Mixed subdivisions suitable for the Canny-Emiris formula

TL;DR

This paper advances the computation of sparse resultants using mixed subdivisions, optimizing matrix sizes for specific polytopes, and introduces applications in computer vision and surface implicitization, along with a tropical proof approach.

Contribution

It improves matrix size bounds for the Canny-Emiris formula in multihomogeneous and zonotopal cases and provides new applications and a tropical proof method.

Findings

Matrix sizes are reduced for zonotopes and multihomogeneous systems.

New cases of mixed subdivisions are identified and conjectured for minimal matrices.

Applications demonstrated in computer vision and surface implicitization.

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.