On the mixed cayley-sylvester resultant matrix
Authors: 
Weikun Sun, Hongbo LiAuthors Info & Claims
Pages 146 - 159
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.
References
[1]
D. N. Bernstein. The number of roots of a system of equations, Func. Anal. and Appl., 9(2):183-185, 1975.
[2]
E. Bézout. Théorie Générale des Équations Algébriques, Paris, 1770.
[3]
A. Cayley. On the theory of elimination, Dublin Math. J., II:116-120, 1848.
[4]
E. W. Chionh, M. Zhang and R. N. Goldman. Block structure of three Dixon resultants and their accompanying transformation Matrices, Technical Report, TR99-341, Department of Computer Science, Rice University, 1999.
[5]
E. W. Chionh, M. Zhang and R. N. Goldman. Fast computations of the Bezout and the Dixon resultant matrices, J. Symb. Comp., 33(1):13-29, 2002.
[6]
D. Cox, J. Little and D. O'shea. Using Algebraic Geometry, Springer-Verlag, New York, 1998.
[7]
A. L. Dixon. The elimination of three quantics in two independent variables. Proc. London Math. Soc., 6(5):49-69, 473-492, 1908.
[8]
I. Emiris and B. Mourrain. Matrices in elimination theory. J. Symb. Comp., 28(1):3-43, 1999.
[9]
I. M. Gelfand, M. M. Kapranov and A. Z. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
[10]
C. M. Hoffman. Algebraic and numeric techniques for offsets and blends. In W. Dahmen, M. Gasca and C. Micchelli editors, Computations of Curves and Surfaces, Kluwer Academic Publishers, 1990.
[11]
D. Kapur, T. Saxena and L. Yang. Algebraic and geometric reasoning using the Dixon resultants. In ACM ISSAC 94, pages 99-107, Oxford, England, 1994.
[12]
P. Pedersen and B. Sturmfels. Product formulas for resultants and chow forms. Math. Zeitschrift, 214:377-396, 1993.
[13]
J. Sylvester. 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.
[14]
S. Zhao and H. Fu. An extended fast algorithm for constructing the Dixon resultant matrix. Science in China Series A: Mathematics, 48(1):131-143, 2005.
- On the mixed cayley-sylvester resultant matrix
Recommendations
The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution
Blind image deconvolution ($$\text {BID}$$BID) is one of the most important problems in image processing, and it requires the determination of an exact image $$\mathcal {F}$$F from a degraded form of it $$\mathcal {G}$$G when little or no information ...
Cayley-Dixon resultant matrices of multi-univariate composed polynomials
CASC'05: Proceedings of the 8th international conference on Computer Algebra in Scientific ComputingThe 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 ...
Applications of the generalized Sylvester matrix
Recently a Sylvester matrix for several polynomials has been defined, establishing the relative primeness and the greatest common divisor of polynomials. Using this matrix, we perform qualitative analysis of several polynomials regarding the inners, the ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Sponsors
- State Key Laboratory of Software Development Environment: State Key Laboratory of Software Development Environment
- Key Laboratory of Mathematics: Key Laboratory of Mathematics
- Ministry of Education of China: Ministry of Education of China
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 20 September 2006
Author Tags
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 02 Feb 2025