  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.
