Gröbner Bases: A Short Introduction for Systems Theorists
In this paper, we give a brief overview on Gröbner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Gröbner approach, we develop the concept of Gröbner bases by studying uniquenss of polynomial division (“reduction”). For explicitly constructing Gröbner bases, the crucial notion of S—polynomials is introduced, leading to the complete algorithmic solution of the construction problem. The algorithm is applied to examples from polynomial equation solving and algebraic relations. After a short discussion of complexity issues, we conclude the paper with some historical remarks and references.
KeywordsCommutative Algebra Symbolic Computation Short Introduction Univariate Polynomial Multivariate Polynomial
Unable to display preview. Download preview PDF.
- 1.W. W. Adams, P. Loustaunau. Introduction to Gröbner Bases. Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 1994.Google Scholar
- 3.B. Buchberger. An Algorithm for Finding the Bases Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal (German). PhD thesis, Univ. of Innsbruck (Austria), 1965.Google Scholar
- 6.B. Buchberger. A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner Bases. In Edward W. Ng, editor, Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (EUROSAM’ 79), Marseille, France, volume 72 of Lecture Notes in Computer Science, pages 3–21. Springer, 1979.Google Scholar
- 7.B. Buchberger. Gröbner-Bases: An Algorithmic Method in Polynomial Ideal Theory. In N. K. Bose, editor, Multidimensional Systems Theory, chapter 6, pages 184–232. Reidel Publishing Company, Dodrecht, 1985.Google Scholar
- 9.B. Buchberger and F. Winkler, editors. Gröbner Bases and Applications, volume 251 of London Mathematical Society Series. Cambridge University Press, 1998. Proc. of the International Conference “33 Years of Groebner Bases”.Google Scholar
- 10.B. Buchberger. Introduction to Gröbner Bases, pages 3–31 in , Cambridge University Press, 1998.Google Scholar
- 11.B. Buchberger. Gröbner-Bases and System Theory. To appear as Special Issue on Applications of Gröbner Bases in Multidimensional Systems and Signal Processing, Kluwer Academic Publishers, 2001.Google Scholar
- 12.A. Capani, G. Niesi, and L. Robbiano. CoCoA: A System for Doing Computations in Commuatative Algebra, 1998. Available via anonymous ftp from http://cocoa.dima.uniqe.it.
- 17.D. Grayson and M. Stillman. Macaulay 2: A Software System for Algebraic Geometry and Commutative Algebra. Available over the web at http://www.math.uiuc.edu/-Macaulay2.
- 18.G.-M. Greuel and G. Pfister and H. Schönemann. Singular Reference Manual. Reports On Computer Algebra, Number 12, Centre for Computer Algebra, University of Kaiserslautern, 1997. Available over the web at http://www.mathematik.uni?kl.de/~zca/Singular.