Compact Formulae in Sparse Elimination

TL;DR

This paper surveys compact formulae in sparse elimination, including root bounds, resultants, discriminants, and implicit equations, highlighting recent advances and introducing new determinantal formulas for specific systems.

Contribution

It introduces new determinantal formulas for the discriminant of sparse multilinear systems and discusses efficient computation of Newton polytopes for various algebraic objects.

Findings

Generated a new determinantal formula for the discriminant of sparse multilinear systems.

Presented efficient methods for computing Newton polytopes of algebraic objects.

Compared recent formulae for root bounds and mixed volume calculations.

Abstract

It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-B{\'e}zout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.