Comparison of various multivariate resultant formulations

Three most important resultant formulations are the Macaulay, Dixon and sparse resultant formulations. For most polynomial systems, however, the matrices constructed in these formulations become singular and the projection operator vanishes identically. In such cases, perturbation techniques for Macaulay formulation such as generalized characteristic polynomial ( GCP) and a method based on rank submatrix computation (RSG,), applicable to all three formulations, can be used, giving four methods, Macaulay/ GCP, Macaulay/RSC, Dixon/RSC and Sparse/RSC, for computing nontrivial projection operators. In this paper, these four methods are compared. It is shown that the Dixon matrix is (by a factor up to O(e’) for a certain class) smaller than the sparse resultant matrix which is (by a factor up to O (en) for a certain class) smaller than the Macaulay matrix. Empirical results confirm that Dixon/RSC’ is the most efficient, followed by Sparse/RSC then Macaulay/RSC and finally Macaulay/ GCP, which is found to be almost impractical. All four methods are found to generate extraneous factors in the projection operator. Efficient heuristics for interpolation, used to expand the resultant matrices. are also discussed.