Aichinger equation on commutative semigroups

Abstract

We consider Aichinger?s equation f (x1 +... + xm+1) = Xm+1 i=1 1i(x1, x2, ...,bxi,..., xm+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 x1,..., xm+1 ? Rp +, then there exists a unique polynomial function F defined on Rp such that F|R p + = f . In particular, if f is also bounded on a set A ? Rp + with positive Lebesgue measure then its unique polynomial extension F is an ordinary polynomial in p variables with total degree ? m, and the functions 1i are also restrictions to Rpm + of ordinary polynomials of total degree ? m defined on Rpm.