Aichinger equation on commutative semigroups

Abstract

We consider Aichinger's equation f px 1 `¨¨¨`x m`1 q " m`1 ÿ i"1for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be extended to a polynomial function defined on a larger group. For example, if f solves Aichinger's equation under the additional restriction that x 1 , ¨¨¨, x m`1 P R p `, then there exists a unique polynomial function F defined on R p such that F |R p `" f . In particular, if f is also bounded on a set A Ď R p `with positive Lebesgue measure then its unique polynomial extension F is an ordinary polynomial of p variables with total degree ď m, and the functions g i are also restrictions to R pm `of ordinary polynomials of total degree ď m defined on R pm .

Using Aichinger’s equation to characterize polynomial functions

“…, m + 1} for all k). Indeed, this fact completely characterises polynomial functions, as was proved by Aichinger and Moosbauer [1] for functions defined on abelian groups and by Almira [2] for functions defined on commutative semigroups S which satisfy that S + S = S and 0 ∈ S. Concretely [1, Lemma 4.1] states that, when (S, +) is also a group, the function f : S → H is a generalized polynomial of degree ≤ m if and only if it solves Aichinger's equation:…”