Abstract
In this paper we give a detailed description of an algorithm for computation of polynomial Janet bases and present its implementation in C, C++ and Reduce. This algorithm extends to polynomial ideals the algorithm for computation of monomial Janet bases presented in the first part of our paper and improves the specialization to Janet division of the general algorithm for computation of involutive polynomial bases. The computational efficiency of our codes is compared with that of computer algebra system Singular which provides one of the best implementations of the Buchberger algorithm for computation of Gröbner bases.
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Gerdt, V. P., Blinkov, Yu. A.: Involutive bases of polynomial ideals. Math. Comp. Simul. 45 (1998) 519–542
Gerdt, V. P., Blinkov, Yu. A.: Minimal involutive bases. Math. Comp. Simul. 45 (1998) 543–560
Gerdt, V. P.: Completion of linear differential systems to involution. In: Computer Algebra in Scientific Computing / CASC’99, V.G. Ganzha, E.W. Mayr and E.V. Vorozhtsov (Eds.), Springer-Verlag, Berlin (1999) 115–137
Buchberger, B.: Grobner bases: an algorithmic method in polynomial ideal theory. In: Recent Trends in Multidimensional System Theory, N.K. Bose (ed.), Reidel, Dordrecht (1985) 184–232
Becker, T., Weispfenning, V., Kredel, H.: Gröbner bases. A computational approach to commutative algebra. Graduate Texts in Mathematics 141, Springer-Verlag, New York (1993)
Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles, Cahiers Scientifiques, IV, Gauthier-Villars, Paris (1929)
Pommaret, J.-F.: Partial Differential Equations and Group Theory. New Perspectives for Applications, Kluwer, Dordrecht (1994)
Seiler, W. M.: Analysis and Application of the Formal Theory of Partial Diferential Equations. PhD thesis, School of Physics and Materials, Lancaster University (1994)
Gerdt, V. P.: On the relation between Pommaret and Janet bases. In: Computer Algebra in Scientific Computing/ CASC 2000, V.G. Ganzha, E.W. Mayr, E.V. Vorozhtsov (Eds.), Springer-Verlag, Berlin (2000) 167–181
Gerdt, V. P.: Involutive division technique: Some generalizations and optimizations, Zapiski Nauchnykh Seminarov POMI (St.Petersburg) 258 (1999) 185–206. To be published in J. Math. Sci.
Gerdt, V. P., Blinkov, Yu. A., Yanovich, D. A.: Construction of Janet bases. I. Monomial bases. This volume.
Greuel, G.-M., Pfister, G., Schoenemann, H.: Singular: A Computer Algebra System for Polynomial Computation, Department of Mathematics, University of Keiserslautern (2001) http://www.singular.uni-kl.de/Manual/2-0-0;.
Bini, D., Mourrain, B.: Polynomial Test Suite (1996) http://www-sop.inria.fr/saga/POL.
Verschelde, J.: The Database with Test Examples. http//www.math.uic.edurjan/demo.html
The GNU Multiple Precision Arithmetic Library. Edition 3.1.1, 18 September 2000 http://www.gnu.org/manual/gmp/.
Bachmann, O., Schönemann, H.: Monomial representations for Gröbner bases computations. In: Proceedings of ISSAC’98, ACM Press (1998) 309–321
Gerdt, V. P.: On Algorithmic Optimization in Computation of Involutive Bases. In preparation.
Seiler, W. M.: A Combinatorial Approach to Involution and δ-regularity. Preprint, Universität Mannheim (2000)
Blinkov, Yu. A.: The Method of Separative Monomials for Involutive Divisions. Programming and Computer Software 3 (2001) 43–45
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Gerdt, V.P., Blinkov, Y.A., Yanovich, D.A. (2001). Construction of Janet Bases II. Polynomial Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC 2001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56666-0_19
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DOI: https://doi.org/10.1007/978-3-642-56666-0_19
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