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

We provide a unifying framework for the treatment of equations of the form ∑ 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 integ...

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Bibliographic Details
Published inAequationes mathematicae Vol. 89; no. 3; pp. 685 - 718
Main Author Ebanks, Bruce
Format Journal Article
LanguageEnglish
Published Basel Springer Basel 01.06.2015
Springer Nature B.V
Subjects
Analysis
Combinatorics
Mathematical functions
Mathematical problems
Mathematics
Mathematics and Statistics
Primary 39B52
Secondary 13N15
Derivation of higher order
16W25
Homogeneous function
Integral domain
Additive map
39B72
Ring derivation
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Summary:We provide a unifying framework for the treatment of equations of the form ∑ 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.
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-014-0256-8