Schwaiger, Jens Functional equations for homogeneous polynomials arising from multilinear mappings and their stability. (English) Zbl 0820.39011 Pr. Nauk. Uniw. Śląsk. Katowicach 1444, Ann. Math. Silesianae 8, 157-171 (1994). 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) Cited in 9 Documents MSC: 39B52 Functional equations for functions with more general domains and/or ranges Keywords:difference equations; Abelian semigroup; Abelian group; commutative ring; \(A\)-modules; stability; Banach spaces; non-Archimedean valued fields PDF BibTeX XML Cite \textit{J. Schwaiger}, Pr. Nauk. Uniw. Śląsk. Katowicach, Ann. Math. Silesianae 1444(8), 157--171 (1994; Zbl 0820.39011) OpenURL