PurityFiltration
Grade filtration of linear functional systems. The grade (purity) 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 using sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, or 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. 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 Maple package PurityFiltration and the GAP package homalg, but not by the standard PDE solvers of computer algebra systems such as Maple.
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References in zbMATH (referenced in 10 articles )
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Sorted by year (- Guan, Jiancheng; Li, Weiqing; Ouyang, Baiyu: On minor prime factorizations for multivariate polynomial matrices (2019)
- Bourlès, H.; Marinescu, B.; Oberst, U.: Exponentially stable linear time-varying discrete behaviors (2015)
- Oberst, Ulrich: The computation of purity filtrations over commutative Noetherian rings of operators and their application to behaviors (2015)
- Pommaret, Jean-Francois: Relative parametrization of linear multidimensional systems (2015)
- Robertz, Daniel: Recent progress in an algebraic analysis approach to linear systems (2015)
- Böckle, Gebhard: Cohomological theory of crystals over function fields and applications (2014)
- Quadrat, A.; Robertz, D.: A constructive study of the module structure of rings of partial differential operators. (2014)
- Quadrat, Alban: Grade filtration of linear functional systems. (2013)
- Quadrat, Alban: Purity filtration of 2-dimensional linear systems. (2010)
- Menezes, Carlos: An upper bound for the characteristic variety of an induced (\mathcalD)-module (1997)