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Functional equations satisfied by additive functions have a special interestnot only in the theory of functional equations, but also in the theory of(commutative) algebra because the fundamental notions such as derivations andautomorphisms are additive functions satisfying some further functionalequations as well. It is an important question that how these morphisms can becharacterized among additive mappings in general.  The paper contains some multivariate characterizations of higher orderderivations. The univariate characterizations are given as consequences by thediagonalization of the multivariate formulas. The method allows us to refinethe process of computing the solutions of univariate functional equations ofthe form \[ \sum_{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \] where $p_k$ and $q_k$($k=1, \ldots, n$) are given nonnegative integers and the unknown functions$f_{1}, \ldots, f_{n}\colon R\to R$ are supposed to be additive on the ring$R$. It is illustrated by some explicit examples too.  As another application of the multivariate setting we use spectral analysisand spectral synthesis in the space of the additive solutions to prove thatsuch a functional equation characterizes derivations of higher order. Theresults are uniformly based on the investigation of the multivariate version ofthe functional equations.

On functional equations characterizing derivations: methods and examples