A Greedy Approach to the Canny-Emiris Formula

Abstract

The Canny-Emiris formula [3] gives the sparse resultant as a ratio between the determinant of a Sylvester-type matrix and a minor of it, by a subdivision algorithm. The most complete proof of the formula was given by D'Andrea et al. in [9] under general conditions on the underlying mixed subdivision. Before the proof, Canny and Pedersen had proposed [5] a greedy algorithm which provides smaller matrices, in general. The goal of this paper is to give an explicit class of mixed subdivisions for the greedy approach such that the formula holds, and the dimensions of the matrices are reduced compared to the subdivision algorithm. We measure this reduction for the case when the Newton polytopes are zonotopes generated by n line segments (where n is the rank of the underlying lattice), and for the case of multihomogeneous systems. This article comes with a JULIA implementation of the treated cases.