ar X iv : 0 70 4 . 12 57 v 1 [ m at h . A P ] 1 0 A pr 2 00 7 Complexity of Janet basis of a D - module

We prove a double-exponential upper bound on the degree and on the complexity of constructing a Janet basis of a D-module. This generalizes a well known bound on the complexity of a Gröbner 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. 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 ≤ i ≤ 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. Historically, it was first introduced in [9]. In more recent times of developing computer algebra Janet bases were studied in [5], [13], [10]. Janet bases generalize Gröbner bases which were widely elaborated in the algebra of polynomials (see e. g.[3]). For Gröbner bases a double-exponential complexity bound was obtained in [12], [6] relying on [1] and which was made more precise (with a self–contained proof) in [4]. Surprisingly, no complexity bound on Janet bases was established so far; in the present paper we fill this gap and prove a double-exponential complexity bound. On the other hand, a double-exponential complexity lower bound on Gröbner bases [12], [14] provides by the same token a bound on Janet bases. There is a folklore opinion that the problem of constructing a Janet basis is easily reduced to the commutative case by considering the associated graded module, and, on the other hand, in the commutative case [6], [12], [4] the double– exponential upper bound is well known. But it turns out to be a fallacy! From a known system of generators of a D-module one can not obtain immediately any system of generators (even not necessarily a Gröbner basis) of the associated graded module. The main problem here is to construct such a system of generators of the graded module. It may have the elements of degrees (dl) O(n) ,