Aichinger equation on commutative semigroups

TL;DR

This paper studies solutions to Aichinger's functional equation on commutative semigroups, showing they are generalized polynomials and exploring their properties and extensions to larger groups.

Contribution

It proves that solutions are generalized polynomials, demonstrates stability under composition and product, and establishes conditions for unique polynomial extensions from semigroups to groups.

Findings

Solutions are generalized polynomials under mild conditions.

Compositions and products of solutions remain generalized polynomials.

Boundaries for degrees are natural and optimal in this context.

Abstract

We consider Aichinger's equation $$f(x_1+\cdots+x_{m+1})=\sum_{i=1}^{m+1}g_i(x_1,x_2,\cdots, \widehat{x_i},\cdots, x_{m+1})$$ for 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 $x_1,\cdots,x_{m+1}\in \mathbb{R}_+^p$, then there exists a unique polynomial function $F$ defined on $\mathbb{R}^p$ such that $F_{|\mathbb{R}_+^p}=f$. In particular, if $f$ is also bounded on a set $A\subseteq \mathbb{R}_+^p$ with positive Lebesgue measure then its unique polynomial extension $F$ is an ordinary polynomial of $p$ variables with total degree $\leq m$, and the functions $g_i$ are also restrictions to $\mathbb{R}_+^{pm}$ of ordinary polynomials of total degree $\leq m$ defined on $\mathbb{R}^{pm}$.