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

  • Polynomial Algorithms
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Eurocal '87 (EUROCAL 1987)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 378))

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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

  • 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 

  • 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 

  • 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 

  • Buchberger, B. (1983a), A note on the complexity of constructing Gröbner bases, Proc. EUROCAL 83, Springer LNCS 162, 137–145.

    Google Scholar 

  • 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 

  • Giusti, M. (1985), A note on the complexity of constructing standard bases, Proc. EUROCAL 85, Springer LNCS 204, 411–412.

    Google Scholar 

  • 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 

  • Lazard, D. (1983), Gröbner bases, Gaussian elimiation and resolution of systems of algebraic equations, Proc. EUROCAL 83. Springer LNCS 162, 146–156.

    Google Scholar 

  • Mora, F. (1982), An algorithm to compute the equations of tangent cones, Proc. EUROCAM 82, Springer LNCS 144, 158–165.

    Google Scholar 

  • Robbiano, L. (1985), Term orderings on the polynomial ring, Proc. EUROCAL 85, Springer LNCS 204, 513–517.

    Google Scholar 

Remark

  • Mora,F. and Robbiano,L. (1987), The Gröbner Fan of an Ideal, preprint.

    Google Scholar 

  • Weispfenning,V. (1987), Constructing universal Gröbner bases, extended abstract.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Technology Dresden, GDR

    Klaus-Peter Schemmel

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  1. Klaus-Peter Schemmel
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James H. Davenport

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© 1989 Springer-Verlag Berlin Heidelberg

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Schemmel, KP. (1989). An extension of buchberger's algorithm to compute all reduced gröbner bases of a polynomial ideal. In: Davenport, J.H. (eds) Eurocal '87. EUROCAL 1987. Lecture Notes in Computer Science, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51517-8_130

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  • DOI: https://doi.org/10.1007/3-540-51517-8_130

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51517-3

  • Online ISBN: 978-3-540-48207-9

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