Resultants for unmixed bivariate polynomial systems produced using the Dixon formulation

Abstract

A necessary and sufficient condition on the support of a generic unmixed bivariate polynomial system is identified such that for polynomial systems with such support, the Dixon resultant formulation produces their resultants. It is shown that Sylvester-type matrices, called Dixon dialytic matrices, can also be obtained for such polynomial systems. These results are shown to be a generalization of related results reported by Chionh as well as Zhang and Goldman. For a support not satisfying the above condition, the degree of the extraneous factor in the projection operator computed by the Dixon formulation is calculated by analyzing how much the support deviates from a related rectangular support satisfying the condition. This degree estimate is an upper bound on the degree of the extraneous factor in the case when the polynomial system with a given unmixed support is generic. The concept of a support hull interior point of a support is introduced. A generic inclusion of terms corresponding to support hull interior points in an unmixed polynomial system is shown not to affect the degree of the projection operator computed by the Dixon construction. It is shown that the proposed construction for Dixon dialytic matrices works especially well for mixed bivariate systems. “Good” Sylvester-type matrices can be constructed by solving an optimization problem on their supports (by translating supports so that they have maximal overlap). The determinant of such a matrix gives a projection operator with a low degree extraneous factor, i.e. the degree of the extraneous factor in a projection operator computed from the original polynomial system is at least as much as the degree of the extraneous factor in a projection operator computed from the corresponding “translated” polynomial system. The results are illustrated on a variety of examples.