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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 $f= g\circ \delta\sb n$, where $g: H\sp n\to G$ is symmetric and $n$-additive and $\delta\sb n(x)= (x, x,\dots, x)$, if and only if $\overset n\to {\underset u\to\Delta} f(x)= n! f(u)$ for all $u\in H$. Let now $A$ be a commutative ring with identity, uniquely divisible by $n!$, $n\ge 2$ 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 $f= g\circ \delta\sb n$, 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.

39B52Functional equations for functions with more general domains and/or ranges