Complexity of the 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 elaborated for constructing all the solutions of a linear system over a homogeneous version of a Weyl algebra.