Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants

Resultants are defined in the sparse (or toric) context in order to exploit the structure of the polynomials as expressed by their Newton polytopes. Since determinantal formulae are not always possible, the most efficient general method for computing resultants is as the ratio of two determinants. This is made possible by Macaulay’s seminal result [15] in the dense homogeneous case, extended by D’Andrea [6] to the sparse case. However, the latter requires a lifting of the Newton polytopes, defined recursively on the dimension. Our main contribution is a single lifting function of the Newton polytopes, which avoids recursion, and yields a simpler algorithm for computing Macaulay-type formulae of sparse resultants, in the case of generalized unmixed systems, where all Newton polytopes are scaled copies of each other. In the mixed subdivision used to construct the matrices, our algorithm defines significantly fewer cells than D’Andrea’s, and is easier to implement and analyze, though the matrices are same in both cases. Our approach provably extends to mixed systems of up to 4 polynomials, and those whose Newton polytopes have a sufficiently different face structure, but it should be generalizable to any mixed system. Our Maple implementation is applied to study a full example.