Using Aichinger's equation to characterize polynomial functions

TL;DR

This paper leverages Aichinger's equation to provide simplified proofs for characterizations of polynomial functions and applies it to solve various functional equations across different groups.

Contribution

It introduces a unified approach using Aichinger's equation to solve multiple classical functional equations in a broad algebraic setting.

Findings

Aichinger's equation characterizes polynomial functions.

Solutions to several functional equations are obtained using this characterization.

The approach applies to arbitrary commutative groups.

Abstract

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye-Olkin's functional equation, Wilson's functional equation, the Kakutani-Nagumo-Walsh functional equation, and a general version of Fr\'echet's unmixed functional equation.