Weak Bezout inequality for D-modules
Let {w(i),j} (I) <= i <= n,1 <= j <= s subset of L-m, = F(X-1,X-...(,) X-m)[partial derivative/partial derivative X-1 ,...,partial derivative/partial derivative X-m] be linear partial differential operators of orders with respect to partial derivative/partial derivative X-1 ,..., partial derivative/partial derivative X-m at most d. We prove an upper bound n(4m(2)d 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 <= j <= s) subset of 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 algebras of fractions of the form L-m(F[X-1,..., X-m, partial derivative/partial derivative X-1 ,..., partial derivative/partial derivative X-k])(-1). (c) 2005 Elsevier Inc. All rights reserved.
bezout inequality d-module hilbert-kolchin polynomial