Weak Bezout inequality for D-modules
نویسنده
چکیده
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 bound n(4mdmin{n, s}) (2(m−t)) on the leading coefficient of the Hilbert-Kolchin polynomial of the left Lm-module 〈{w1,j , . . . , wn,j}1≤j≤s〉 ⊂ L n m 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 Lm(F [X1, . . . , Xm, ∂ ∂X1 , . . . , ∂ ∂Xk ]). Introduction Denote the derivatives Di = ∂ ∂Xi , 1 ≤ i ≤ m and by Am = F [X1, . . . ,Xm,D1, . . . ,Dm] the Weil algebra [2] over an infinite field F . It is well-known that Am is defined by the following relations: XiXj = XjXi,DiDj = DjDi,XiDi = DiXi − 1,XiDj = DjXi, i 6= j (1) For a family {wi,j}1≤i≤n,1≤j≤s ⊂ Lm of elements of the algebra of linear partial differential operators one can consider a system
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عنوان ژورنال:
- J. Complexity
دوره 21 شماره
صفحات -
تاریخ انتشار 2005