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Grade filtration of linear functional systems

Alban Quadrat 1, 2
1 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France
Abstract : The grade filtration of a finitely generated left module M over an Auslander regular ring D is a built-in classification of the elements of M in terms of their grades (or their (co)dimensions if D is also a Cohen-Macaulay ring). In this paper, we show how grade filtration can be explicitly characterized by means of elementary methods of homological algebra. Our approach avoids the use of sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, and Spencer cohomology used in the literature of algebraic analysis. Efficient implementations dedicated to the computation of grade filtration can then be easily developed in the standard computer algebra systems (see the Maple package PurityFiltration and the GAP4 package AbelianSystems). Moreover, this characterization of grade filtration is shown to induce a new presentation of the left D-module M which is defined by a block-triangular matrix formed by equidimensional diagonal blocks. The linear functional system associated with the left D-module M can then be integrated in cascade by successively solving inhomogeneous linear functional systems defined by equidimensional homogeneous linear systems of increasing dimension. This equivalent linear system generally simplifies the computation of closed-form solutions of the original linear system. In particular, many classes of underdetermined/overdetermined linear systems of partial differential equations can be explicitly integrated by the packages PurityFiltration and AbelianSystems, but not by computer algebra systems such as Maple.
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https://hal.inria.fr/inria-00632281
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Submitted on : Friday, October 14, 2011 - 12:36:08 AM
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Alban Quadrat. Grade filtration of linear functional systems. [Research Report] RR-7769, INRIA. 2011, pp.86. ⟨inria-00632281⟩

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