Advertisement

An algorithm for constructing detaching bases in the ring of polynomials over a field

  • Franz Winkler
Algorithms 2 — Polynomial Ideal Bases
Part of the Lecture Notes in Computer Science book series (LNCS, volume 162)

Abstract

Most ideal theoretic problems in a polynomial ring are extremely hard to solve, if the ideal is given by an arbitrary basis. B. Buchberger, 1965, was the first to show that for polynomials over a field it is possible to construct a "detaching" basis from a given arbitrary one, such that the problems mentioned above become easily soluble. Other authors (e.g. M. Lauer, 1976, and S.C. Schaller, 1979) have considered different coefficient domains. In this paper we investigate a method, developed by C.Sims and C.Ayoub, for constructing "detaching" bases in the ring of polynomials over Z, where the power products are ordered lexicographically. We show that the method also works for polynomials over a field, with only weak conditions on the ordering of the power products. New proofs of correctness and termination are presented. Furthermore we are able to improve the complexity behaviour of Ayoub's algorithm for the case of polynomials over a field.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ay80]
    C.Ayoub: On Constructing Bases for Ideals in Polynomial Rings over the Integers, Research Report, Dept.Math., Pennsylvania State Univ., 1980Google Scholar
  2. [Bu65]
    B. Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Dissertation, Univ. Innsbruck, 1965Google Scholar
  3. [Bu70]
    B. Buchberger: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequ.math., vol.4/3, pp.374–383, 1970Google Scholar
  4. [Bu76]
    B. Buchberger: A Theoretical Basis for the Reduction of Polynomials to Canonical Forms, ACM SIGSAM Bull. 39, pp.19–29, Aug.1976Google Scholar
  5. [Bu79]
    B.Buchberger: A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner-Bases, Proc. EUROSAM'79, pp.3–21, June 1979Google Scholar
  6. [BW79]
    B.Buchberger, F.Winkler: Miscellaneous Results on the Construction of Gröbner-Bases for Polynomial Ideals, Techn.Rep. Nr. 137, Inst. f. Math., Univ. Linz, June 1979Google Scholar
  7. [La76a]
    M.Lauer: Canonical Representatives for Residue Classes of a Polynomial Ideal, Proc. 1976 ACM Symp. on Symbolic and Algebraic Computation, pp.339–345, Aug.1976Google Scholar
  8. [La76b]
    M. Lauer: Kanonische Repräsentanten für die Restklassen nach einem Polynomideal, Diplomarbeit, Univ. Kaiserslautern, 1976Google Scholar
  9. [Sc79]
    S.C.Schaller: Algorithmic Aspects of Polynomial Residue Class Rings, Ph.D. Dissertation, Univ. Wisconsin-Madison, 1979Google Scholar
  10. [Si78]
    C.Sims: The Role of Algorithms in the Teaching of Algebra, in: M.F. Newman (ed.): Topics in Algebra, Springer Lecture Notes in Math., Nr.697, pp.95–107, 1978Google Scholar
  11. [vdW70]
    B.L. van der Waerden: Modern Algebra, vol.2, New York, Ungar, 1970Google Scholar
  12. [Wi78]
    F.Winkler: Implementierung eines Algorithmus zur Konstruktion von Gröbner-Basen, Diplomarbeit, Univ. Linz, 1978Google Scholar
  13. [Wi82]
    F.Winkler: An Algorithm for Constructing Detaching Bases in the Ring of Polynomials over a Field, Techn.Rep. Nr. CAMP 82-20.0, Inst. f. Math., Univ. Linz, December 1982Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Franz Winkler
    • 1
  1. 1.Institut für Mathematik Arbeitsgruppe CAMPJohannes Kepler UniversitätLinzAustria

Personalised recommendations