Polynomial Solutions to Constant Coefficient Differential Equations

Let Dx, ... , Dr e C[d/dxx, ... , d/dxn) be constant coefficient differential operators with zero constant term. Let S = {fe C[xx,... , x„]\Dj(f) = 0 for all 1 < j < r) be the space of polynomial solutions to the system of simultaneous differential equations Dj(f) = 0. It is proved that S is a module over 3¡(V), the ring of differential operators on the affine scheme V with coordinate ring C[d/dxx, ... , d/dXnl/(Dx, ... , Dr) .If V is smooth and irreducible, then S is a simple ^(K)-module, S = 1. 31 (V), and the generators for 3(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 3>'( K)-module for certain curves V , and certain homogeneous spaces V , and this allows one to obtain a basis for 5 , through knowledge of 2l(V).