Abstract
Let U be a submodule of P 8, P a ring of multivariate polynomials, let G 1, …, G r generate U, and let F 1:= Σb 1j G j ,…, F m := Σb ij G j with b mj ∈ P. If a set of generators for the module of syzygies w. r. t. (G 1,…, G r ) is given, then one problem is to compute a set of generators for the module of syzygies w. r. t. (F 1,…, F m ) and to decide whether F 1, …, F m also generate U. For this purpose an algorithm is presented which is similar to the Gauß-Jordan algorithm in linear algebra. It uses Gröbner bases techniques for modules and allows to solve some constructive problems in connection with modules.
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References
Armbruster, D., Bifurcation theory and Computer Algebra: An initial approach, in “EUROCAL’ 85,” Springer Lecture Notes in Comp. Sci. 204, 1985, pp. 126–137.
Furukawa, A., Sasaki, T., Kobayashi, H., Gröbner basis of a module over K[x 1, …, x n]_and polynomial solutions of a system of linear equations, in “SYMSAC’ 86,” proceedings, 1986, pp. 222–224.
Gebauer, R., Möller, H. M., On an installation of Buchberger’s algorithm, J. Symb. Comp. 6 (1988), 275–286.
Lazard, D., Ideal bases and primary decomposition: case of two variables, J. Symb. Comp. 1 (1985), 261–270.
Maeß, G., “Vorlesungen über numerische Mathematik,” Vol. 1, section 2.2.2, Akademie-Verlag Berlin, 1984.
Melenk, H., Möller, H.M., Neun, W., On Gröbner basis computation on a supercomputer using REDUCE, Preprint SC 88-2 of Konrad-Zuse-Zentrum für Informationstechnik, Berlin 1988.
Möller, H.M., Mora, F., New constructive methods in classical ideal theory, J. Algebra 100 (1986), 138–178.
Traverso, C., Grobner trace algorithms, in “ISSAC’ 88,” Springer Lecture Notes in Comp. Sci., 1988.
Zacharias, G., Generalized Gröbner bases in commutative polynomial rings, Thesis at M.I.T. Dept. Comp. Sci. 1978.
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© 1991 Springer Science+Business Media New York
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Möller, H.M. (1991). Computing Syzygies à la Gauß-Jordan. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_22
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DOI: https://doi.org/10.1007/978-1-4612-0441-1_22
Publisher Name: Birkhäuser, Boston, MA
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