On the Computation of Newton Polytopes of Eliminants

TL;DR

This paper presents a new algorithm for computing Newton polytopes of eliminants in polynomial systems, leveraging mixed fiber polytopes and tropical geometry to improve practical performance over existing methods.

Contribution

The authors develop an algorithm based on mixed subdivisions and tropical geometry to efficiently compute Newton polytopes of eliminants, advancing computational algebraic geometry.

Findings

Algorithm shows improved performance over existing methods

Utilizes mixed fiber polytopes and tropical geometry techniques

Applicable to differential elimination problems

Abstract

For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input equations. We use their results in combination with mixed subdivisions to design an algorithm computing these special polytopes. We demonstrate the increase in practical performance of our algorithm compared to existing methods using tropical geometry and discuss the differences that lead to this increase in performance. We also demonstrate an application of our work to differential elimination.