Cayley-Dixon Resultant Matrices of Multi-univariate Composed Polynomials

  • Arthur D. Chtcherba
  • Deepak Kapur
  • Manfred Minimair
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


The behavior of the Cayley-Dixon resultant construction and the structure of Dixon matrices are analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. It is shown that a Dixon projection operator (a multiple of the resultant) of the composed system can be expressed as a power of the resultant of the outer polynomial system multiplied by powers of the leading coefficients of the univariate polynomials substituted for variables in the outer system. The derivation of the resultant formula for the composed system unifies all the known related results in the literature. A new resultant formula is derived for systems where it is known that the Cayley-Dixon construction does not contain any extraneous factors. The approach demonstrates that the resultant of a composed system can be effectively calculated by considering only the resultant of the outer system.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sederberg, T., Goldman, R.: Algebraic geometry for computer-aided design. IEEE Computer Graphics and Applications 6, 52–59 (1986)CrossRefGoogle Scholar
  2. 2.
    Hoffman, C.: Geometric and Solid modeling. Morgan Kaufmann Publishers, Inc., San Mateo (1989)Google Scholar
  3. 3.
    Morgan, A.: Solving polynomial systems using continuation for Scientific and Engineering problems. Prentice-Hall, Englewood Cliffs (1987)zbMATHGoogle Scholar
  4. 4.
    Chionh, E.: Base points, resultants, and the implicit representation of rational Surfaces. PhD dissertation, Univ. of Waterloo, Dept. of Computer Science (1990)Google Scholar
  5. 5.
    Zhang, M.: Topics in Resultants and Implicitization. PhD thesis, Rice University, Dept. of Computer Science (2000)Google Scholar
  6. 6.
    Bajaj, C., Garrity, T., Warren, J.: On the application of multi-equational resultants. Technical Report CSD-TR-826, Dept. of Computer Science, Purdue (1988)Google Scholar
  7. 7.
    Ponce, J., Kriegman, D.: Elimination Theory and Computer Vision: Recognition and Positioning of Curved 3D Objects from Range. In: Donald, K., Mundy (eds.) Symbolic and Numerical Computation for AI. Academic Press, London (1992)Google Scholar
  8. 8.
    Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using the Dixon resultants. In: ACM ISSAC 1994, Oxford, England, pp. 99–107 (1994)Google Scholar
  9. 9.
    Rubio, R.: Unirational Fields. Theorems, Algorithms and Applications. PhD thesis, University of Cantabria, Santander, Spain (2000)Google Scholar
  10. 10.
    Cheng, C.C., McKay, J.H., Wang, S.S.: A chain rule for multivariable resultants. Proceedings of the American Mathematical Society 123, 1037–1047 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jouanolou, J.P.: Le formalisme du résultant. Adv. Math. 90, 117–263 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Minimair, M.: Factoring resultants of linearly combined polynomials. In: Sendra, J.R. (ed.) Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 207–214. ACM, New York (2003); ISSAC 2003, Philadelphia, PA, USA (August 3-6 2003)CrossRefGoogle Scholar
  13. 13.
    Minimair, M.: Computing resultants of partially composed polynomials. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2004, St. Petersburg, Russia, pp. 359–366. TUM München (2004)Google Scholar
  14. 14.
    Hong, H., Minimair, M.: Sparse resultant of composed polynomials I. J. Symbolic Computation 33, 447–465 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Buse, L., Elkadi, M., Mourrain, B.: Generalized resultants over unirational algebraic varieties. J. Symbolic Computation 29, 515–526 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kapur, D., Saxena, T.: Comparison of various multivariate resultants. In: ACM ISSAC 1995, Montreal, Canada (1995)Google Scholar
  17. 17.
    Hong, H.: Subresultants under composition. J. Symb. Comp. 23, 355–365 (1997)zbMATHCrossRefGoogle Scholar
  18. 18.
    Kapur, D., Saxena, T.: Extraneous factors in the Dixon resultant formulation. In: ISSAC, Maui, Hawaii, USA, pp. 141–147 (1997)Google Scholar
  19. 19.
    McKay, J.H., Wang, S.S.: A chain rule for the resultant of two polynomials. Arch. Math. 53, 347–351 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Zhang, M., Goldman, R.: Rectangular corner cutting and sylvester \(\mathcal{A}\)-resultants. In: Proc. of the ISSAC, St. Andrews, ScotlandGoogle Scholar
  21. 21.
    Chtcherba, A.D.: A new Sylvester-type Resultant Method based on the Dixon-Bézout Formulation. PhD dissertation, University of New Mexico, Department of Computer Science (2003)Google Scholar
  22. 22.
    Dixon, A.: The eliminant of three quantics in two independent variables. Proc. London Mathematical Society 6, 468–478 (1908)CrossRefGoogle Scholar
  23. 23.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, 1st edn. Springer, New York (1998)zbMATHGoogle Scholar
  24. 24.
    Chtcherba, A.D., Kapur, D., Minimair, M.: Cayley-dixon construction of resultants of multi-univariate composed polynomials. Technical Report TR-CS-2005-15, Dept. of Computer Science, University of New Mexico (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Arthur D. Chtcherba
    • 1
  • Deepak Kapur
    • 2
  • Manfred Minimair
    • 3
  1. 1.Dept. of Computer ScienceUniversity of Texas – Pan AmericanEdinburgUSA
  2. 2.Dept. of Computer ScienceUniversity of New MexicoAlbuquerqueUSA
  3. 3.Dept. of Mathematics and Computer ScienceSeton Hall UniversitySouth OrangeUSA

Personalised recommendations