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 rational formulae. This is made possible by Macaulay’s famous determinantal formula in the dense homogeneous case, extended by D’Andrea 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 method for computing Macaulay-type formulae of sparse resultants. We focus on the case of generalized unmixed systems, where all Newton polytopes are scaled copies of each other, and sketch how our approach may extend to mixed systems of up to four polynomials, as well as those whose Newton polytopes have a sufficiently different face structure. In the mixed subdivision used to construct the matrices, our algorithm defines significantly fewer cells than D’Andrea’s, though the matrix formulae are same. We discuss asymptotic complexity bounds and illustrate our results by fully studying a bivariate example.