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Yokoyama, Kazuhiro; Noro, Masayuki; Takeshima, Taku

Solutions of systems of algebraic equations and linear maps on residue class rings. (English) Zbl 0757.13013

J. Symb. Comput. 14, No. 4, 399-417 (1992).
The authors present new mathematical results and several new algorithms for solving a system of algebraic equations algebraically. They translate some theoretical arguments for the problem into their counterparts in the theory of linear maps, then they give a new description for the \(U\)- resultant and forms of solutions of systems straighforwardly. New algorithms proposed in the paper apply algorithms of linear algebra to avoid repeated computations of Gröbner bases under lexicographic order, and they require computations of a Gröbner basis, under arbitrary order, only once in principle. The new algorithms improve the efficiency of computation. Among others, the paper contains sections of a new presentation of the theory of solutions of systems and construction of the solutions in simple form. Comparison with other algorithms is given at the end of the paper.
Reviewer: Y.Kuo (Knoxville)

Cited in 1 Review
Cited in 14 Documents

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
12E12 Equations in general fields

Keywords:

resultant; algorithms for solving a system of algebraic equations; Gröbner basis; efficiency
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\textit{K. Yokoyama} et al., J. Symb. Comput. 14, No. 4, 399--417 (1992; Zbl 0757.13013)
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References:

[1] Auzinger, W.; Stetter, H. J., An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations (1989), Preprint · Zbl 0658.65047
[2] Buchberger, B., Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungs-system, Aequationes Mathematicae, 4, 374-383 (1970) · Zbl 0212.06401
[3] Buchberger, B., Gröbner bases: an algorithmic method in polynomial ideal theory, (Bose, N. K., Multidimensional System Theory (1985), Reidel Publ. Comp.,: Reidel Publ. Comp., Dordrecht), 184-232 · Zbl 0587.13009
[4] Buchberger, B., Applications of Gröbner bases in non-linear computational geometry, Springer Lecture Notes in Computer Science, 296, 52-80 (1988)
[5] Faugère, J. C.; Gianni, P.; Lazard, D.; Mora, T., Efficient computation of zero-dimensional Gröbner bases by change of ordering, Technical Report of Laboratoire Informatique Theorique et Programmation (LITP) 89-52 (1989)
[6] Gianni, P.; Trager, B.; Zacharias, G., Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Computation, 6, 149-167 (1988) · Zbl 0667.13008
[7] Hartshorne, R., Algebraic Geometry (1977), Springer-Verlag · Zbl 0367.14001
[8] Hintenaus, P., Decomposing and parameterizing the solution set of an algebraic system, (Ph.D. Thesis (1989), Johannes Kepler University), RISC-LINZ Report No. 89-27
[9] Kobayashi, H.; Fujise, T.; Furukawa, A., Solving systems of algebraic equations by a general elimination method, J. Symbolic Computation, 5, 303-320 (1988) · Zbl 0648.12017
[10] Kobayashi, H.; Moritsugu, S.; Hogan, W., Solving systems of algebraic equations. Symbolic and Algebraic Computation, (Gianni, P., International Symposium ISSAC’88. International Symposium ISSAC’88, Springer Lecture Notes in Computer Science, 358 (1988)), 139-149
[11] Kobayashi, H.; Moritsugu, S.; Hogan, W., On radical zero-dimensional ideals, J. Symbolic Computation, 8, 545-552 (1989) · Zbl 0693.68020
[12] Lazard, D., Résolution des systèmes d’équations algébriques, Theor. Comp. Sci, 15, 77-110 (1981) · Zbl 0459.68013
[13] Lazard, D., Gröbner bases, Gaussian elimination and resolution of system of algebraic equations, (Proc. EUROCAL ’83. Proc. EUROCAL ’83, Springer Lecture Notes in Computer Science, 162 (1983)), 146-156 · Zbl 0539.13002
[14] Lazard, D., Solving zero-dimensional algebraic systems, J. Symbolic Computation, 13, 117-131 (1992) · Zbl 0753.13012
[15] Seidenberg, A., Construction in algebra, Trans. Am. Math. Soc, 197, 273-313 (1974) · Zbl 0356.13007
[16] Trinks, W. L., Über Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen, J. Number Theory, 10, 475-488 (1978) · Zbl 0404.13004
[17] van der Waerden, B. L., Moderne Algebra II (1936), Springer-Verlag · Zbl 0002.00804
[18] van der Waerden, B. L., Algebra II (1991), Springer-Verlag · Zbl 0724.12002
[19] Yokoyama, K.; Noro, M.; Takeshima, T., Solutions of systems of algebraic equations and linear maps on residue class rings, (IIAS-SIS Research Report No. 94 (1989), FUJITSU LIMITED) · Zbl 0757.13013
[20] Yokoyama, K.; Noro, M.; Takeshima, T., Computing primitive elements of extension fields, J. Symbolic Computation, 8, 553-580 (1989) · Zbl 0697.68054
[21] Yokoyama, K.; Noro, M.; Takeshima, T., Solutions of systems of algebraic equations and linear maps on residue class rings, (IIAS-SIS Research Report No. 103 (1990), FUJITSU LIMITED) · Zbl 0757.13013
[22] Zariski, O.; Samuel, P., Commutative Algebra II (1960), Springer-Verlag · Zbl 0121.27801
[23] Zippel, R., Interpolating polynomials from their values, J. Symbolic Computation, 9, 375-403 (1990) · Zbl 0702.65011
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