Tentative Title: Algebraic Algorithms for D-modules and Numerical Analysis

Title: Multivariate Resultants based on Cayley-Dixon's Method

Multivariate resultant formulations based on Cayley-Dixon's construction which was generalized by Kapur, Saxena and Yang (ISSAC'94) is presented. A new way of generating Sylvester-type multivariate resultant matrices based on Cayley-Dixon construction, called the Dixon dialytic method, is proposed. For a large class of polynomial systems, these resultant formulations are shown to compute resultants exactly (i.e., without any extraneous factors). The concept of a support hull of an unmixed polynomial system (in which all polynomials have the same set of terms with nonzero coefficients) is introduced. The degree of a projection operator of an unmixed polynomial system constructed using the Dixon formulation can be analyzed in terms of its support hull. Support hull thus plays the same role for the resultant constructions based on the Dixon formulation as the convex hull of the support of the polynomial system for determining the degree of the resultant. A method for discovering a variable order for which the Dixon formulation either produces the resultant or yields a projection operator with a minimal degree extraneous factor is proposed.

This work is jointly done with my former student Arthur Chtcherba.