Complexity of a Standard Basis of a D-module

A double-exponential upper bound is obtained for the degree and for the complexity of constructing a standard basis of a D-module. This generalizes a well-known bound for the complexity of a Gröbner basis of a module over the algebra of polynomials. It should be emphasized that the bound obtained cannot be deduced immediately from the commutative case. To get the bound in question, a new technique is developed for constructing all the solutions of a linear system over a homogeneous version of a Weyl algebra. Introduction Let A be the Weyl algebra F [X1, . . . , Xn, ∂ ∂X1 , . . . , ∂ ∂Xn ] (or the algebra of differential operators F (X1, . . . , Xn)[ ∂ ∂X1 , . . . , ∂ ∂Xn ]). For brevity, we denote Di = ∂ ∂Xi , 1 ≤ i ≤ n. Any A-module is called a D-module. It is well known that an A-module that is a submodule of a free finitely generated A-module has a Janet basis (if A is a Weyl algebra, it is often called a standard basis, but in this paper it is natural and convenient to call it a Janet basis also in that case). Historically, it was first introduced in [9]. In the more recent time of developing computer algebra, Janet bases were studied in [5, 14, 10]. The Janet bases generalize the Gröbner bases, which were widely used in the algebra of polynomials (see, e.g., [3]). For the Gröbner bases, a double-exponential complexity bound was obtained in [12, 6] with the help of [1]. Later, sharper results on the same subject (with independent and self-contained proofs) were obtained in [4]. Surprisingly, no complexity bound for Janet bases has been established so far. The reason is clear: the problem is not easy. In the present paper we fill this very essential gap and prove a double-exponential upper bound for complexity. On the other hand, a double-exponential complexity lower bound for Gröbner bases [12, 15] provides by the same token a bound for Janet bases. Notice also that there has been a folklore opinion that the problem of constructing a Janet basis reduces easily to the commutative case by considering the associated graded module, and, on the other hand, in the commutative case [6, 12, 4], the doubleexponential upper bound is well known. But this turns out to be a fallacy! From a known system of generators of a D-module, no system of generators (even not necessarily a Gröbner basis) of the associated graded module can be obtained immediately. The main problem here is to construct such a system of generators of the graded module. It may have elements of degrees (dl) O(n) ; see the notation below. Then, indeed, to the last system of generators of large degrees, one can apply the result known in the commutative case and get the bound ((dl) O(n) ) O(n) = (dl) O(n) . Thus, some new ideas specific to the noncommutative case are needed. 2000 Mathematics Subject Classification. Primary 16Z05.