Arithmetic analogues of derivations
In Invent. Math. 122, No. 2, 309-340 (1995; Zbl 0841.14037), \textit{A. Buium} introduced an arithmetic analogue of derivations, called \(\pi\)-derivations; this concept was used to prove a series of arithmetic analogues of results about algebraic differential equations. The question arises of how ``natural'' these \(\pi\)-derivation operators are and what other choices one has in defining arithmetic analogues of ``usual'' derivations. The present note is an attempt to answer this question: We will introduce an a priori quite general notion of jet operator on a ring and prove that any such operator on a local domain of characteristic zero is ``equivalent'' to one of the following: a difference operator, a derivation operator, a \(\pi\)-difference operator, or a \(\pi\)-derivation operator in the sense of the author's paper cited above. The first three kinds of operators are always trivial on the rational integers; so one is left with the fourth kind, i.e., with the \(\pi\)-derivations of the paper cited above, as the ``only'' possibility, within our paradigm, for an arithmetic analogue of usual derivations.
scientific article; zbMATH DE number 1106641