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Aequationes mathematicae

, Volume 89, Issue 3, pp 685–718 | Cite as

Characterizing ring derivations of all orders via functional equations: results and open problems

  • Bruce Ebanks
Article

Abstract

We provide a unifying framework for the treatment of equations of the form
$$\sum_{k=1}^n x^{p_k} f_k (x^{q_k}) = 0$$
for additive maps f k and integers p k , q k (1 ≤  k ≤  n). We show how to solve many equations of this type, and we present some open problems. In general our unknown functions map an integral domain of characteristic zero into itself. When negative exponents appear, we restrict our attention to fields of characteristic zero. All of the results could be formulated for integral domains or fields of sufficiently large characteristic as well.

Mathematics Subject Classification (2010)

Primary 39B52 39B72 Secondary 13N15 16W25 

Keywords

Ring derivation Derivation of higher order Additive map Homogeneous function Integral domain 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

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