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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 $f=g\circ {\delta }_{n}$, where $g:{H}^{n}\to G$ is symmetric and $n$-additive and ${\delta }_{n}\left(x\right)=\left(x,x,\cdots ,x\right)$, if and only if $\stackrel{n}{\underset{u}{{\Delta }}}f\left(x\right)=n!f\left(u\right)$ 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 $f=g\circ {\delta }_{n}$, 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.

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