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An extension of Buchberger’s criteria for Gröbner basis decision

  • John Perry (a1)
Abstract
Abstract

Two fundamental questions in the theory of Gröbner bases are decision (‘Is a basis G of a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known that G is a Gröbner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m−1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m−1)/2 S-polynomials to m−1.

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References
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[2] J. Backelin and R. Fröberg , ‘How we proved that there are exactly 924 cyclic-7 roots’, ISSAC ’91: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, NY, 1991) 103111, ISBN:0-89791-437-6.

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[7] B. Buchberger , ‘Gröbner-bases: an algorithmic method in polynomial ideal theory’, Multidimensional systems theory — progress, directions and open problems in multidimensional systems, (ed. N. K. Bose ; Reidel, Dordrecht, 1985) 184232.

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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