## 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)

### MSC:

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