications using computer algebra systems.

Abstract

You can invent striking and challenging problems with unique solution by building some symmetry into functional equations. Some are suitable for high school; others could generate college-level projects involving computer algebra. The problems are functional equations with group actions in the background. Interesting examples arise even from small finite groups. Whether a given problem ``works" with a given choice of constant coefficients depends on whether a related multilinear form is nonzero. These forms are essentially the classical group determinants studied by Frobenius in the nineteenth century.