Extended Dixon's resultant and its applications

  • Quoc-Nam Tran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1360)


Dixon's resultant method is an efficient way of simultaneously eliminating several variables from a system of nonlinear polynomial equations at a time. However, the method only works for systems of n + 1 generic n-degree polynomials in n variables and does not work for most algebraic and geometric problems. In this paper, by using techniques from pseudoinverse theory and linear transformations, the author extends Dixon's resultant method to an arbitrary system of n + 1 nontrivial polynomials in n variables where the Dixon matrix can be singular or even nonsquare. The extended method does not require any precondition — this is the main contribution of the paper. The extended method can be used efficiently as an elimination method in geometric reasoning, computer aided geometric design (CAGD) and solid modeling. Several examples show that the new method works well also in situations where other methods (of the same subject) may fail to give a correct answer.


Power Product Generalize Inverse Full Column Rank Common Root Arbitrary System 
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  1. [BIG74]
    A. Ben-Israel and T. N. E. Greville. Generalized Inverses: Theory and Applications. John Wiley & sons, New York, 1974.Google Scholar
  2. [Bje51a]
    A. Bjerhammar. Applications of calculus of matrices to method of least squares with special reference to geodetic calculations. Technical Report 49, Kungl. Tekn. Högsk. Handl., 1951.Google Scholar
  3. [Bje51b]
    A. Bjerhammar. Rectangular reciprocal matrices with special reference to geodetic calculations. Bull. Géodésique, pages 188–220, 1951.Google Scholar
  4. [Bje68]
    A. Bjerhammar. A generalized matrix algebra. Technical Report 124, Kungl. Tekn. Högsk. Handl., 1968.Google Scholar
  5. [BO71]
    T. L. Boullion and P. L. Odell. Generalized Inverse Matrices. Wiley-Interscience, New York, 1971.Google Scholar
  6. [Buc85]
    B. Buchberger. Gröbner Bases: An algorithmic method in polynomial ideal theory. In N. K. Bose, editor, Multidimensional Systems Theory, chapter 6, pages 184–232. Reidel Publishing Company, Dodrecht, 1985.Google Scholar
  7. [Cay65]
    A. Cayley. On the theory of elimination. Cambridge and Dublin Math. J., 111:210–270, 1865.Google Scholar
  8. [CG95]
    E.-W. Chionh and R. N. Goldman. Elimination and resultants — multivariate resultants. IEEE Computer Graphics and Applications, 15(2):60–69, 1995.CrossRefGoogle Scholar
  9. [Chi90]
    E.-W Chionh. Base points, resultants, and the implicit representation of rational Surfaces. PhD thesis, Dept. Comp. Sci., Uni. of Waterloo, 1990.Google Scholar
  10. [Cho88]
    S.-C. Chou. Mechanical Geometry Theorem Proving. Reidel Publishing Company, Dordrecht, 1988.Google Scholar
  11. [Dix08a]
    A.-L. Dixon. The eliminant of three quantics in two independent variables. Proc. London Math. Soc., 7:49–69, 473–492, November 1908.Google Scholar
  12. [Dix08b]
    A.-L. Dixon. On a form of the elimination of two quantics. Proc. London Math. Soc., 6:468–478, June 1908.Google Scholar
  13. [Dix09a]
    A.-L. Dixon. The eliminant of the equations of four quadric surfaces. Proc. London Math. Soc., 8:340–352, December 1909.Google Scholar
  14. [Dix09b]
    A.-L. Dixon. Symbolic expressions for the eliminant of two binary quantics. Proc. London Math. Soc., 8:265–276, June 1909.Google Scholar
  15. [Far88]
    G. Farin. Curves and Surfaces for Computer Aided Geometric Design — A Practical Guide. Academic Press, New York, 1988.Google Scholar
  16. [Fre03]
    I. Fredholm. Sur une classe d'équations fonctionnelles. Acta Math., 27:365–390, 1903.Google Scholar
  17. [GW95]
    X. S. Gao and D. K. Wang. On the automatic derivation of a set of geometric formulae. J. Geometry, 53:79–88, 1995.CrossRefGoogle Scholar
  18. [Her75]
    I. N. Herstein. Topics in Algebra. John Wiley & Sons, USA, second edition, 1975.Google Scholar
  19. [Hof89]
    C. M. Hoffmann. Geometric and Solid Modeling — An Introduction. Morgan Kaufmann Publishers, Inc., San Mateo, California, 1989.Google Scholar
  20. [HWW95]
    H. Hong, D. Wang, and F. Winkler, editors. Algebraic Approaches to Geometric Reasoning. Baltzer Science Publisher, Amsterdam, 1995. Special issue of Ann. of Math. and Artif. Intell., 13(1,2).Google Scholar
  21. [Kal91]
    M. Kalkbrener. Three Contributions to Elimination Theory. PhD thesis, Research Institute for Symbolic Computation, Univ. Linz, Linz, Austria, 1991.Google Scholar
  22. [KS95]
    D. Kapur and T. Saxena. Comparison of various multivariate resultant formulations. In The proceedings of ISSAC'95, pages 187–194, 1995.Google Scholar
  23. [KSY94]
    D. Kapur, T. Saxena, and L. Yang. Algebraic and geometric reasoning using Dixon resultants. In The proceedings of ISSAC'94, pages 99–107, 1994.Google Scholar
  24. [Moo20]
    E. H. Moore. On the reciprocal of the general algebraic matrix (abstract). Bull. Amer. Math. Soc., 26:394–395, 1920.Google Scholar
  25. [Pen55]
    R. Penrose. A generalized inverse for matrices. Proc. Cambridge Philos. Soc., 51:406–413, 1955.Google Scholar
  26. [PW95]
    J. Pfalzgraf and D. Wang, editors. Automated Practical Reasoning. Texts and Monographs in Symbolic Computation. Springer-Verlag, Wien, 1995.Google Scholar
  27. [Rit50]
    J. F. Ritt. Differential Algebra, volume 33. AMS Colloquium Publications, New York, 1950.Google Scholar
  28. [Sa164]
    George Salmon. Lessons Introductory to the Modern Higher Algebra. Chelsea Publishing Company, Bronx, New York, fifth edition, 1964.Google Scholar
  29. [Tra95]
    Quoc-Nam Tran. A hybrid symbolic-numerical method for tracing surface-to-surface intersections. In The proceedings of ISSAC'95, pages 51–58, 1995.Google Scholar
  30. [Tra96]
    Quoc-Nam Tran. A Hybrid Symbolic-Numerical Approach in Computer Aided Geometric Design (CAGD) and Visualization. PhD thesis, Research Institute for Symbolic Computation (RISC-Linz), University of Linz, Austria, 1996.Google Scholar
  31. [Tra97a]
    Quoc-Nam Tran. Extending Dixon's resultant using pseudoinverse matrices and its apllications to geometric reasonning. Technical Report 97-11, RISC-Linz, The University of Linz, Austria, 1997. Talk at “Automated Deduction in Geometry”, September 1996, Toulouse, FranceGoogle Scholar
  32. [Tra97b]
    Quoc-Nam Tran. Extending Newton's method for finding the roots of an arbitrary system of equations and its applications. International Journal of Modeling and Simulation, 17(4), 1997. To appear.Google Scholar
  33. [TW97a]
    Quoc-Nam Tran and Franz Winkler. Casa reference manual (version 2.3). Technical report, RISC-Linz, The University of Linz, Austria, 1997.Google Scholar
  34. [TW97b]
    Quoc-Nam Tran and Franz Winkler. An overview of CASA — a system for computational algebra and constructive algebraic geometry. In T. Racio T. V. Effelterre and F. Winkler, editors, The Symbolic and Algebraic Computation (SAC) Newsletter, volume 2. Stichting CAN, Computer Algebra Nederland, Universiteit van Amsterdam, the Netherlands, 1997.Google Scholar
  35. [vdW40]
    B. L. van der Waerden. Moderne Algebra, volume 2. Springer Verlag, Berlin, second edition, 1940.Google Scholar
  36. [Wu94]
    W.-T. Wu. Mechanical Theorem Proving in Geometries. Texts and Monographs in Symbolic Computation. Springer-Verlag, Wien, 1994. Translated from Chinese by X. Jin and D. Wang.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Quoc-Nam Tran
    • 1
  1. 1.Research Institute for Symbolic Computation (RISC-Linz)Johannes Kepler UniversityLinzAustria

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