Abstract
In a preceding paper [9] we reported on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities. Here we discuss, how this approach may be refined to produce triangular systems in the sense of [12] and [13]. Such a refinement guarantees, different to the usual Gröbner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in [9] our method weakens the usual restriction to lexicographic term orders.
Triangular systems are a very helpful tool between factorization at a heuristical level and full decomposition into prime components. Our approach grew up from a consequent interpretation of the algorithmic ideas in [5] as a delayed quotient computation in favour of early use of (multivariate) factorization. It is implemented in version 2.2 of the REDUCE package CALI [8].
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Becker, T., Weispfenning, V., Kredel, H.: A computational approach to commutative algebra. Graduate Texts in Math. 141, Springer, New York, 1993.
Boege, W., Gebauer, R., Kredel, H.: Some examples for solving systems of algebraic equations by calculating Gröbner bases. J. Symb. Comp. 2 (1986), 83–98.
Chou, S. C.: Automated theorem reasoning in geometries using the characteristic set method and Gröbner basis method. In: Proc. ISSAC'90, ACM Press 1990, 255–260.
Faugere, J., Gianni, P., Lazard, D., Mora, T.: Efficient computations of zerodimensional Gröbner bases by change of ordering. J. Symb. Comp. 16 (1993), 329–344.
Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomial ideals. J. Symb. Comp. 6 (1988), 149–167.
Giovini, A. et al.: “One sugar cube, please” or selection strategies in the Buchberger algorithm. In: Proc. ISSAC'91, ACM Press, 1991, 49–54.
Gräbe, H.-G.: Two remarks on independent set. J. Alg. Comb. 2 (1993), 137–145.
Gräbe, H.-G.: CALI — A REDUCE package for commutative algebra. Version 2.2, Febr. 1995. Available through the REDUCE library e.g. at redlib@rand.org.
Gräbe, H.-G.: On factorized Gröbner bases. To appear in: Proc. “Computer algebra in Science and Engineering”, Bielefeld 1994.
Kalkbrener, M.: A generalized Euclidean algorithm for geometry theorem proving. J. Symb. Comp. 15 (1993), 143–167.
Kredel, H.: Primary ideal decomposition. In: Proc. EUROCAL-87, LNCS 378 (1989), 270–281.
Lazard, D.: Solving zero dimensional algebraic systems. J. Symb. Comp. 13 (1992), 117–131.
Lazard, D.: A new method for solving algebraic systems of positive dimension. Discr. Appl. Math. 33 (1991), 147–160.
Möller, H.-M.: On decomposing systems of polynomial equations with finitely many solutions. J. AAECC 4 (1993), 217–230.
Traverso, C., Donati, L.: Experimenting the Gröbner basis algorithm with the AlPi system. In: Proc. ISSAC'89, ACM Press 1989.
Wang, D.: An elimination method for solving polynomial systems. J. Symb. Comp. 16 (1993), 83–114.
Winkler, F.: Gröbner bases in geometry theorem proving and simplest non degeneracy conditions. Math. Pannonica 1 (1990), 15–32.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gräbe, HG. (1995). Triangular systems and factorized Gröbner bases. In: Cohen, G., Giusti, M., Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1995. Lecture Notes in Computer Science, vol 948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60114-7_18
Download citation
DOI: https://doi.org/10.1007/3-540-60114-7_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60114-2
Online ISBN: 978-3-540-49440-9
eBook Packages: Springer Book Archive