A sparse overview on sparse resultants

TL;DR

This survey reviews recent advances in the theory and computation of sparse resultants, focusing on determinantal formulas, Koszul complex methods, and Newton polytope computation within algebraic geometry.

Contribution

It provides a comprehensive overview of the construction, comparison, and computational techniques related to sparse resultants, highlighting recent progress and methodologies.

Findings

Analysis of the Canny-Emiris formula for sparse resultants

Comparison of determinantal formulas with Koszul complex approaches

Techniques for computing Newton polytopes of sparse resultants

Abstract

In this survey, we give an overview of advances in the theory and computation of sparse resultants. First, we examine the construction and proof of the Canny-Emiris formula, which gives a rational determinantal formula. Second, we discuss and compare the latter with the computation of the sparse resultant as the determinant of the Koszul complex given by $n + 1$ nef divisors in a toric variety. Finally, we cover techniques for computing the Newton polytope of sparse resultants.