By Dung Trang Le

ISBN-10: 9814273236

ISBN-13: 9789814273237

Blending straight forward effects and complex tools, Algebraic method of Differential Equations goals to accustom differential equation experts to algebraic tools during this niche. It provides fabric from a college equipped through The Abdus Salam foreign Centre for Theoretical Physics (ICTP), the Bibliotheca Alexandrina, and the overseas Centre for natural and utilized arithmetic (CIMPA).

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**Example text**

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 . (2) I = D(Fp , Fq ). Proof. We reproduce the proof of prop. 1. For part (2), let us consider the left D-module M = I/D(Fp , Fq ). For any A ∈ I, there are unique elements Qp , . . , 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 ).

The following proposition is taken from prop. 6 in,2 and gives a very precise information about the spaces of solutions E(I) and F (I). 5. Let I ⊂ D be a non-zero left ideal and Fp , . . , Fq a Gr¨ obner basis of I. Then the following properties hold: (1) (2) (3) (4) E(I) = ker(Fp : A → A)(= E(DFp )). F (I) = ker(Fq : M → M)(= E(DFq )). dimC E(I) = p(I)(= p = ord(Fp )), dimC F (I) = ν(I)(= ν(Fq )). P ∈ I ⇔ P f = 0, ∀f ∈ E(I) and P g = 0, ∀g ∈ F (I). Proof. 3, (1) and the fact that A has no O-torsion.

Ince. Ordinary Differential Equations. Dover, New York, 1950. 9. B. Iversen. Cohomology of Sheaves. Universitext. Springer Verlag, Heidelberg, 1986. 10. M. Katz. Nilpotent connections and the monodromy theorem: Applications ´ of a result of Turritin. Inst. Hautes Etudes Sci. Publ. , 39 (1970), 175– 232. 11. H. Komatsu. On the index of ordinary differential operators. J. Fac. Sci. Univ. Tokyo Sect. , 18 (1971), 379–398. 12. Lˆe D˜ ung Tr´ ang (editor). Introduction a ` la th´eorie alg´ebrique des syst`emes diff´erentiels.