|
Structure of Gröbner bases with respect to block orders
Author(s):
M'Hammed
El Kahoui;
Said
Rakrak.
Journal:
Math. Comp.
76
(2007),
2181-2187.
MSC (2000):
Primary 13P10, 12Y05
Posted:
April 17, 2007
MathSciNet review:
2336290
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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 http://www.math.clemson.edu/~sgao/pub.html (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, http://www.singular.uni-kl.de.
- 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)
Similar Articles:
Retrieve articles in Mathematics of Computation
with
MSC (2000):
13P10, 12Y05
Retrieve articles in all Journals with
MSC (2000):
13P10, 12Y05
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
Email:
elkahoui@mpi-sb.mpg.de
Said
Rakrak
Affiliation:
Department of Mathematics and Computer Science, Faculty of Sciences and Techniques, Cadi Ayyad University, P.O. Box 549 Marrakech, Morocco
Email:
rakrak@fstg-marrakech.ac.ma
DOI:
10.1090/S0025-5718-07-01972-2
PII:
S 0025-5718(07)01972-2
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
Posted:
April 17, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
