Advertisement

Current Challenges in Developing Open Source Computer Algebra Systems

  • Janko BöhmEmail author
  • Wolfram Decker
  • Simon Keicher
  • Yue Ren
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

This note is based on the plenary talk given by the second author at MACIS 2015, the Sixth International Conference on Mathematical Aspects of Computer and Information Sciences. Motivated by some of the work done within the Priority Programme SPP 1489 of the German Research Council DFG, we discuss a number of current challenges in the development of Open Source computer algebra systems. The main focus is on algebraic geometry and the system Singular.

Keywords

Algebraic Geometry Computer Algebra System Free Resolution Rational Polyhedral Cone Computational Algebraic Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Arnold, E.A.: Modular algorithms for computing Gröbner bases. J. Symbolic Comput. 35(4), 403–419 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arzhantsev, I.V., Hausen, J.: Geometric invariant theory via Cox rings. J. Pure Appl. Algebra 213(1), 154–172 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barakat, M.: Computations of unitary groups in characteristic \(2\) (2014). http://www.mathematik.uni-kl.de/~barakat/forJPSerre/UnitaryGroup.pdf
  4. 4.
    Behrends, R.: Shared memory concurrency for GAP. Comput. Algebra Rundbrief 55, 27–29 (2014)Google Scholar
  5. 5.
    Behrends, R., Hammond, K., Janjic, V., Konovalov, A., Linton, S., Loidl, H.-W., Maier, P., Trinder, P.: HPC-GAP: engineering a 21st-century high-performance computer algebra system. Concurrency Comput. Pract. Experience (2016). cpe.3746Google Scholar
  6. 6.
    Behrends, R., Konovalov, A., Linton, S., Lübeck, F., Neunhöffer, M.: Parallelising the computational algebra system GAP. In: Proceedings of the 4th International Workshop on Parallel and Symbolic Computation, PASCO 2010, pp. 177–178. ACM, New York (2010)Google Scholar
  7. 7.
    Berchtold, F., Hausen, J.: GIT equivalence beyond the ample cone. Michigan Math. J. 54(3), 483–515 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bernal Guillén, M.M.: Relations in the Cox Ring of \(\overline{M}_{0,6}\). Ph.D. thesis, University of Warwick (2012)Google Scholar
  9. 9.
    Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Böhm, J., Decker, W., Laplagne, S., Pfister, G.: Computing integral bases via localization and Hensel lifting (2015). http://arxiv.org/abs/1505.05054
  11. 11.
    Böhm, J., Decker, W., Laplagne, S., Pfister, G.: Local to global algorithms for the Gorenstein adjoint ideal of a curve (2015). http://arxiv.org/abs/1505.05040
  12. 12.
    Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S.: locnormal.lib - A Singular library for a local-to-global approach to normalization (2013). Available in the Singular distribution, http://www.singular.uni-kl.de
  13. 13.
    Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S.: modnormal.lib - A Singular library for a modular approach to normalization (2013). Available in the Singular distribution, http://www.singular.uni-kl.de
  14. 14.
    Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S.: Parallel algorithms for normalization. J. Symbolic Comput. 51, 99–114 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Böhm, J., Decker, W., Laplagne, S., Seelisch, F.: paraplanecurves.lib - A Singular library for the parametrization of rational curves (2013). Available in the Singular distribution, http://www.singular.uni-kl.de
  16. 16.
    Böhm, J., Keicher, S., Ren, Y.: gitfan.lib - A Singular library for computing the GIT fan (2015). Available in the Singular distribution, http://www.mathematik.uni-kl.de/~boehm/gitfan
  17. 17.
    Böhm, J., Keicher, S., Ren, Y.: Computing GIT-fans with symmetry and the Mori chamber decomposition of \(\overline{M}_{0,6}\) (2016)Google Scholar
  18. 18.
    Boku, D.K., Decker, W., Fieker, C., Steenpass, A.: Gröbner bases over algebraic number fields. In: Proceedings of the International Workshop on Parallel Symbolic Computation, PASCO 2015, pp. 16–24. ACM, New York (2015)Google Scholar
  19. 19.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bravo, A.M., Encinas, S., Villamayor U., O.: A simplified proof of desingularization and applications. Rev. Mat. Iberoamericana 21(2), 349–458 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bruns, W., Ichim, B.: Normaliz: algorithms for affine monoids and rational cones. J. Algebra 324(5), 1098–1113 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Rest-klassenring nach einem nulldimensionalen Polynomideal. Dissertation, Universität Innsbruck (1965)Google Scholar
  23. 23.
    de Jong, T.: An algorithm for computing the integral closure. J. Symbolic Comput. 26(3), 273–277 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Decker, W., de Jong, T., Greuel, G.-M., Pfister, G.: The normalization: a new algorithm, implementation and comparisons. In: Dräxler, P., Ringel, C.M., Michler, G.O. (eds.) Computational Methods for Representations of Groups and Algebras. Progress in Mathematics, vol. 173, pp. 177–185. Birkhäuser, Basel (1999)CrossRefGoogle Scholar
  25. 25.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2 — A computer algebra system for polynomial computations (2015). http://www.singular.uni-kl.de
  26. 26.
    Dolgachev, I.V., Hu, Y.: Variation of geometric invariant theory quotients. (With an appendix: “An example of a thick wall” by Nicolas Ressayre). Publ. Math. Inst. Hautes Étud. Sci. 87, 5–56 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  28. 28.
    Eisenbud, D.: The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry. Graduate Texts in Mathematics, vol. 229. Springer, New York (2005)zbMATHGoogle Scholar
  29. 29.
    Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf cohomology and free resolutions over exterior algebras. Trans. Am. Math. Soc. 355(11), 4397–4426 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Encinas, S., Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77(4), 821–845 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Erocal, B., Motsak, O., Schreyer, F.-O., Steenpass, A.: Refined algorithms to compute syzygies. J. Symb. Comput 74, 308–327 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Frühbis-Krüger, A.: Computational aspects of singularities. In: Singularities in Geometry and Topology, pp. 253–327. World Sci. Publ., Hackensack (2007)Google Scholar
  33. 33.
    Frühbis-Krüger, A.: resolve.lib - A Singular library for the resolution of singularities (2015). Available in the Singular distribution, http://www.singular.uni-kl.de
  34. 34.
    The GAP Group. GAP - Groups, Algorithms, and Programming, Version 4.7.9 (2015)Google Scholar
  35. 35.
    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes – Combinatorics and Computation, pp. 43–74. Birkhäuser, Basel (2000)CrossRefGoogle Scholar
  36. 36.
    Grauert, H., Remmert, R., Stellenalgebren, A.: Analytische Stellenalgebren. Springer, New York (1971). Unter Mitarbeit von O. Riemenschneider, Die Grundlehren derCrossRefzbMATHGoogle Scholar
  37. 37.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  38. 38.
    Greuel, G.-M., Laplagne, S., Seelisch, F.: Normalization of rings. J. Symbolic Comput. 45(9), 887–901 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Greuel, G.-M., Laplagne, S., Seelisch, F.: normal.lib - A Singular library for normalization (2010). Available in the Singular distribution, http://www.singular.uni-kl.de
  40. 40.
    Hampe, S.: a-tint: a polymake extension for algorithmic tropical intersection theory. European J. Combin. 36, 579–607 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Hart, B.: ANTIC: Algebraic number theory in C. Comput. Algebra Rundbrief 56, 10–12 (2015)Google Scholar
  42. 42.
    Hart, W., Johansson, F., Pancratz, S.: FLINT: Fast Library for Number Theory (2013). Version 2.4.0, http://flintlib.org
  43. 43.
    Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  44. 44.
    Keicher, S.: Computing the GIT-fan. Internat. J. Algebra Comput. 22(7), 11 (2012). Article ID 1250064MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Ren, Y.: polymake.so - A Singular module for interfacing with polymake (2015). Available in the Singular distribution, http://www.singular.uni-kl.de
  46. 46.
    Schreyer, F.-O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstraßschen Divisionssatz und eine Anwendung auf analytische Cohen-Macaulay-Stellenalgebren minimaler Multiplizität. Diploma thesis, Universität Hamburg (1980)Google Scholar
  47. 47.
    Serre, J.-P.: Bases normales autoduales et groupes unitaires en caractéristique 2. Transform. Groups 19(2), 643–698 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Steenpaß, A.: parallel.lib - A Singular library for parallel computations (2015). Available in the Singular distribution, https://www.singular.uni-kl.de
  49. 49.
    The homalg project authors. The \(\mathtt{{homalg}}\) project - Algorithmic Homological Algebra (2003–2014). http://homalg.math.rwth-aachen.de/

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Janko Böhm
    • 1
    Email author
  • Wolfram Decker
    • 1
  • Simon Keicher
    • 2
  • Yue Ren
    • 1
  1. 1.University of KaiserslauternKaiserslauternGermany
  2. 2.Universidad de ConcepciónConcepciónChile

Personalised recommendations