Polynomial solutions to constant coefficient differential equations

Abstract:

Let ${D_1}, \ldots ,{D_r} \in \mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]$ be constant coefficient differential operators with zero constant term. Let \[ S = \{ f \in \mathbb {C}[{x_1}, \ldots ,{x_n}]|{D_j}(f) = 0\;{\text {for all }}1 \leqslant j \leqslant r\} \] be the space of polynomial solutions to the system of simultaneous differential equations ${D_j}(f) = 0$. It is proved that $S$ is a module over $\mathcal {D}(V)$, the ring of differential operators on the affine scheme $V$ with coordinate ring $\mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left \langle {{D_1}, \ldots ,{D_r}} \right \rangle$. If $V$ is smooth and irreducible, then $S$ is a simple $\mathcal {D}(V)$-module, $S = 1.\mathcal {D}(V)$, and the generators for $\mathcal {D}(V)$ yield an algorithm for obtaining a basis for $S$. If $V$ is singular, then $S$ need not be simple. However, $S$ is still a simple $\mathcal {D}(V)$-module for certain curves $V$, and certain homogeneous spaces $V$, and this allows one to obtain a basis for $S$, through knowledge of $\mathcal {D}(V)$.