Reference : Derivations and differential operators on rings and fields
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Derivations and differential operators on rings and fields
Kiss, Gergely mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica
[en] derivations of any order ; differential operators
[en] Let R be an integral domain of characteristic zero. We prove that a
function D : R → R is a derivation of order n if and only if D belongs to the
closure of the set of differential operators of degree n in the product topology
of R^R, where the image space is endowed with the discrete topology. In
other words, f is a derivation of order n if and only if, for every finite set
F ⊂ R, there is a differential operator D of degree n such that f = D
on F. We also prove that if d1, . . . , dn are nonzero derivations on R, then
d1 ◦ . . . ◦ dn is a derivation of exact order n.
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