Weak Bezout inequality for D-modules

Abstract

Let $\{w_{i,j}\}_{1\leq i\leq n, 1\leq j\leq s} \subset L_m=F(X_1,...,X_m)[{\partial \over \partial X_1},..., {\partial \over \partial X_m}]$ be linear partial differential operators of orders with respect to ${\partial \over \partial X_1},..., {\partial \over \partial X_m}$ at most $d$. We prove an upper bound n(4m^2d\min\{n,s\})^{4^{m-t-1}(2(m-t))} on the leading coefficient of the Hilbert-Kolchin polynomial of the left $L_m$-module $<\{w_{1,j}, ..., w_{n,j}\}_{1\leq j \leq s} > \subset L_m^n$ 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 {\it algebras of fractions} of the form $$L_m(F[X_1,..., X_m, {\partial \over \partial X_1},..., {\partial \over \partial X_k}])^{-1}.$$