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On the Use of Gröbner Bases for Computing the Structure of Finite Abelian Groups

  • M. Borges-Quintana
  • M. A. Borges-Trenard
  • E. Martínez-Moro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

Some algorithmic properties are obtained related with the computation of the elementary divisors and a set of canonical generators of a finite abelian group, this properties are based on Gröbner bases techniques used as a theoretical framework. As an application a new algorithm for computing the structure of the abelian group is presented.

Keywords

Abelian Group Great Common Divisor Elementary Divisor Residue Number System Canonical Structure 
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.

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References

  1. 1.
    Borges Quintana, M.: On some Gröbner Bases Techniques and their Applications (Spanish). Phd Thesis. Universidad de Oriente, Santiago de Cuba, Cuba (2002)Google Scholar
  2. 2.
    Buchberger, B., Winkler, F.: Gröbner Bases and Applications. In: Proc. of the International Conference 33 Years of Gröbner Bases. London Mathematical Society Series, vol. 251. Cambridge University Press, Cambridge (1998)Google Scholar
  3. 3.
    Buchberger, B.: Introduction to Gröbner Bases. In: [2], pp. 3–31 (1998)Google Scholar
  4. 4.
    Buchmann, J., Jacobson Jr., M.J., Teske, E.: On Some Computational Problems in Finite Abelian Groups. Math. Comput. 66(220), 1663–1687 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, H.: A Course in Computational Algebraic Number Theory (3rd corrected printing), New York. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Cohen, H., Díaz y Díaz, F., Olivier, M.: Algorithmic Methods for Finitely Generated Abelian Groups. J. Symbolic Computation 31(1-2), 133–147 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Dumas, J.G., Saunders, B.D., Villard, G.: On Efficient Sparse Integer Matrix Smith Normal Form Computations. J. Symbolic Computation 32(1-2), 71–99 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eberly, W., Giesbrecht, M.W., Villard, G.: Computing the determinant and Smith form of an integer matrix. In: The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA (2000)Google Scholar
  9. 9.
    Havas, G., Majewski, B.S.: Integer Matrix Diagonalization. J. Symbolic Computation 24(3-4), 399–408 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Iliopoulos, C.S.: Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix. Siam J. Comput. 18/4, 658–669 (1989)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lübeck, F.: On the Computation of Elementary Divisors of Integer Matrices. J. Symbolic Computation 33(1), 57–65 (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Mora, T.: An Introduction to Commutative and Noncommutative Gröbner Bases. Theoretical Computer Science 134, 131–173 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Storjohann, A.: Near optimal algorithms for computing Smith normal forms of integer matrices. In: Lakshman, Y.N. (ed.) ISSAC 1996: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, Zurich, Switzerland, July 24-26, 1996, pp. 267–274. ACM Press, New York (1996)CrossRefGoogle Scholar
  14. 14.
    Storjohann, A.: Algorithms for Matrix Canonical Forms. Ph.D. Thesis, Institut für Wissenschaftliches Rechnen, ETH-Zentrum, Zürich, Switzerland (2000)Google Scholar
  15. 15.
    Terras, A.: Fourier analysis on finite groups and applications. LMS Student Texts, 43. Cambridge University Press, Cambridge (1999)Google Scholar
  16. 16.
    Teske, E.: A Space Efficient Algorithm for Group Structure Computation. Math. Comput. 67, 1637–1663 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Teske, E.: The Pohlig-Hellman Method Generalized for Group Structure Computation. J. Symbolic Computation 27(6), 521–534 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • M. Borges-Quintana
    • 1
  • M. A. Borges-Trenard
    • 1
  • E. Martínez-Moro
    • 2
  1. 1.Dpto. de MatemáticaFCMC, U. de OrienteSantiago de CubaCuba
  2. 2.Dpto. de Matemática AplicadaU. de ValladolidSpain

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