Manuel Bronstein
ABSTRACT
We apply the elimination property of Gröbner bases with respect to pure lexicographic ordering to solve systems of algebraic equations. We suggest reasons for this approach to be faster than the resultant technique, and give examples and timings that show that it is indeed faster, and more correct, than MACSYMA's solve.
- 1.Buchberger, B., A Theoretical Basis for the Reduction of Polynomials to Canonical Forms, ACM-SIGSAM Bulletin, 39, August 1976, pp. 19-29. Google Scholar
Digital Library
- 2.Buchberger, B., A Note on the complexity of computing Gr'dbner bases, Proceedings of EUROCAL '83, Lecture Notes in Computer Science, vol. 162. Springer-Verlag, 1983, pp. 137-145. Google ScholarDigital Library
- 3.Buchberger, B., Gr~bner bases: An Algorithmic Method in Polynomial ideal Theory, Recent Trends in Multidimensional Systems Theory, N.K. Bose (ed.), D. Reidel Publ. Comp., 1985.Google Scholar
- 4.Collins, G.E., The Calculation of Multivariate Polynomial Resultants, J. ACM 18 (1971), pp. 515.532. Google ScholarDigital Library
- 5.Davenport, J.H., Computer Algebra for Cylindrical Algebraic Decomposition, Report TRITA- NA-8511, NADA, Royal Institute of Technology, Stockholm.Google Scholar
- 6.Lauer, M., Canonical Representatives for Residue Classes of a Polynomial Ideal, SYMSAC, 1976, pp. 339-345. Google ScholarDigital Library
- 7.Trinks, W., Uber B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu i~sen, Journal of Number Theory 10, 1978, pp. 475- 488.Google ScholarCross Ref
- 8.Zacharias, G., Generalized GrSbner Bases in Commutative Polynomial Rings, Bachelor Thesis, Lab. for Computer Science, MIT, 1978.Google Scholar
Index Terms
Gsolve: a faster algorithm for solving systems of algebraic equations
Recommendations
Representations of Algebraic Domains and Algebraic L-domains by Information Systems
Information systems play an important role in characterizing order structures. In this paper, we introduce the notions of the algebraic information system and algebraic L-information system. They are of the same logic-oriented style as the information ...
Solving algebraic systems which arise as necessary integrability conditions for polynomial-nonlinear evolution equations
The investigation of the problem of integrability of polynomial-nonlinear evolution equations, in particular, verifying the existence of the higher symmetries and conservation laws can often be reduced to the problem of finding the exact solution of a ...
Toward Solving Equations in Kleene Algebras
This work aims to investigate conditions under which program analysis can be viewed as algebraically solving equations involving terms of subclasses of Kleene algebras and variables. In this paper, we show how to solve a kind of linear equations in ...
Reviews
Katherine A. Yelick
This paper compares two methods for finding a set of solutions of a system of algebraic equations. Both methods proceed by calculating a set of univariate equations and then solving those equations; they differ in the technique used to produce the univariate equations. The first method calculates the univariate equations by the resultant method, and the second finds a Gro¨bner basis and then uses Buchberger's elimination procedure. Although the resultant approach is faster at computing the univariate equations, the equations produced by it are more difficult to solve than those produced with the Gro¨bner basis method. Thus, the Gro¨bner basis approach appears to be faster overall. The comparison was done experimentally: examples were taken from typical user input, and programs implementing each method were applied. The Gro¨bner basis approach was successful more often than the resultant approach, and when both succeeded the former was faster. The paper is only three pages long. It contains some interesting examples and should prove worthwhile to those interested in the pragmatics of the automatic solution of algebraic equations. The reader is warned, however, that familiarity with both of the methods is assumed, and the work is in other ways incomplete. It does not, for example, consider variations of the methods or draw general conclusions from the performance numbers. Furthermore, it is not clear whether the differences in performance are artifacts of the methods or of their implementations.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments