Abstract
For a generic n-degree polynomial system which contains n+1 polynomials in n variables, there are two classical resultant matrices, Sylvester resultant matrix and Cayley resultant matrix, lie at the two ends of a gamut of n+1 resultant matrices. This paper gives the construction of the n–1 resultant matrices which lie between the two pure resultant matrices by the combined method of Sylvester dialytic and Cayley quotient. Since the construction involves two steps, Cayley quotient and Sylvester dialytic, the block structure of these mixed resultant matrices are similar to that of Sylvester resultant matrix in large scale, and the detailed submatrices are similar to Dixon resultant matrix.
Supported partially by NKBRSF 2004CB318001 and NSFC 10471143.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bernstein, D.N.: The number of roots of a system of equations. Func. Anal. and Appl. 9(2), 183–185 (1975)
Bézout, E.: Théorie Générale des Équations Algébriques, Paris (1770)
Cayley, A.: On the theory of elimination. Dublin Math. J. II, 116–120 (1848)
Chionh, E.W., Zhang, M., Goldman, R.N.: Block structure of three Dixon resultants and their accompanying transformation Matrices, Technical Report, TR99-341, Department of Computer Science, Rice University (1999)
Chionh, E.W., Zhang, M., Goldman, R.N.: Fast computations of the Bezout and the Dixon resultant matrices. J. Symb. Comp. 33(1), 13–29 (2002)
Cox, D., Little, J., O’shea, D.: Using Algebraic Geometry. Springer, New York (1998)
Dixon, A.L.: The elimination of three quantics in two independent variables. Proc. London Math. Soc. 6(5), 49–69, 473–492(1908)
Emiris, I., Mourrain, B.: Matrices in elimination theory. J. Symb. Comp. 28(1), 3–43 (1999)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.Z.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston (1994)
Hoffman, C.M.: Algebraic and numeric techniques for offsets and blends. In: Dahmen, W., Gasca, M., Micchelli, C. (eds.) Computations of Curves and Surfaces. Kluwer Academic Publishers, Dordrecht (1990)
Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using the Dixon resultants. In: ACM ISSAC 1994, pp. 99–107. Oxford, England (1994)
Pedersen, P., Sturmfels, B.: Product formulas for resultants and chow forms. Math. Zeitschrift 214, 377–396 (1993)
Sylvester, J.: On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of sturms functions, and that of the greatest algebraic common measure. Philosophical Trans. 143, 407–548 (1853)
Zhao, S., Fu, H.: An extended fast algorithm for constructing the Dixon resultant matrix. Science in China Series A: Mathematics 48(1), 131–143 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sun, W., Li, H. (2006). On the Mixed Cayley-Sylvester Resultant Matrix. In: Calmet, J., Ida, T., Wang, D. (eds) Artificial Intelligence and Symbolic Computation. AISC 2006. Lecture Notes in Computer Science(), vol 4120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11856290_14
Download citation
DOI: https://doi.org/10.1007/11856290_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-39728-1
Online ISBN: 978-3-540-39730-4
eBook Packages: Computer ScienceComputer Science (R0)