By Dung Trang Le

Blending basic 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 foreign Centre for Theoretical Physics (ICTP), the Bibliotheca Alexandrina, and the foreign Centre for natural and utilized arithmetic (CIMPA).

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**Example text**

Pm1 y1 + · · · + Pmr yr P11 y1 .. + .. = .. g1 .. (9) = gm we can consider the left sub-D-module I ⊂ Dr generated by P i = (Pi1 , . . , Pir ), i = 1, . . , m, and we have as above an isomorphism (for the solutions of the associated homogeneous system) ker(P : S r → S m ) HomD (Dr /I, S), where P is the matrix of linear differential operators (Pij ). But if we are interested in the non-homogeneous system (9), it is not reasonable to try to solve it for any choice of g1 , . . , gm , since the existence of a solution would imply that any time we have a syzygy Q1 P 1 + · · · + Qm P m = 0, with Qi ∈ D, then Q1 g1 + · · · + Qm gm = 0.

In particular An is a noncommutative ring (for n ≥ 1). More generally, for any f, g ∈ C[x] and 1 ≤ i ≤ n we have (∂i ◦ φg )(f ) = ∂i (gf ) = ∂i (g)f + g∂i (f ) = φ∂i (g) (f ) + (φg ◦ ∂i )(f ). That is: the equality ∂i ◦ φg = φg ◦ ∂i + φ∂i (g) holds in EndC (C[x]) and therefore in An . The last equality is known as Leibniz’s rule. 1. Prove that the following equalities hold in An : ∂i ◦ φxj = φxj ◦ ∂i for all 1 ≤ i, j ≤ n with i = j. ∂i ◦ ∂j = ∂j ◦ ∂i for all 1 ≤ i ≤ j ≤ n. φxi ◦ φxj = φxj ◦ φxi for all 1 ≤ i ≤ j ≤ n.

By Nakayama’s lemma, the set of classes B = {e1 , . . , ep } is a basis of the (O/m =)C-vector space M/mM and so we have ω > 0. Let u ∈ S be a non-vanishing syzygy with ν(u) = ν(uj0 ) = ω. We have p p ui e i = · · · = 0=∂ i=1 p wj ej , j=1 with wj = ∂(uj ) + ui vij , i=1 March 31, 2010 14:8 WSPC - Proceedings Trim Size: 9in x 6in 01˙macarro 29 but ν(∂(uj0 )) = ν(uj0 ) − 1 and so ν(wj0 ) = ω − 1, which contradicts the minimality of ω. 3. 6), and Fp , . . , Fq ∈ I a Gr¨ obner basis of I. Then, the following properties hold: (1) For any A ∈ I, there is an integer r ≥ 0 such that xr A ∈ DFp .