The Block Structure of Three Dixon Resultants and Their Accompanying Transformation Matrices

Abstract

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.