A Complete Analysis of Resultants and Extraneous Factors for Unmixed Bivariate Polynomial Systems using the Dixon formulation

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 can also be obtained for such polynomial systems. These results are shown to be a generalization of related results recently 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. The concept of the interior of a support is introduced; a generic inclusion of terms corresponding to support interior points in a polynomial system does not affect the nature of the projection operator computed by the Dixon construction. For generic mixed bivariate systems, ”good” Sylvester type matrices can be constructed by solving an optimization problem on their supports. The determinant of such a matrix gives a projection operator with a low degree extraneous factor. The results are illustrated on a variety of examples.