Differential Gröbner bases in one variable and in the partial case
Let \(K\) be a field of characteristic zero. Let \(\Delta =\{\delta_1,\ldots,\delta_m\}\) be a set of commuting derivation operators and \(\Theta =\{\delta_1^{a_1}\cdots\delta_m^{a_m}| \;(a_1,\ldots,a_m) \in {\mathbb N}^m\}\) the semigroup of derivatives generated by \(\Delta\). The ring of differential polynomials is defined to be the ring \(K\{Z\} = K[\theta Z| \;\theta\in\Theta]\), where \(Z\) is an indeterminate. This is a commutative polynomial ring with infinite variables. The author [\textit{G. Carrá Ferro}, in: Applied Algebra, algebraic algorithms and error-correcting codes, Proc. 5th Int. Conf., AAECC-5, Menorca 1987, Lect. Notes Comput. Sci. 356, 129-140 (1989; Zbl 0717.13017)] and \textit{F. Ollivier} [ibid., AAECC-8, Lect. Notes Comput. Sci. 508, 304-321 (1991; Zbl 0737.13009)] have introduced the notion of a Gröbner basis for an ideal in \(K\{Z\}\) with respect to a term ordering compatible with the derivations. Differential Gröbner bases do not behave like the ones of ideals in a polynomial ring of finite variables. This paper gives characterizations for differential Gröbner bases. It is also a good survey on this topic with many illustrating examples.
scientific article; zbMATH DE number 1067229