Description
InvolutiveBases is a package which provides routines for dealing with Janet and Pommaret bases.
Janet bases can be constructed from given Gr\"obner bases. It can be checked whether a Janet basis is a Pommaret basis. Involutive reduction modulo a Janet basis can be performed. Syzygies and free resolutions can be computed using Janet bases. A convenient way to use this strategy is to use an optional argument for resolution, see Involutive.
Some references:
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J. Apel, The theory of involutive divisions and an application to Hilbert function computations. J. Symb. Comp. 25(6), 1998, pp. 683-704.
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V. P. Gerdt, Involutive Algorithms for Computing Gr\"obner Bases. In: Cojocaru, S. and Pfister, G. and Ufnarovski, V. (eds.), Computational Commutative and Non-Commutative Algebraic Geometry, NATO Science Series, IOS Press, pp. 199-225.
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V. P. Gerdt and Y. A. Blinkov, Involutive bases of polynomial ideals. Minimal involutive bases. Mathematics and Computers in Simulation 45, 1998, pp. 519-541 resp. 543-560.
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M. Janet, Leçons sur les systèmes des équations aux dérivées partielles. Cahiers Scientifiques IV. Gauthiers-Villars, Paris, 1929.
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J.-F. Pommaret, Partial Differential Equations and Group Theory. Kluwer Academic Publishers, 1994.
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W. Plesken and D. Robertz, Janet's approach to presentations and resolutions for polynomials and linear pdes. Archiv der Mathematik 84(1), 2005, pp. 22-37.
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D. Robertz, Janet Bases and Applications. In: Rosenkranz, M. and Wang, D. (eds.), Gr\"obner Bases in Symbolic Analysis, Radon Series on Computational and Applied Mathematics 2, de Gruyter, 2007, pp. 139-168.
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W. M. Seiler, A Combinatorial Approach to Involution and delta-Regularity: I. Involutive Bases in Polynomial Algebras of Solvable Type. II. Structure Analysis of Polynomial Modules with Pommaret Bases. Preprints, arXiv:math/0208247 and arXiv:math/0208250.