A General Solver Based on Sparse Resultants

TL;DR

This paper introduces a general solver for polynomial systems using sparse resultants, leveraging Newton polytopes to reduce root-finding to an eigenproblem with high efficiency and accuracy.

Contribution

It presents a novel eigenproblem reduction method for sparse resultants that does not increase problem dimension, enabling a versatile solver for polynomial systems.

Findings

Efficient and accurate solutions for systems from vision, robotics, and biology.

Reduction to eigenproblem without increasing problem dimension.

Sparse elimination is effective for moderate-sized systems.

Abstract

Sparse (or toric) elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its degree is a function of the mixed volumes of the Newton polytopes. We sketch the sparse resultant constructions of Canny and Emiris and show how they reduce the problem of root-finding to an eigenproblem. A novel method for achieving this reduction is presented which does not increase the dimension of the problem. Together with an implementation of the sparse resultant construction, it provides a general solver for polynomial systems. We discuss the overall implementation and illustrate its use by applying it to concrete problems from vision, robotics and structural biology. The high efficiency and accuracy of the solutions suggest that sparse elimination may be the method of choice for systems of moderate size.