Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ acts on $R$ as, \begin{equation} \label{eq:1} x_i \bullet f=x_i f,\quad \partial_i \bullet f=\frac{\partial f}{\partial x_i},\quad \forall f \in R \end{equation} Let $\mathcal D=\{D_1, \ldots, D_k\}$ be a finite subset of $W$. Denote $\mathcal D \bullet R$ be the $\mathbb K$-linear space, \begin{equation} \label{eq:2} \mathcal D \bullet R\equiv {\rm span}_{\mathbb K}\{ D_i \bullet f |1\leq i\leq k,\ \forall f\in R\} \end{equation} The $\mathbb K$-linear quotient space $R/(\mathcal D \bullet R)$ is the object of our interest. We want a practical way (1) to calculate $\dim_{\mathbb K} R/(\mathcal D \bullet R)$, if it is finite, (2) to determine if a given polynomial $F\in R$ is in $(\mathcal D \bullet R)$ or not.

Note that if $D_i$'s contain no differential operator, i.e. if $D_i\in R$, $1\leq i \leq k$, then both questions can be solved by the (commutative) Gr\"obner basis computation of a polynomial ideal. For the general case, is there a way to solve them by computing Gr\"obner basis for an ideal of the Weyl algebra?

finite set, or theleft ideal in $W$ generated bythat finite set? $\endgroup$ – Ketil Tveiten Jun 1 '16 at 8:22