Abstract
Recently, several authors studied methods based on endomorphisms for localizing and computing the common zeros of systems of polynomial equations f i (x 1,..., x n )=0, i=1,..., s, in case the ideal I generated by f 1,..., f s has dimension zero. The main idea is to consider the trace and the eigenvalues of the endomorphisms Ф f: [g] ↦ [g · f], where [·] denotes equivalence classes modulo I in the polynomial ring. In this paper we give discuss some of these methods and combine them with the concept of dual bases for describing zero dimensional ideals.
This work is supported in part by the CEC, ESPRIT Basic Research Action 6846 (PoSSo)
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Auzinger, W., Stetter, H.J.: An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. Birkhäuser International Series in Numerical Mathematics 86 (1988) 11–30.
Becker, E.: Sums of squares and trace forms in real algebraic geometry. Cahier du Séminaire d'Histoire des Mathématiques, 2e Série, Vol. 1, Université Pierre et Marie Curie, 1991.
Becker, T., Weispfenning, V.: The Chinese remainder problem, multivariate interpolation, and Gröbner bases. Proc. ISSAC'91 (ed.: S.M.Watt), acm press (1991) 64–69.
Becker, E., Wörmann, T.: On the trace formula for quadratic forms and some applications. Proc. RAGSQUAD'92 (ed.: T.Lam), to appear.
Buchberger, B.: An algorithm for finding a basis for the residue class ring of a zero dimensional polynomial ideal (German). Ph.D. thesis, Univ. of Innsbruck, Austria, 1965.
Buchberger, B.: Application of Gröbner bases in non-linear computational geometry. Springer Lecture Notes in Computer Sciences. 296 (1988) 52–80.
Faugère, J.C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. Unpublished manuscript 1989.
Gantmacher, F.R.: The Theory of Matrices, vol.1, Chelsea Publ. Comp. 1977.
Gröbner, W.: Algebraische Geometrie I/II. B.I.Hochschultaschenbücher 273, 737, Bibliogr. Institut. 1970.
Lazard, D.: Algèbre linéaire sur K[X1,..., Xn] et élimimation. Bull. Soc. Math. France 105 (1977), 165–190.
Lazard, D.: Résolution des systèmes d'équations algébriques. Theoret. Comp. Sci. 15 (1981), 77–110.
Marinari, M.G., Möller, H.M., Mora, T.: Gröbner bases of ideals given by dual bases. Proc. ISSAC'91 (ed.: S.M.Watt), acm press (1991) 55–63.
Marinari, M.G., Möller, H.M., Mora, T.: Gröbner bases of ideals defined by functionals with an application to ideals of projective points. J. of AAECC (to appear).
Pedersen, P.: Counting real zeros. Thesis, Courant institute, N.Y. University (1991).
Pedersen, P., Roy, M.-F., Szpirglas, A.: Counting real zeros in the multivariate case. Proc. of MEGA'92 (ed.:A. Galligo), 1992 (to appear).
Stetter, H.J.: Verification in computer algebra systems. In: Validation Numerics — Theory and Applications Computing Suppl.8 (1993) (to appear).
Stetter, H.J.: Multivariate polynomial equations as matrix eigenproblems. Preprint.
Yokoyama, K., Noro, M., Takeshima, T.: Solutions of systems of algebraic equations and linear maps on residue class rings. J. Symb. Comp. 14 (1992) 399–417.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Möller, H.M. (1993). Systems of algebraic equations solved by means of endomorphisms. In: Cohen, G., Mora, T., Moreno, O. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1993. Lecture Notes in Computer Science, vol 673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56686-4_32
Download citation
DOI: https://doi.org/10.1007/3-540-56686-4_32
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56686-1
Online ISBN: 978-3-540-47630-6
eBook Packages: Springer Book Archive