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Comparison of Some Multivariable Hybrid Resultant Matrix Formulations


  • Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310,UTM Johor Bahru, Johor, Malaysia


Objective: To evaluate and compare different hybrid resultant formulations in relation to computational complexity, performance and optimality condition. Methods/Statistical Analysis: Hybrid matrices are evaluated using computer algebra system. Findings: we have shown that, none of the hybrid formulation works well with the exception of HDP4. However, after deleting the zero rows and columns the resulting matrix may not be a square matrix, on the other hand, we studied and established that none of the hybrid formulation produces a square matrix in general and likewise predicted that, the density or sparseness of the polynomial equations does not influence the performance of these hybrid matrix formulations. Applications/Improvements: This comparison reveals that, the existing hybrid methods for computing resultant are not efficient and therefore there is need for another formulations with will focus on the current limitations described in this paper.


Hybrid Matrix, Hybrid Resultant, Resultant, Resultant Matrix, System of Polynomials.

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  • Wang W, Lian X. Computations of multi-resultant with mechanization. Applied mathematics and computation.2005; 170(1):237–57.
  • Sylvester JJ. On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. Philosophical transactions of the Royal Society of London. 1853,143. p.407–548.
  • Zippel R. Effective polynomial computation. Kluwer Academic Publishers, Boston; 1993.
  • Kapur D, Lakshman YN. Elimination Methods: an Introduction. Symbolic and Numerical Computation for Artificial Intelligence B. Donald et. al. Academic Press; 1992.
  • Cayley A. On the theory of elimination. Cambridge and Dublin Mathematical Journal. 1848, 3. p.116–20.
  • Chtcherba AD, Kapur D. Resultants for unmixed bivariate polynomial systems produced using the Dixon formulation.Journal of Symbolic Computation. 2004; 38(2):915–58.
  • Kapur D, Saxena T. Comparison of various multivariate resultant formulations. Proceedings of the 1995 international symposium on Symbolic and algebraic computation; ACM. 1995.
  • Kapur D, Saxena T, Yang L, editors. Algebraic and geometric reasoning using Dixon resultants. Proceedings of the international symposium on Symbolic and algebraic computation; ACM. 1994.
  • Kapur D, Saxena T, editors. Extraneous factors in the Dixon resultant formulation. Proceedings of the 1997 international symposium on Symbolic and algebraic computation; ACM. 1997.
  • Dixon AL. The eliminant of three quantics in two independent variables. Proceedings of the London Mathematical Society. 1909; 2(1):49–69.
  • Weyman J, Zelevinsky A. Determinantal formulas for multigraded resultants. Journal of Algebraic Geometry. 1994; 3(4):569–98.
  • Chionh E-W, Zhang M, Goldman R. Transformation and Transitions from the Sylvester to the Bézout Resultant.Citeseer, 1999.
  • D’Andrea C, Emiris IZ. Hybrid sparse resultant matrices for bivariate polynomials. Journal of Symbolic Computation.2002; 33(5):587–608.
  • Khetan A. The resultant of an unmixed bivariate system.Journal of Symbolic Computation. 2003; 36(3):425–42.
  • Ahmad SN. Construction and implementation of a hybrid resultant matrix algorithm based on the sylvester-bezout formulation [PhD Thesis]. University of Technology Malaysia 2016.
  • Macaulay F. Some formulae in elimination. Proceedings of the London Mathematical Society. 1902; 1(1):3–27.
  • Karimisangdehi S. New algorithms for optimizing the sizes of dixon and Dixon dialytic matrices: Universiti Teknologi Malaysia, Faculty of Science. 2012.
  • Chionh E-W, Zhang M, Goldman RN, editors. The block structure of three Dixon resultants and their accompanying transformation matrices. Journal of Symbolic Computation – Elsevier. 1999.
  • Chionh E-W, Zhang M, Goldman RN. Fast computation of the Bezout and Dixon resultant matrices. Journal of Symbolic Computation. 2002; 33(1):13–29.
  • Zhang M, Chionh E, Goldman R. Hybrid dixon resultants.The Mathematics of Surfaces. 1998, 8, 193–212.
  • Awange JL, Grafarend EW, Paláncz B, Zaletnyik P. Algebraic geodesy and geoinformatics: Springer Science & Business Media; 2010.
  • Paláncz B, Zaletnyik P, Awange JL, Grafarend EW. Dixon resultant’s solution of systems of geodetic polynomial equations.Journal of Geodesy. 2008; 82(8):505–11.
  • Chtcherba AD, Kapur D. Conditions for exact resultants using the Dixon formulation. Proceedings of the 2000 international symposium on Symbolic and algebraic computation. ACM. 2000.
  • Emiris IZ, Kalinka T, Konaxis C. Geometric operations using sparse interpolation matrices. Graphical Models.2015, 82:99–109.
  • Li W, Yuan C-M, Gao X-S. Sparse difference resultant.Journal of Symbolic Computation. 2015; 68:169–203.


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