= Solution
{c}
Use the <Weyl algebra> convention $[y_i,x_j]=\delta_{ij}$, with $[x_i,x_j]=[y_i,y_j]=0$, where the brackets are <commutators>. Its <order filtration of a Weyl algebra> is
$$
F_jR=\sum_{|\beta|\leq j}\mathbb C[x_1,\ldots,x_n]y^\beta\quad(j\geq0),\qquad F_jR=0\quad(j<0).
$$
Thus $x_i$ has degree zero and $y_i$ degree one. By the <ordered monomial basis of a Weyl algebra>, the <associated graded ring> is
$$
A=\operatorname{gr}R\cong\mathbb C[x_1,\ldots,x_n,\xi_1,\ldots,\xi_n],\qquad \deg x_i=0,\quad\deg\xi_i=1,
$$
where $\xi_i$ is the symbol of $y_i$.
The <Rees ring of a filtered algebra> is the <graded ring>
$$
\boxed{\widetilde R=\bigoplus_{j\geq0}(F_jR)t^j\subseteq R[t],}
$$
where $t$ is a central indeterminate. Here it is generated by $x_i$, $Y_i=y_it$, and $t$, with degrees $0,1,1$, respectively, and relations
$$
[Y_i,x_j]=\delta_{ij}t,\qquad [x_i,x_j]=[Y_i,Y_j]=[t,x_i]=[t,Y_i]=0.
$$
It interpolates between the <associated graded ring> and the original <filtered ring>:
$$
\widetilde R/(t)\cong\operatorname{gr}R,\qquad \widetilde R/(t-1)\cong R.
$$
A <good filtration for the order filtration> on $M$ is an increasing exhaustive family of complex <vector subspaces> $(M_j)_{j\in\mathbb Z}$, zero for sufficiently negative $j$, with $F_iR\,M_j\subseteq M_{i+j}$, such that the <associated graded module>
$$
\operatorname{gr}M=\bigoplus_{j\in\mathbb Z}M_j/M_{j-1}
$$
is a <finitely generated module> over $A$. Equivalently, there are finitely many $m_1,\ldots,m_s$ and integer shifts $d_1,\ldots,d_s$ such that $M_j=\sum_{\ell=1}^s(F_{j-d_\ell}R)m_\ell$. In particular, choose any finite generating family $m_1,\ldots,m_s$ of $M$ and put
$$
\boxed{M_j=\sum_{\ell=1}^s(F_jR)m_\ell.}
$$
This is a <good filtration for the order filtration>.
For $M\ne0$, its <dimension of a Weyl algebra module for the order filtration> is
$$
\boxed{d(M)=\dim\operatorname{Supp}_A(\operatorname{gr}M)
=\dim_{\mathrm{Krull}}\bigl(A/\operatorname{Ann}_A(\operatorname{gr}M)\bigr).}
$$
Here <support of a module>, <annihilator of a module>, and <Krull dimension> are taken over the commutative <polynomial ring> $A$. The corresponding <algebraic set> is the <order characteristic variety of a Weyl algebra module>. This dimension is independent of the chosen <good filtration for the order filtration>; one may set $d(0)=-\infty$.
The pieces $F_jR$ generally have infinite complex <dimension>, since they contain all polynomials in the $x_i$. Thus a polynomial-growth definition based directly on $\dim_{\mathbb C}M_j$ would be inappropriate for this <order filtration of a Weyl algebra>.
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