Mixed subdivisions suitable for the Canny-Emiris formula
Carles Checa, Ioannis Z. Emiris
TL;DR
The paper tackles efficient computation of sparse resultants via the Canny-Emiris framework by introducing a new family of affine liftings ρ, determined by a vector $v$ outside the hyperplane arrangement, that yield mixed subdivisions for which the Canny-Emiris determinant-ratio formula remains valid while reducing matrix size, particularly when Newton polytopes are zonotopes and systems are multihomogeneous. It establishes a tropical refinement theorem showing that refining mixed subdivisions corresponds to moving $v$ within chambers, thereby proving the Canny-Emiris formula for these liftings and providing a novel, purely tropical proof of the refinement result. In the zonotope/multihomogeneous setting, the authors derive a combinatorial expression for the size of the greedy matrix $\mathcal{H}_{\mathcal{G}}$ in terms of type functions $\varphi_b$, giving explicit row-count bounds and a conjecture that ties minimal matrix size to degree reverse lexicographic orders. The work also connects to practical applications in computer vision (e.g., 5-point problem) and surface implicitization, and includes Julia code to implement the constructions; together, these contributions offer a path to smaller, more tractable resultant matrices while preserving the fundamental determinant-ratio structure.
Abstract
The Canny-Emiris formula gives the sparse resultant as the ratio of the determinant of a Sylvester-type matrix over a minor of it, both obtained via a mixed subdivision algorithm. The same authors gave an explicit class of mixed subdivisions for the greedy approach so that the formula holds, and the dimension of the constructed matrices is smaller than that of the subdivision algorithm, following the approach of Canny and Pedersen. Our method improves upon the dimensions of the matrices when the Newton polytopes are zonotopes and the systems are multihomogeneous. In this text, we provide more such cases, and we conjecture which might be the liftings providing minimal size of the resultant matrices. We also describe two applications of this formula, namely in computer vision and in the implicitization of surfaces, while offering the corresponding JULIA code. We finally introduce a novel tropical approach that leads to an alternative proof of one of the results.