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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:HG has the form f=gδ n , where g:H n G is symmetric and n-additive and δ n (x)=(x,x,,x), if and only if Δ u nf(x)=n!f(u) for all uH.

Let now A be a commutative ring with identity, uniquely divisible by n!, n2 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:MN of the form f=gδ 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:
39B52Functional equations for functions with more general domains and/or ranges