A new algorithm for computing regular representations for radicals of parametric differential ideals. (English) Zbl 1438.12004
Summary: The regular representation of the radical of a differential ideal has various applications such as solving the membership problem, computing Taylor expansion of solutions, finding the Lie symmetries, and solving dynamical systems. Presently, there is no algorithm giving all regular representations for all possible values of the parameters for a polynomial differential ideal with parametric coefficients. In this article, we propose a new algorithm that computes all different regular representations with respect to all possible states of the parameters. Also, we present an efficient criterion to reduce some ineffectual computations. Implementing the algorithm in Maple and several examples reported in this article demonstrate the high efficiency of the algorithm.
MSC:
| 12H05 | Differential algebra |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 68W30 | Symbolic computation and algebraic computation |
References:
| [1] | Becker, T.; Weispfenning, V., Gröbner bases: A computational approach to commutative algebra, graduate texts in mathematics, 141 (1991), New York, NY: Springer, New York, NY |
| [2] | Boulier, F., Étude et Implantation de Quelques Algorithmes en Algébre Différentielle (1994), Université Lille I, 59655, Villeneuve d’Ascq: Université Lille I, 59655, Villeneuve d’Ascq, France |
| [3] | Boulier, F., Differential elimination and biological modelling, radon series on computational and applied mathematics, Gröbner Bases in Symbolic Analysis (2007) |
| [4] | Boulier, F., The management of parameters in the MAPLE differentialalgebra package (2018) |
| [5] | Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M., ISSAC’95, International Symposium on Symbolic and Algebraic Computation, Representation for the radical of a finitely generated differential ideal, 158-166 (1995) · Zbl 0911.13011 |
| [6] | Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M., Computing representations for radicals of finitely generated differential ideals, Applications Algebra Engrg Commission Computation, 20, 273-321 (2009) |
| [7] | Buchberger, B., A criterion for detecting unnecessary reductions in the construction of Gröbner bases, In symbolic and algebraic computation, EUROSAM’79, Internat, Symposium Marseille Lecture Notes in Computer Science, Springer, Berlin, 72, 3-21 (1979) · Zbl 0417.68029 |
| [8] | Cox, D.; Little, J.; O’Shea, D., Ideals, varieties and algorithms. an introduction to computational algebraic geometry and commutative algebra, undergraduate texts in mathematics (1992), New York, NY: Springer, New York, NY · Zbl 0756.13017 |
| [9] | Harrington, H. A.; Van Gorder, R. A., Reduction of dimension for nonlinear dynamical systems, Nonlinear Dynam, 88, 715-734 (2017) · doi:10.1007/s11071-016-3272-5 |
| [10] | Hubert, E., Factorization free decomposition algorithms in differential algebra, Journal Symbolic Computation, 29, 4-5, 641-662 (2000) · Zbl 0984.12004 |
| [11] | Hydon, P. E., Symmetry methods for differential equations (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 1035.35005 |
| [12] | Jauberthie, C.; Trave-Massuyes, L.; Verdiere, N., set-membership identifiaility of nonlinear models and related parameter estimation properties, International Journal Applications Mathematical Computation Sciences, 26, 2083-8492 (2016) · Zbl 1355.93048 · doi:10.1515/amcs-2016-0057 |
| [13] | Kolchin, E. R., Differential algebra and algebraic groups (1973), New York, NY: Academic Press, New York, NY · Zbl 0264.12102 |
| [14] | Mansfield, E. L.; Clarkson, P. A., Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations, Journal Symbolic Computation, 23, 517-533 (1997) · doi:10.1006/jsco.1996.0105 |
| [15] | Montes, A., A new algorithm for discussing Gröbner bases with parameters, Journal Symbolic Computation, 33, 183-208 (2002) · Zbl 1068.13016 · doi:10.1006/jsco.2001.0504 |
| [16] | Ritt, J. F., Differential algebra (1950), New York, NY: Dover, New York, NY · Zbl 0037.18501 |
| [17] | Rosenfeld, A., Specializations in differential algebra, Transactions Amer Mathematical Social, 90, 394-407 (1959) · Zbl 0192.14001 · doi:10.1090/S0002-9947-1959-0107642-2 |
| [18] | Sabzevari, M.; Hashemi, A.; Alizadeh, B. M.; Merker, J., Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-Generic submanifolds of \(C^N\), FiloMat, 30, 6, 1387-1411 (2016) · Zbl 1474.32051 |
| [19] | Sit, W. Y.; Guo, L.; Keigher, W. F.; Cassidy, P. J.; Sit, W. Y., Differential algebra and related topics, The Ritt-Kolchin theory for differential polynomials, 1-70 (2002), World Scientific Publishing Co. Inc · Zbl 1011.12007 |
| [20] | Umirbaev, U., Algorithmic problems for differential polynomial algebras, Journal Algebra, 455, 77-92 (2016) · Zbl 1371.12005 · doi:10.1016/j.jalgebra.2016.02.010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
