Weyl Closure of a Linear Differential Operator

We study the Weyl closure Cl(L) = K(x) ∂ L ∩ D for an operator L of the first Weyl algebra D = K x, ∂ . We give an algorithm to compute Cl(L) and we describe its initial ideal under the order filtration. Our main application is an algorithm for constructing a Jordan–Hölder series for a holonomicD -module and a formula for its length. Using the closure, we also reproduce a result ofStrömbeck (1978), who described the initial ideals of left ideals of D under the order filtration, and a result ofCannings and Holland (1994), who described the isomorphism classes of right ideals of D.