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Structure of Gröbner bases with respect to block orders

Authors: M'Hammed El Kahoui and Said Rakrak
Journal: Math. Comp. 76 (2007), 2181-2187
MSC (2000): Primary 13P10, 12Y05
Published electronically: April 17, 2007
MathSciNet review: 2336290
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Abstract: In this paper we study the structure of Gröbner bases with respect to block orders. We extend Lazard's theorem and the Gianni-Kalkbrenner theorem to the case of a zero-dimensional ideal whose trace in the ring generated by the first block of variables is radical. We then show that they do not hold for general zero-dimensional ideals.

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  • 1. W. W. Adams, A. Boyle, and P. Loustaunau, Transitivity for weak and strong Gröbner bases, J. Symbolic Comput. 15 (1993), no. 1, 49-65. MR 1210447 (94b:68064)
  • 2. W. W. Adams and P. Loustaunau, An introduction to Gröbner Bases, Graduate Texts in Mathematics, vol. 3, AMS, 1994. MR 1287608 (95g:13025)
  • 3. -, Gröbner bases and primary decomposition in polynomial rings in one variable over Dedekind domains, J. Pure and Applied Algebra 121 (1997), 1-15. MR 1471120 (98i:13049)
  • 4. T. Becker, On Gröbner bases under specialization, Appl. Algebra Engrg. Comm. Comput. 5 (1994), no. 1, 1-8. MR 1250930 (94h:13019)
  • 5. T. Becker and V. Weispfenning, Gröbner Bases: A computational approach to commutative algebra, Springer-Verlag, Berlin and New York, 1993. MR 1213453 (95e:13018)
  • 6. B. Buchberger, Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal, Ph.D. thesis, Inst. University of Insbruck, Innsbruck, Austria, 1965.
  • 7. -, Gröbner bases: An algorithmic method in polynomial ideal theory, In Recent trends in multidimensional system theory. Bose Ed. Reidel (1985), 184-232.
  • 8. D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms, second ed., Springer-Verlag, 1997. MR 1417938 (97h:13024)
  • 9. S. Gao, V. M. Rodrigues, and J. Stroomer, Gröbner basis structure of finite sets of points, Preprint, Available at (2003).
  • 10. P. Gianni, Properties of Gröbner bases under specialization, Lect. N. Comp. Sci. Berlin, Heidelberg, New York: Springer 378 (1987), 293-297. MR 1033305 (91g:13032)
  • 11. G.-M. Greuel, G. Pfister, and H. Schönemann, SINGULAR 3.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, June, 2005,
  • 12. M. Kalkbrenner, Solving systems of algebraic equations using Gröbner bases, Lect. N. Comp. Sci. Berlin, Heidelberg, New York: Springer 378 (1987), 282-292.
  • 13. D. Lazard, Ideal bases and primary decomposition: Case of two variables, J. Symbolic Computation 1 (1985), 261-270. MR 849035 (87k:13001)
  • 14. M. G. Marinari and T. Mora, A remark on a remark by Macaulay or enhancing Lazard structural theorem, Bull. Iranian Math. Soc. 29 (2003), no. 1, 1-45, 85. MR 2046304 (2004m:13071)
  • 15. B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2002. MR 1925796 (2003i:13037)

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Additional Information

M'Hammed El Kahoui
Affiliation: Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany
Address at time of publication: Department of Mathematics, Faculty of Sciences Semlalie, Cadi Ayyad University, P.O. Box 2390, Marrakech, Morocco

Said Rakrak
Affiliation: Department of Mathematics and Computer Science, Faculty of Sciences and Techniques, Cadi Ayyad University, P.O. Box 549 Marrakech, Morocco

Keywords: Gr\"obner basis, Lazard structure theorem, Gianni-Kalkbrenner structure theorem.
Received by editor(s): October 28, 2004
Received by editor(s) in revised form: March 22, 2006
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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