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By Dung Trang Le

Blending hassle-free effects and complex tools, Algebraic method of Differential Equations goals to accustom differential equation experts to algebraic tools during this niche. It offers fabric from a faculty prepared via 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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2. 1. The above corollary shows that the finite dimensional vector space ker Lp is a measure of the non-regularity (or the irregularity) of the singular point p of L. This point of view is the first step of the notion of irregularity complexes of holonomic D-modules in higher dimension (see the paper21 ). 7. 2,13 In this section we work over the ring of convergent power series in one variable O = C{z}, that we can think as the ring of germs at 0 of holomorphic functions defined on a open neighborhood of the origin.

2) 0 1 · · · → 0 → HomD (M, A) → HomD (M, M) → 0 → · · · , but HomD (M, A) HomD (D/I, A) E(I) and F (I) HomD (M, M) HomD (D/I, M). 5. 1. For a holonomic D-module it is relatively easy to give a formula for χ(R HomD (M, O)) = dimC HomD (M, O) − dimC Ext1D (M, O) in terms of two integers algebraically associated with M : the multiplicity e0 of the “null section” and the multiplicity e1 of the “conormal of 0 in the “characteristic variety” defined by means of filtrations and the theory of Hilbert polynomials (cf.

2. The above results are the precursors of the irregularity complexes along a hypersurface and the notion of regular holonomic module in higher dimension (see the papers20,21 ). 29 11. A Local Version of the Riemann-Hilbert Correspondence in One Variable (in Collaboration with F. 5 in13 using the description of simple objects in the category C instead of the more involved description of indecomposable objects (see the master thesis7 ). 1. Let us call C0 the category defined in the following way: (1) The objets of C0 are the diagrams (E, F, u, v) where E, F are complex vector spaces and u : E → F and v : F → E are linear maps such that IdE + v ◦ u and IdF + u ◦ v are automorphisms.

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