Let $K$ be a field of characteristic zero. Let $\Omega = K[x\_1, \dots, x\_n, dx\_1, \dots, dx\_n]$ be the differential ring of algebraic differential forms over $K[X\_1, \dots, X\_n]$.


Is there an algorithm (e.g. by Gröbner basis-like techniques) that solves the ideal membership problem for $\Omega$? That is: given a finitely generated differential ideal $I \subseteq \Omega$, is it decidable whether $f \in I$ or not for a given $f \in \Omega$?


Is calculating the differential radical of $\Omega$ or eliminating variables like in ordinary commutative algebra with Gröbner bases equally possible?


(The reason why I am interested in these things is, of course, algebraic handling of partial differential equations in exterior form.)



Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?
Yes. There is such an algorithm. This ring is a finitely generated free module over a polynomial ring, and it is sufficient to solve the submodule membership problem for these modules. But this is easy - the Grobner basis idea works perfectly. Just order the monomials times generators and find a basis for the initial submodule.


