Functional equations for homogeneous polynomials arising from multilinear mappings and their stability. (English) Zbl 0820.39011

Let \(H\) be an Abelian semigroup and \(G\) be an Abelian group uniquely divisible by \(n!\). A well-known result of Djokovic says that a function \(f: H\to G\) has the form \(f= g\circ \delta_ n\), where \(g: H^ n\to G\) is symmetric and \(n\)-additive and \(\delta_ n(x)= (x, x,\dots, x)\), if and only if \(\overset {n} {\underset {u}\Delta} f(x)= n! f(u)\) for all \(u\in H\).
Let now \(A\) be a commutative ring with identity, uniquely divisible by \(n!\), \(n\geq 2\) fixed; let \(M\), \(N\) be \(A\)-modules. One of the results of the present paper gives a characterization through a functional equation connected with the differences, of the functions \(f: M\to N\) of the form \(f= g\circ \delta_ n\), where \(g\) is \(n\)-linear and symmetric.
The stability of this functional equation and of other similar equations is then studied in Banach spaces over non-Archimedean valued fields.
Reviewer: G.L.Forti (Milano)


39B52 Functional equations for functions with more general domains and/or ranges