Browsing by Author "Chionh, Eng-Wee"

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    A Set of Convolution Identities Relating the Blocks of Two Dixon Resultant Matrices
    (1999-06-16) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Resultants for bivariate polynomials are often represented by the determinants of very big matrices. Properly grouping the entries of these matrices into blocks is a very effective tool for studying the properties of these resultants. Here we derive a set of convolution identities relating the blocks of two Dixon bivariate resultant representations.
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    Hybrid Dixon Resultants
    (1998-05-13) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Dixon [1908] describes three distinct homogeneous determinant representations for the resultant of three bivariate polynomials of bidegree(m,n). These Dixon resultants are the determinants of matrices of orders 6mn, 3mn and 2mn, and the entries of these matrices are respectively homogeneous of degrees 1, 2, and 3 in the coefficients of the original three polynomial equations. Here we mix and match columns from these three Dixon matrices to construct a large assortment of new hybrid determinant representations of orders ranging from 2mn to 6mn for the resultant of three bivariate polynomials of bidegree (m,n).
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    The Block Structure of Three Dixon Resultants and Their Accompanying Transformation Matrices
    (1999-06-16) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Dixon [1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m,n) . The first technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn . The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix of order 2mn . The third method employs a combination of Cayley's determinant device with Sylvester's dialytic method to build the resultant as the determinant of a matrix of order 3mn . Here relations between these three resultant formulations are derived and the structure of the transformations between these resultant matrices is investigated. In particular, it is shown that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves have elegant symmetry properties. Elementary entry formulas for the transformation matrices are also provided. In light of these results, the three Dixon resultant matrices are reexamined and shown to have natural block structures compatible with the block structures of the transformation matrices. These block structures are analyzed here and applied along with the block structures of the transformation matrices to simplify the calculation of the entries of the Dixon resultants of order 2mn and 3mn and to make these calculations more efficient by removing redundant computations.
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    Transformations and Transitions from the Sylvester to the Bezout Resultant
    (1999-06-17) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    A simple matrix transformation linking the resultant matrices of Sylvester and Bezout is derived. This transformation matrix is then applied to generate an explicit formula for each entry of the Bezout resultant, and this entry formula is used, in turn, to construct an efficient recursive algorithm for computing all the entries of the Bezout matrix. Hybrid resultant matrices consisting of some columns from the Sylvester matrix and some columns from the Bezout matrix provide natural transitions from the Sylvester to the Bezout resultant, and allow as well the Bezout construction to be generalized to two polynomials of different degrees. Such hybrid resultants are derived here, employing again the transformation matrix from the Sylvester to the Bezout resultant.