An Algorithm for Solving Systems of Algebraic Equations in Three Variables

  • Michael Kalkbrener
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)


The most famous algorithm for computing the greatest common divisor (gcd) of two univariate polynomials over a field is, undoubtedly, the algorithm of Euclid. In attempting to generalize it to the multivariate case one easily arrives at the concept of polynomial remainder sequences and discovers the phenomenon of explosive coefficient growth (see, e.g., Brown 1971). To overcome this problem primitive polynomial remainder sequences (pprs) have been introduced. However, the classical primitive polynomial remainder sequence algorithm for computing gcds (Brown 1971) has one rather obvious disadvantage. In order to make a polynomial primitive, its content, which is the gcd of its coefficients, has to be computed. Therefore, additional gcd computations in the coefficient domain are necessary for computing a pprs. Fortunately, the costs of these content computations can be considerably reduced by subresultant techniques (Collins 1967, Brown and Traub 1971, Brown 1978) or trial division (Hearn 1979, Stoutemyer 1985).


Algebraic Equation Induction Hypothesis Great Common Divisor Input Specification Common Zero 
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  1. Brown, W. S. (1971): On Euclid’s algorithm and the computation of polynomial greatest common divisors. J. ACM 18: 478–504.MATHCrossRefGoogle Scholar
  2. Brown, W. S. (1978): The subresultant PRS algorithm. ACM Trans. Math. Software 4: 237–249.MathSciNetMATHCrossRefGoogle Scholar
  3. Brown, W. S., Traub, J. F. (1971): On Euclid’s algorithm and the theory of subresultants. J. ACM 18:505–514.MathSciNetMATHCrossRefGoogle Scholar
  4. Buchberger, B. (1970): Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems (in German). Aeq. Math. 4: 374–383.MathSciNetMATHCrossRefGoogle Scholar
  5. Buchberger, B. (1985): Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N. K. (ed.): Multidimensional systems theory. D. Reidel, Dordrecht, pp. 184–232.CrossRefGoogle Scholar
  6. Canny, J. (1988): The complexity of robot motion planning. MIT Press, Cambridge, MA.Google Scholar
  7. Collins, G. E. (1967): Subresultants and reduced polynomial remainder sequences. J. ACM 14: 128–142.MATHCrossRefGoogle Scholar
  8. Gatermann, K. (1990): Symbolic solution of polynomial equation systems with symmetry. In: Proc. ISSAC’ 90, Tokyo, Japan, 1990, pp. 112–119.Google Scholar
  9. Gebauer, R., Kalkbrener, M., Wall, B., Winkler, F. (1991): CASA: a computer algebra package for constructive algebraic geometry. In: Proc. ISSAC’ 91, Bonn, Germany, 1991, pp. 403–410.Google Scholar
  10. Grigor’ev, D. Y., Vorobjov, N. N. (1988): Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput. 5: 37–64.MathSciNetCrossRefGoogle Scholar
  11. Hearn, A. C. (1979): Non-modular computation of polynomial gcds using trial division. In: Proc. EUROSAM’ 79, Marseille, France, 1979, pp. 227–239.Google Scholar
  12. Hentzelt, K. (1922): Zur Theorie der Polynomideale und Resultanten (in German). Math. Ann. 88: 53–79.MathSciNetMATHCrossRefGoogle Scholar
  13. Hurwitz, A. (1913): Über die Trägheitsformen eines algebraischen Moduls (in German). Ann. Math. Pura Appl. (3a) 20: 113–151.CrossRefGoogle Scholar
  14. Kalkbrener, M. (1991): Three contributions to elimination theory. Ph.D. thesis, Research Institute for Symbolic Computation, University of Linz, Linz, Austria.Google Scholar
  15. Kalkbrener, M. (1992): Primitive polynomial remainder sequences in elimination theory. Tech. Rep. MSI-Ser. 92-12, Cornell University, Ithaca, USA.Google Scholar
  16. Lazard, D. (1981): Résolution des systèmes d’équationes algébriques (in French). Theor. Comput. Sci. 15: 77–110.MathSciNetMATHCrossRefGoogle Scholar
  17. Macaulay, F. S. (1916): Algebraic theory of modular systems. Cambridge University Press, Cambridge.MATHGoogle Scholar
  18. Noonburg, V. W. (1989): A neural network modeled by an adaptive Lotka-Volterra system. SIAM J. Appl. Math. 49: 1779–1792.MathSciNetMATHCrossRefGoogle Scholar
  19. Ritt, J. F. (1950): Differential algebra. American Mathematical Society, New York.MATHGoogle Scholar
  20. Stoutemyer, D. R. (1985): Polynomial remainder sequence greatest common divisors revisited. In: Proc. Second RIKEN Int. Symp. on Symbolic and Algebraic Computation by Computers. World Scientific Publishing, Singapore, pp. 1–12.Google Scholar
  21. van der Waerden, B. L. (1940): Moderne Algebra II, 2nd edn. (in German). Springer, Berlin.MATHGoogle Scholar
  22. Wu, W.-t. (1984): Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4: 207–235.Google Scholar

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© Springer-Verlag Wien 1995

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  • Michael Kalkbrener

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