An Algorithm of Constructing Cohomological Series Solutions of Holonomic Systems

Let D be the sheaf of analytic differential operators of n variables x1, . . . , xn. We consider a left ideal I of D generated by {l1, . . . , lm} which are in the Weyl algebra D = C〈x1, . . . , xn, ∂1, . . . , ∂n〉. If no confusion arises, we also denote by I the left ideal D ·{l1, . . . , ln} in D. Assume that D/I is holonomic. It was proved by M.Kashiwara that the germs of the k-th extension group ExtD(D/I, Ô) is a finite dimensional vector space over the field of complex numbers C [3]. We note that the vector space is called k-th order (cohomological) solution space. In [5], an algorithm to determine the dimension of the vector space was given. In this paper, we will give an algorithm to construct a basis of this vector space in a free module over the formal power series. In [5], we studied the adapted free resolution of D/I and an algorithm of computing restrictions of D-modules. The algorithm of evaluating the dimension of the germ of the k-th extension group was an immediate application of Cauchy-Kowalevski-Kashiwara’s theorem on the restriction of the D-module D/I to the origin and the k-th extension group. In this paper, we will explicitly construct matrix representations of boundary operators of complexes appearing in a proof of the CKK Theorem to construct series solutions. Let · · · ψi+1 → Di ψi → Di−1 → · · · → D → M → 0