On the relationship between the Dixon-based resultant construction and the supports of polynomial systems

Abstract

Different matrix based resultant formulations use the support of the polynomials in a polynomial system in various ways for setting up resultant matrices for computing resultants. Every formulation suffers, however, from the fact that for most polynomial systems, the output is not a resultant, but rather a nontrivial multiple of the resultant, called a projection operator. It is shown that for the Dixon-based resultant methods, the degree of the projection operator of unmixed polynomial systems is determined by the support hull of the support of the polynomial system. This is similar to the property that the Newton polytope of a support determines the degree of the resultant for toric zeros.The support hull of a given support is similar to its convex hull (Newton polytope) except that instead of the Euclidean distance, the support hull is defined using relative quadrant (octant) position of points. The concept of a support hull interior point with respect to a support is defined. It is shown that for unmixed polynomial systems, generic inclusion of terms corresponding to support hull interior points does not change the size of the Dixon matrix (hence, the degree of the projection operator). The support hull of a support is the closure of the support with respect to support-interior points.The above results are shown to hold both for the generalized Dixon formulation as well as for Sylvester-type Dixon dialytic matrices constructed using the Dixon formulation.It is proved that for an unmixed polynomial system, the size of the Dixon matrix is less than or equal to the Minkowski sum of the alternating sums of the successive projections of the support of the polynomial system. This is a refinement of the result in Kapur and Saxena 1996 about the size of the Dixon matrix of a polynomial system, where it was shown that for the unmixed polynomial system, the size of the Dixon matrix is less than or equal to the Minkowski sum of the successive projection of the support.Many other combinatorial properties of the size of the Dixon matrix and the structure of the Dixon polynomial of a given polynomial system are related to the support hull of the polynomial system and their projections along different dimensions.This research is supported in part by NSF grant nos. CCR-0203051, CDA-9503064 and a grant from the Computer Science Research Institute at Sandia National Labs.