Skip to main content

Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Standard bases of ideals of the polynomial ring R[X] = R[x 1, …, x k ] over a commutative Artinian chain ring R that are concordant with the norm on R have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from R. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family L R (I), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram \(\mathcal{F}\) and a full system of \(\mathcal{F}\)-monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Gröbner basis of a monic ideal to a standard basis with the same cardinality.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, American Mathematical Society, Providence (1994), Graduate Studies in Mathematics, Vol. 3.

    MATH  Google Scholar 

  2. 2.

    J. Apel, “Computational ideal theory infinitely generated extension rings,” J. Theoretical Computer Science, 244, 1–33 (2000).

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley (1969).

  4. 4.

    E. Byrne and P. Fitzpatrick, “Gröbner bases over Galois rings with an application to decoding alternant codes,” J. Symbolic Comput., 31, 565–584 (2001).

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, Berlin-New York (1992).

    MATH  Google Scholar 

  6. 6.

    C. Faith, Algebra II. Ring Theory, Springer, Berlin (1976).

    MATH  Google Scholar 

  7. 7.

    E. V. Gorbatov, “Standard basis of a polynomial ideal over commutative Artinian chain ring,” Discrete Math. Appl., 14, No. 1, 75–101 (2004).

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    E. V. Gorbatov and A. A. Nechaev, “A criterion of the cyclicity of a family of linear recurring sequences over a QF-module,” Usp. Mat. Nauk, 56, No. 4 (2001).

    Google Scholar 

  9. 9.

    F. Kasch, Moduln und Ringe, Teubner, Stuttgart (1977).

    MATH  Google Scholar 

  10. 10.

    L. Kronecker, Vorlesungen über Zahlentheorie, Bd. 1, Teubner, Leipzig (1901).

    MATH  Google Scholar 

  11. 11.

    W. Krull, “Algebraische Theorie der Ringe, II,” Math. Ann., 91, 1–46 (1923).

    MathSciNet  Article  Google Scholar 

  12. 12.

    V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, and A. A. Nechaev, “Linear recurring sequences over rings and modules,” J. Math. Sci., 76, No. 6, 2793–2915 (1995).

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    V. L. Kurakin, A. V. Mikhalev, A. A. Nechaev, and V. N. Tsypyschev, “Linear and polylinear recurring sequences over Abelian groups and modules,” J. Math. Sci., 102, No. 6, 4598–4626 (2000).

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    V. N. Latyshev, Combinatorial Ring Theory, Standard Bases [in Russian], Izd. Mosk. Univ., Moscow (1988).

    MATH  Google Scholar 

  15. 15.

    P. Lu, A Criterion for Annihilating Ideals of Linear Recurring Sequences over Galois Rings, AAECC-417.

  16. 16.

    D. A. Mikhailov and A. A. Nechaev, “Canonical generating system of a monic polynomial ideal over a commutative Artinian chain ring,” Diskret. Mat., 13, No. 4, 3–42 (2001).

    MATH  MathSciNet  Google Scholar 

  17. 17.

    D. A. Mikhailov and A. A. Nechaev, “Solving systems of polynomial equations over Galois-Eisenstein rings with the use of the canonical generating systems of polynomial ideals,” Discrete Math. Appl., 14, No. 1, 41–73 (2004).

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    T. Mora, Seven Variations on Standard Bases, Preprint, Univ. de Genova, Dip. di Mathematica, No. 45 (1986).

  19. 19.

    A. A. Nechaev, “Linear recurring sequences over commutative rings,” Diskret. Mat., 3, No. 4, 105–127 (1991).

    MATH  MathSciNet  Google Scholar 

  20. 20.

    A. A. Nechaev, “Linear recurring sequences over quasi-Frobenius modules,” Russian Math. Surveys, 48, No. 3 (1993).

    Google Scholar 

  21. 21.

    A. A. Nechaev, “Finite quasi-Frobenius modules, applications to codes and linear recurrences,” Fund. Prikl. Mat., 1, No. 1, 229–254 (1995).

    MATH  MathSciNet  Google Scholar 

  22. 22.

    A. A. Nechaev, “Polylinear recurring sequences over modules and quasi-Frobenius modules,” in: Proc. First Int. Tainan-Moscow Algebra Workshop, 1994, Walter de Gruyter, Berlin-New York (1996), pp. 283–298.

    Google Scholar 

  23. 23.

    G. H. Norton and A. Salagean, “Strong Gröbner bases and cyclic codes over a finite-chain ring,” in: Proceedings of the International Workshop on Coding and Cryptography, Paris (2001), pp. 8–12.

  24. 24.

    L. Robbiano, “On the theory of graded structures,” J. Symbolic Comput., 2, 139–170 (1986).

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 23–71, 2004.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gorbatov, E.V. Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences. J Math Sci 139, 6672–6707 (2006). https://doi.org/10.1007/s10958-006-0384-3

Download citation

Keywords

  • Standard Basis
  • Shift Register
  • Polynomial Ideal
  • Polynomial System
  • Coset Representative