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Plural, a Non–commutative Extension of Singular: Past, Present and Future

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Book cover Mathematical Software - ICMS 2006 (ICMS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4151))

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Abstract

We describe the non–commutative extension of the computer algebra system Singular, called Plural. In the system, we provide rich functionality for symbolic computation within a wide class of non–commutative algebras. We discuss the computational objects of Plural, the implementation of main algorithms, various aspects of software engineering and numerous applications.

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References

  1. Apel, J.: Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung. Dissertation, Universität Leipzig (1988)

    Google Scholar 

  2. Apel, J., Klaus, U.: FELIX, a Special Computer Algebra System for the Computation in Commutative and Non-commutative Rings and Modules (1998), available from: http://felix.hgb-leipzig.de/

  3. Brickenstein, M.: Neue Varianten zur Berechnung von Gröbnerbasen. Diplomarbeit, Universität Kaiserslautern (2004)

    Google Scholar 

  4. Brickenstein, M.: Slimgb: Gröbner Bases with Slim Polynomials. In: Reports On Computer Algebra No. 35, Centre for Computer Algebra, University of Kaiserslautern (2005), available from: http://www.mathematik.uni-kl.de/~zca/

  5. Bueso, J., Gómez–Torrecillas, J., Verschoren, A.: Algorithmic methods in non-commutative algebra. Applications to quantum groups. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  6. Castro-Jiménez, F.J., Ucha, J.M.: On the computation of Bernstein–Sato ideals. J. of Symbolic Computation 37, 629–639 (2004)

    Article  MATH  Google Scholar 

  7. Chyzak, F.: The GROEBNER Package for MAPLE (2003), available from: http://algo.inria.fr/libraries/

  8. Chyzak, F., Salvy, B.: Non–commutative Elimination in Ore Algebras Proves Multivariate Identities. J. of Symbolic Computation 26(2), 187–227 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chyzak, F., Quadrat, A., Robertz, D.: Linear control systems over Ore algebras. Effective algorithms for the computation of parametrizations. In: Proc. of Workshop on Time-Delay Systems (TDS 2003), INRIA (2003)

    Google Scholar 

  10. Cohen, A.M., Gijsbers, D.A.H.: GBNP, a Non–commutative Gröbner Bases Package for GAP 4 (2003), available from: http://www.win.tue.nl/~amc/pub/grobner

  11. Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf algorithms using the exterior algebra. Trans. Am. Math. Soc. 355(11), 4397–4426 (2003)

    Article  MATH  Google Scholar 

  12. García Román, M., García Román, S.: Gröbner bases and syzygies on bimodules over PBW algebras. J. of Symbolic Computation 40(3), 1039–1052 (2005)

    Article  MATH  Google Scholar 

  13. Gerdt, V.P.: Involutive Algorithms for Computing Groebner Bases. In: Pfister, G., Cojocaru, S., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry. IOS Press, Amsterdam (2005)

    Google Scholar 

  14. Gómez–Torrecillas, J., Lobillo, F.J.: Auslander-regular and Cohen-Macaulay quantum groups. J. Algebr. Represent. Theory 7(1), 35–42 (2004)

    Article  MATH  Google Scholar 

  15. Granger, M., Oaku, T., Takayama, N.: Tangent cone algorithm for homogenized differential operators. J. of Symbolic Computation 39(3–4), 417–431 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Grayson, D., Stillman, M.: MACAULAY 2, a Software System for Research in Algebraic Geometry (2005), available from: http://www.math.uiuc.edu/Macaulay2/

  17. Green, E., Heath, L., Keller, B.: Opal: A System for Computing Noncommutative Gröbner Bases. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 331–334. Springer, Heidelberg (1997)

    Google Scholar 

  18. Greuel, G.-M., Pfister, G., Bachmann, O., Lossen, C., Schönemann, H.: A SINGULAR Introduction to Commutative Algebra. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  19. Greuel, G.-M., Levandovskyy, V., Schönemann, H.: PLURAL. A SINGULAR 3.0 Subsystem for Computations with Non–commutative Polynomial Algebras. Centre for Computer Algebra, University of Kaiserslautern (2006)

    Google Scholar 

  20. Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra. University of Kaiserslautern (2006), available from: http://www.singular.uni-kl.de

  21. Helton, J.W., Stankus, M.: NCGB 3.1, a Noncommutative Gröbner Basis Package for Mathematica (2001), available from: http://www.math.ucsd.edu/~ncalg/

  22. Backelin, J., et. al. The Gröbner basis calculator BERGMAN (2006), available from: http://servus.math.su.se/bergman/

  23. Kandri-Rody, A., Weispfenning, V.: Non–commutative Gröbner bases in algebras of solvable type. J. of Symbolic Computation 9(1), 1–26 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kredel, H.: Solvable polynomial rings. Shaker (1993)

    Google Scholar 

  25. Kredel, H., Pesch, M.: MAS, Modula-2 Algebra System (1998), available from: http://krum.rz.uni-mannheim.de/mas.html

  26. Levandovskyy, V.: Non–commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Doctoral Thesis, Universität Kaiserslautern (2005), available from: http://kluedo.ub.uni-kl.de/volltexte/2005/1883/

  27. Levandovskyy, V.: On preimages of ideals in certain non–commutative algebras. In: Pfister, G., Cojocaru, S., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry. IOS Press, Amsterdam (2005)

    Google Scholar 

  28. Levandovskyy, V.: PBW Bases, Non–Degeneracy Conditions and Applications. In: Buchweitz, R.-O., Lenzing, H. (eds.) Representation of algebras and related topics. Proceedings of the ICRA X conference, vol. 45, pp. 229–246. AMS. Fields Institute Communications (2005)

    Google Scholar 

  29. Levandovskyy, V.: Intersection of ideals with non–commutative subalgebras. In: Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC 2006). ACM Press, New York (to appear, 2006)

    Google Scholar 

  30. Levandovskyy, V., Schönemann, H.: Plural — a computer algebra system for noncommutative polynomial algebras. In: Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC 2003). ACM Press, New York (2003)

    Google Scholar 

  31. Levandovskyy, V., Zerz, E.: Algebraic systems theory and computer algebraic methods for some classes of linear control systems. In: Proc. of the International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006) (2006)

    Google Scholar 

  32. Li, H.: Noncommutative Gröbner bases and filtered-graded transfer. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  33. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings. AMS (2001)

    Google Scholar 

  34. Mora, T.: Groebner bases in non-commutative algebras. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358, pp. 150–161. Springer, Heidelberg (1989)

    Google Scholar 

  35. Mora, T.: An introduction to commutative and non-commutative Groebner bases. Theor. Comp. Sci. 134, 131–173 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. Oaku, T., Takayama, N.: An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation. Journal of Pure and Applied Algebra 139(1–3), 201–233 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Pearce, R.: The F4 Algorithm for Gröbner Bases, Package for MAPLE (2005), available from: http://www.cecm.sfu.ca/~rpearcea/

  38. Saito, S., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  39. Takayama, N.: kAN/SM1, a Gröbner engine for the ring of differential and difference operators. available from: http://www.math.kobe-u.ac.jp/KAN/index.html

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Author information

Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstrasse 69, 4040, Linz, Austria

    Viktor Levandovskyy

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  1. Viktor Levandovskyy
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Editors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avda. de los Castros, s/n, E-39005, Santander, Spain

    Andrés Iglesias

  2. Faculty of Science Department of Mathematics, Kobe University, 1-1, Rokkodai, Nada-ku, 657-8501, Kobe, Japan

    Nobuki Takayama

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© 2006 Springer-Verlag Berlin Heidelberg

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Levandovskyy, V. (2006). Plural, a Non–commutative Extension of Singular: Past, Present and Future. In: Iglesias, A., Takayama, N. (eds) Mathematical Software - ICMS 2006. ICMS 2006. Lecture Notes in Computer Science, vol 4151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11832225_13

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  • DOI: https://doi.org/10.1007/11832225_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38084-9

  • Online ISBN: 978-3-540-38086-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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