Solving systems of polynomial equations using Gröbner basis calculations with applications to mechanics
Abstract
Solving systems of algebraic equations is presented using the Gröbner Basis Package of the computer algebra system MAPLE V. The Grobner basis computations allow exact conclusions on the solutions of sets of polynomial equations, such as to decide if the given set is solvable, if the set has (at most) finitely many solutions, to determine the exact number of solutions in case there are finitely many, and their actual computation with arbitrary precision. The Gröbner basis computations are illustrated by two examples: computing the global equilibrium paths of a propped cantilever and of a simple arch.
References
[1] T. Becker, V. Weispfenning. Gröbner Bases. A Computational Approach to Commutative Algebra. SpringerVerlag, 1993.
[2] B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Mathematic, 4: 374-383, 1970.
[3] B. Buchberger. A theoretical basis for the reduction of polynomials to canonical form. ACM SIGSAM Bulletin, 10/3: 19-29, 1976.
[4] B. Buchberger. Some properties of Gröbner bases for polynomial ideals. ACM SIGSAM Bulletin, 10/4: 19-24, 1976.
[5] B. Buchberger. A criterion for detecting unnecessary reductions in the construction of Grabner bases. In: Proc. EUROSAM 79, Springer LNCS 72: 3- 21, 1979.
[2] B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Mathematic, 4: 374-383, 1970.
[3] B. Buchberger. A theoretical basis for the reduction of polynomials to canonical form. ACM SIGSAM Bulletin, 10/3: 19-29, 1976.
[4] B. Buchberger. Some properties of Gröbner bases for polynomial ideals. ACM SIGSAM Bulletin, 10/4: 19-24, 1976.
[5] B. Buchberger. A criterion for detecting unnecessary reductions in the construction of Grabner bases. In: Proc. EUROSAM 79, Springer LNCS 72: 3- 21, 1979.
Published
2023-06-22
Issue
pp. 167-178
Section
Articles
License
Copyright © The Author(s).
This work is licensed under the Creative Commons Attribution 4.0 International CC BY 4.0.
How to Cite
Popper, G. (2023). Solving systems of polynomial equations using Gröbner basis calculations with applications to mechanics. Computer Assisted Methods in Engineering and Science, 4(2), 167-178. https://cames3.ippt.pan.pl/index.php/cames/article/view/1399
