Acm Symposium on Theory of Computing (stoc 96) Sparsity Considerations in Dixon Resultants

New results relating the sparsity of nonhomogeneous polynomial systems and computation of their projection operator (a non-trivial multiple of the multivariate resultant) using Dixon's method are developed. It is demonstrated that Dixon's method of computing resultants, despite being classical, implicitly exploits the sparse structure of input polynomials. It is proved that the size of the Dixon matrix, and the complexity of computing the resultant using Dixon's method is not determined by the total degree of the polynomial system, but rather by the structure of the Newton polytopes of the polynomial system. An exact formula for the size of the Dixon matrix of unmixed polynomial systems is derived in terms of their Newton polytopes. This relationship is exploited to tightly bound the size of the Dixon matrices of multi-homogeneous polynomial systems and also to devise an algorithm for constructing their Dixon matrices eeciently using dense polynomial interpolation. This work links the classical Dixon formulation (developed in 1908) to the modern line of sparsity analysis based on Newton poly-topes.