By Dung Trang Le

Blending user-friendly effects and complicated equipment, Algebraic method of Differential Equations goals to accustom differential equation experts to algebraic tools during this niche. It offers fabric from a faculty prepared through The Abdus Salam overseas Centre for Theoretical Physics (ICTP), the Bibliotheca Alexandrina, and the overseas Centre for natural and utilized arithmetic (CIMPA).

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**Example text**

Qq−1 ∈ O, Qq ∈ D such that A = Qp Fp + · · · + Qq−1 Fq−1 + Qq Fq , and so M is generated as O-module by {Fp+1 , . . , Fq−1 }. 2, we deduce that M = 0. Let us note that the ring D is the inductive limit lim D(DR ). 2. Let us see some examples of left D-modules: (1) O is a left D-module, since D is a subring of EndC (O) and then any P ∈ D acts on any a ∈ O by P a = P (a). (2) To any linear differential operator P ∈ D we associate the left D-module D/DP . (3) The field K of fractions of O is a left D-module.

Q−1 q−1 Q p Qq−1 Q p+1 p+2 Qqp+2 Qqp Qqp+1 z αq−1 −αq Fq ··· 0 0 ··· 0 0 .. .. , . . αq−2 −αq−1 · · · Qq−1 z q−2 · · · Qqq−2 Qqq−1 March 31, 2010 14:8 WSPC - Proceedings Trim Size: 9in x 6in 01˙macarro 32 which is a matrix with entries in O. If g = a ∈ ker(Fq : M → M), a ∈ A, then Fq (a) = b ∈ O and so, by evaluating the equation (7) at a we obtain 0 Fp (a) Fp (a) Fp+1 (a) Fp+1 (a) 0 d . .. . + = A . . dz Fq−2 (a) Fq−2 (a) 0 αq−1 −αq z b Fq−1 (a) Fq−1 (a) d and dz (Fi (a)) ∈ O for i = p, .

This proves that B is a generating system for the vector space An . Let us prove now that B is linearly independent. Let us consider P = pαβ xα ∂ β a non trivial linear combination of monomials in B. Let β ∈ Nn be the smallest element, with respect to the lexicographical order,b appearing as the exponent of ∂ in P . ( α pαβ xα ) because if β is strictly smaller than β in the lexicographic order then ∂ β (xβ ) = 0. Because of the choice of β there exists α ∈ Nn such that pαβ = 0 and then P (xβ ) is nonzero.