Effective Localization Using Double Ideal Quotient and Its Implementation

  • Yuki IshiharaEmail author
  • Kazuhiro Yokoyama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)


In this paper, we propose a new method for localization of polynomial ideal, which we call “Local Primary Algorithm”. For an ideal I and a prime ideal P, our method computes a P-primary component of I after checking if P is associated with I by using double ideal quotient (I : (I : P)) and its variants which give us a lot of information about localization of I.


Gröbner basis Primary decomposition Localization 



The authors would like to thank the referees for their helpful comments to improve the presentation of this paper. The authors are also grateful to Masayuki Noro for technical assistance with the computer experiments and coding on Risa/Asir.


  1. 1.
    Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Avalon Publishing, New York (1994)zbMATHGoogle Scholar
  2. 2.
    Eisenbud, D., Huneke, C., Vasconcelos, W.: Direct methods for primary decomposition. Inventi. Math. 110(1), 207–235 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. J. Symb. Comput. 6(2), 149–167 (1988)CrossRefGoogle Scholar
  4. 4.
    Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Heidelberg (2002). Scholar
  5. 5.
    Kawazoe, T., Noro, M.: Algorithms for computing a primary ideal decomposition without producing intermediate redundant components. J. Symb. Comput. 46(10), 1158–1172 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Matzat, B.H., Greuel, G.-M., Hiss, G.: Primary decomposition: algorithms and comparisons. In: Matzat, B.H., Greuel, G.M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 187–220. Springer, Heidelberg (1999). Scholar
  7. 7.
    The Risa/Asir developing team: Risa/Asir. A computer algebra system.
  8. 8.
    Shimoyama, T., Yokoyama, K.: Localization and primary decomposition of polynomial ideals. J. Symb. Comput. 22(3), 247–277 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series. American Mathematical Society, no. 97 (2002)Google Scholar
  10. 10.
    Vasconcelos, W.: Computational Methods in Commutative Algebra and Algebraic Geometry. Algorithms and Computation in Mathematics. Springer, Heidelberg (2004)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Rikkyo UniversityTokyoJapan

Personalised recommendations