Getting smaller Dixon dialytic matrix through system manipulating

Abstract

Resultant is the result of eliminating the variables from a system of polynomials. Compared with other matrix based methods such as Sylvester, Macaulay and Sturmfels et al, Dixon formulation is one of the most efficient tools to compute resultants. Previously it was thought that the Dixon resultant formulation defines a completely different class of resultant formulations. Recently a number of mixed constructions have been proposed where some parts of the matrix are of Sylvester type and some parts are of the Dixon type. Dixon Dialytic matrix is one such formulation. These constructions produce smaller matrices at the expense of more complicated matrix entries. Dixon Dialytic method, similar to other resultant methods, are subjected to producing of extraneous factors along with the exact resultant, however; recognition of these factors takes a long process and sometimes the presence of them can make computing the projection operator, impractical. It is proposed a new efficient method for reducing or eliminating the number of extraneous factors from the Dixon Dialytic resultant construction. In the Dixon Dialytic formulation, there is a parameter which is used to further reduce the number of the extraneous factors. Experimental results suggest success of the new algorithm. The scope of this research falls under computational algebra, algebraic geometry and elimination theory.