Weak Bézout inequality for D-modules

Let {wi,j}1⩽i⩽n,1⩽j⩽s⊂Lm=F(X1,…,Xm)[∂∂X1,…,∂∂Xm] be linear partial differential operators of orders with respect to ∂∂X1,…,∂∂Xm at most d. We prove an upper boundn(4m2dmin{n,s})4m-t-1(2(m-t))on the leading coefficient of the Hilbert-Kolchin polynomial of the left Lm-module 〈{w1,j,…,wn,j}1⩽j⩽s〉⊂Lmn having the differential type t (also being equal to the degree of the Hilbert-Kolchin polynomial). The main technical tool is the complexity bound on solving systems of linear equations over algebras of fractions of the formLmFX1,…,Xm,∂∂X1,…,∂∂Xk-1.