On Solving The Systems of Algebraic Equations Using Gröbner Bases
Abstract
Described and proved the algorithm for finding some solution of algebraic equations over arbitrary field k for zero dimension ideals if Gröbner basis of this ideal over lexicographic order is given. The found Solution lies in the algebraic closure of k. An example for a system of algebraic equations having a unique solution in the main field, and exponentially many solutions of this system is suggested.
References
1. B.L. van der Waerden, Algebra I, Achte Auflage der Modernen Algebra, Springer-Verlag Berlin New York 1971. Algebra II, Fünfte Auflage, Springer-Verlag Berlin New York 1967.
2. Shokurov A.V., Sravnenie slozhnostej zadach nakhozhdeniya bazisa Groybnera ideala i reshenij e’togo ideala [Comparing complexities of problems of determining of Grebner’s basis of ideal and solving this ideal]. Trudy ISP RAN [The Proceedings of ISP RAS], 2012, vol. 22, pp. 463-474. (in Russian)
3. Faugere J.C., Gianni P., Lazard D., Mora T., Efficient computation of zero-dimensional Gröbner bases by change of ordering, Journal of Symbolic Computation, 1993,v.16, issue 4,pp.329-344.
Review
For citations:
Shokurov A. On Solving The Systems of Algebraic Equations Using Gröbner Bases. Proceedings of the Institute for System Programming of the RAS (Proceedings of ISP RAS). 2013;25:195-206. (In Russ.)
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