Complexity of a standard basis of a $D$-module

Abstract

We prove a double-exponential upper bound on the degree and on the complexity of constructing a standard basis of a D-module. This generalizes a well known bound on the complexity of a Grobner basis of a module over the algebra of polynomials. We would like to emphasize that the obtained bound can not be immediately deduced from the commutative case. To get our result we have elaborated a new technique of 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 ]). Denote for brevity Di = ∂ ∂Xi , 1 6 i 6 n. Any A–module is called D–module. It is well known that an A–module which is a submodule of a free finitely generated A-module has a Janet basis (if A is a Weyl algebra it is called often a standard basis; but in this paper it is natural and convenient to call it a Janet basis also in the case of the Weyl algebra). Historically, it was first introduced in [9]. In more recent times of developing computer algebra Janet bases were studied in [5], [14], [10]. Janet bases generalize Grobner bases which were widely elaborated in the algebra of polynomials (see e. g.[3]). For Grobner bases a double-exponential complexity bound was obtained in [12], [6] relying on [1]. Further, more precise results on the same subject (with an independent and self–contained proofs) were obtained in [4]. Surprisingly, no complexity bound on Janet bases was established so far. The reason is unique: 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 on Grobner bases [12], [15] provides by the same token a bound on Janet bases. Notice also that there has been a folklore opinion that the problem of constructing a Janet basis is easily reduced to the commutative case by considering