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See detailDerivations and differential operators on rings and fields
Kiss, Gergely UL

in Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica (2018)

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 ... [more ▼]

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. [less ▲]

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