Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Let K be a field of characteristic zero, n≥1 an integer and An+1=K[X,Y1,…,Yn]〈∂X,∂Y1,…,∂Yn〉 the (n+1)th Weyl algebra over K. Let S∈An+1 be an order-1 differential operator of the type with ai,bi∈K[X] and gi∈K[X,Yi] for every i=1,…,n. We construct an algorithm that allows one to recognize whether S generates a maximal left ideal of An+1, hence also whether An+1/An+1S is an irreducible non-holonomic An+1-module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of An, is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K[X,Y1,…,Yn] and, when they have one, to compute it.