Differential ideal membership problem We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case? 
To be precise, suppose I is a differential ideal in the differential ring k{y_1,...,y_n}, which is generated by a finite set of differential polynomials{f_1,...,f_m},can we choose a base A={g_1,...g_d} in I ,such that I is also generated by A,and we can use it to determine whether a differential polynomial belongs to I or not by a method that is similar to the use of the Grobner Base . 
 A: I am not sure that what you want is decidable. You may want to look at this paper: MR1017781 (it is also explained in Section 7.6.2 of MR1361261). We proved there the following. There exists a system $p_1,...,p_n$ of linear differential operators with polynomial coefficients (over the field of char=0) such that the solvability of the system of PDEs
$$\left\\{\begin{array}{c} p_1(u_1,...,u_m)=r_1\\\
\ldots  \\\
p_n(u_1,...,u_m)=r_n\end{array}\right.$$
in polynomials $u_1,...,u_m$ for a given set of polynomials $v_1,...,v_n$ is undecidable. In other words, given the left hand side of a system of linear differential equations with polynomial coefficients, it is not possible to decide for every right hand side whether the system has a solution. A yet another reformulation is in terms of the Weyl algebra $W_n$. There exists a matrix $D$ over $W_n$, such that the solvability of equation 
$$D\cdot \vec u=\vec r$$
(given  $\vec r$, find $\vec u$) in polynomials is undecidable.  
A: In "Some constructions in rings of differential polynomials" Gallo and Mishra show that for certain infinitely generated differential ideals the membership problem is undecidable.
Which does not exclude that membership is decidable for finitely generated ideals (the case mainly encountered in real life). According to this article from 2006 that problem is still open.
But maybe this is useful: in 2009 Gao, Van der Hoeven, Yuan and Zhang showed that perfect ideal membership is decidable. (I think "perfect" means "radical" in differential algebra terminology)
