Groebner Bases and Differential Algebra

the set T A of all monomlals in R, such that it is compatible with O. Given a finetely A-generated A-ideal I in R, i.e. an ideal generated by a finite number of A-polynomials and by their derlvatires, usin~ Buchberger's al~orlthm it is possible to find for a fixed order s an ascendin~ chain of ideals I k such thai: I ,k=l(s) when k>)O, bein~ l(s) the ideal of all A-polynomials in I of order less than or equal to s. The constructive knowledge of the ideal I(s) is indeed equivalent to being able to decide if a A-polynomial in R of order s is in I. In fact, the definitions of A-reduction and <-reduction of a A-polynomial with respect to another one are introduced and compared with the definition of reduction given by Rift. So the notion of A-Gr6bner basis arises in natural way but unfortunately there are finitely A-generated ~-ideels that have no such a basis, so this approach cannot be used to solve the membership problem.