Abstract
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form
where \(p_k\) and \(q_k\) (\(k=1, \ldots , n\)) are given nonnegative integers and the unknown functions \(f_{1}, \ldots , f_{n}:R\rightarrow R\) 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 analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.
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Dedicated to the 65th birthday of Professor László Székelyhidi.
The research of the first author has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K 111651 and by the ÚNKP-4 New National Excellence Program of the Ministry of Human Capacities. The work of the first and the third author is also supported by the EFOP-3.6.1-16-2016-00022 Project. The project is co-financed by the European Union and the European Social Fund. The second author was supported by the internal research project R-AGR-0500 of the University of Luxembourg and by the Hungarian Scientific Research Fund (OTKA) K 104178.
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Gselmann, E., Kiss, G. & Vincze, C. On Functional Equations Characterizing Derivations: Methods and Examples. Results Math 73, 74 (2018). https://doi.org/10.1007/s00025-018-0833-6
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DOI: https://doi.org/10.1007/s00025-018-0833-6