Solving systems of algebraic equations

  • Hidetsune Kobayashi
  • Shuichi Moritsugu
  • Robert W. Hogan
Gröbner Bases
Part of the Lecture Notes in Computer Science book series (LNCS, volume 358)


This paper shows an algorithm for computing all the solutions with their multiplicities of a system of algebraic equations. The algorithm previously proposed by the authors for the case where the ideal is zero-dimensional and radical seems to have practical efficiency. We present a new method for solving systems which are not necessarily radical. The set of all solutions is partitioned into subsets each of which consists of mutually conjugate solutions having the same multiplicity.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Böge, W., Gebauer, R., Kredel, H. (1986). Some Examples for Solving Systems of Algebraic Equations by Calculating Gröbner Bases. J. Symbolic Computation. 2/1, 83–98.Google Scholar
  2. [2]
    Buchberger, B. (1970). Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Mathematicae. 4/3, 374–383.Google Scholar
  3. [3]
    Buchberger,B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. N.K.Bose. (ed.) Multidimensional Systems Theory: Progress, Directions and Open problems in Multidimensional Systems Theory. D.Reidel Publ.Comp. Chapter 6.Google Scholar
  4. [4]
    Gianni,P., Trager,B., Zacharias,C. (1986). Gröbner bases and primary decomposition of polynomial ideals. Preprint.Google Scholar
  5. [5]
    Gröbner,W. (1949). Moderne Algebraische Geometrie. Springer.Google Scholar
  6. [6]
    Hearn,A.C. (1983). REDUCE USER'S MANUAL Version 3.0. The Rand Corporation.Google Scholar
  7. [7]
    Kobayashi,H., Fujise,T., Furkawa,A. (1987). Solving systems of algebraic equations by a general elimination method. to be appeared in J. Symbolic Computation.Google Scholar
  8. [8]
    Kobayashi,H., Moritsugu,S., Hogan,R.W. (1987). On Solving Systems of Algebraic Equations, submitted to J. Symbolic Computation.Google Scholar
  9. [9]
    Lazard, D. (1981). Resolution des systemes d'equations algebriques, Theor. Comp. Sci. 15, 77–110.Google Scholar
  10. [10]
    Trinks, W.L. (1978). Über Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen. J. Number Theory. 10/4, 475–488.Google Scholar
  11. [11]
    van der Waerden,B.L. (1931). Moderne Algebra. Springer.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Hidetsune Kobayashi
    • 1
  • Shuichi Moritsugu
    • 2
  • Robert W. Hogan
    • 3
  1. 1.Dept. of Math.,College of Sci. & Tech.Nihon Univ.TokyoJapan
  2. 2.Dept. of Inform. Sci.,Faculty of Sci.Univ. of TokyoTokyoJapan
  3. 3.CITIZEN WATCH Co., Ltd.Tokorozawa-shi,SaitamaJapan

Personalised recommendations