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An extension of buchberger's algorithm to compute all reduced gröbner bases of a polynomial ideal

  • Klaus-Peter Schemmel
Polynomial Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 378)

Abstract

In this paper we present for the bivariate case an algorithm to compute all possible reduced Gröbner bases of a polynomial ideal. This algorithm is an extension of Buchberger's one, which is based on the possibility to classify and to handle easy all term orderings in case of two variables. The constructed algorithm is interesting for the study of complexity of constructing Gröbner bases in dependence of the chosen term ordering and may lead to new insights on the question which is the best term ordering for quick termination.

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References

  1. Buchberger,B. (1965), An algorithm for finding a basis for the residue class ring of a zerodimensional polynomial ideal (German), Ph.D. thesis, Univ. of Insbruck (Austria), Math. Inst.Google Scholar
  2. Buchberger, B. (1976), A theoretical basis for the reduction of polynomials to canonical form, ACM SIGSAM Bull. 10/3, 19–29 and 10/4, 19–24.Google Scholar
  3. Buchberger,B. (1983), Gröbner bases: an algorithmic method in polynomial ideal theory, Techn. Report CAMP-83.29, Univ. of Linz, Math. Inst., to appear as chapter 6 in: Recent Trends in Multidimensional System Theory (ed. by N.K.Bose), Reidel, 1985.Google Scholar
  4. Buchberger, B. (1983a), A note on the complexity of constructing Gröbner bases, Proc. EUROCAL 83, Springer LNCS 162, 137–145.Google Scholar
  5. Galligo,A. (1979), The divisition theorem and stability in local analytic geometry (French), Extrait des Annales de l'Institut Fourier, Univ. of Grenoble, 29/2.Google Scholar
  6. Giusti, M. (1985), A note on the complexity of constructing standard bases, Proc. EUROCAL 85, Springer LNCS 204, 411–412.Google Scholar
  7. Kollreider,C. (1978), Polynomial reduction: the influence of the ordering of terms on a reduction algorithm, Techn. Rep. CAMP-78.4, Univ. of Linz, Math. Inst.Google Scholar
  8. Lazard, D. (1983), Gröbner bases, Gaussian elimiation and resolution of systems of algebraic equations, Proc. EUROCAL 83. Springer LNCS 162, 146–156.Google Scholar
  9. Mora, F. (1982), An algorithm to compute the equations of tangent cones, Proc. EUROCAM 82, Springer LNCS 144, 158–165.Google Scholar
  10. Robbiano, L. (1985), Term orderings on the polynomial ring, Proc. EUROCAL 85, Springer LNCS 204, 513–517.Google Scholar

Remark

  1. Mora,F. and Robbiano,L. (1987), The Gröbner Fan of an Ideal, preprint.Google Scholar
  2. Weispfenning,V. (1987), Constructing universal Gröbner bases, extended abstract.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Klaus-Peter Schemmel
    • 1
  1. 1.Department of MathematicsUniversity of Technology DresdenGDR

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