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On the Computation of Newton Polytopes of Eliminants

Rafael Mohr, Yulia Mukhina

TL;DR

This work links the Newton polytope of the eliminant arising from projecting a complete intersection to a mixed fiber polytope of the input Newton polytopes, enabling a vertex-based algorithm for construction. By leveraging a vertex oracle and mixed subdivisions (dimension $k$), it avoids tropical elimination and heavy Minkowski-sum computations, achieving substantial practical gains. The authors provide a rigorous algorithm, implement it in Julia, and demonstrate superior performance on implicitization benchmarks compared to tropical methods, with an additional application to differential elimination. The approach yields concrete, scalable bounds and interpolation strategies for eliminants, with broad relevance to implicitization and differential-algebraic computation.

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.

On the Computation of Newton Polytopes of Eliminants

TL;DR

This work links the Newton polytope of the eliminant arising from projecting a complete intersection to a mixed fiber polytope of the input Newton polytopes, enabling a vertex-based algorithm for construction. By leveraging a vertex oracle and mixed subdivisions (dimension ), it avoids tropical elimination and heavy Minkowski-sum computations, achieving substantial practical gains. The authors provide a rigorous algorithm, implement it in Julia, and demonstrate superior performance on implicitization benchmarks compared to tropical methods, with an additional application to differential elimination. The approach yields concrete, scalable bounds and interpolation strategies for eliminants, with broad relevance to implicitization and differential-algebraic computation.

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.

Paper Structure

This paper contains 12 sections, 7 theorems, 34 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement & Motivation.
  3. Methodology & Main Results.
  4. Related Works & Context.
  5. Organization of the Paper.
  6. Theoretical Foundations
  7. Mixed Fiber Polytopes
  8. Vertices of Mixed Fiber Polytopes
  9. Subdivisions of Point Configurations
  10. Computing Mixed Fiber Polytopes
  11. Implementation & Experiments
  12. An Application to Differential Elimination

Key Result

Lemma 2.2

Fix variables $\lambda_0,\dots,\lambda_k$ and let $\Delta_0,\dots,\Delta_k\subset \mathbb{R}^n$ be a collection of polytopes. Then $F_{p}(\lambda_0\Delta_0+\dots+\lambda_k\Delta_k)$ is a homogeneous polynomial of degree $k+1$ with coefficients polytopes contained in $\mathbb{R}^{n-k}$.

Figures (3)

  • Figure 1: A Fiber Polytope $F_{p}(\Delta)$.
  • Figure 2: A Coherent Mixed Subdivision
  • Figure 3: For $\delta\in \left\{d_0,d_1\right\}$, the covector $\gamma:=-1 \in \mathbb{R}^{*}$ attains its fiberwise minimum on the convex hull of $(\delta,0)$ and $(0,\delta)$.

Theorems & Definitions (29)

  • Example 1.1
  • Definition 2.1: Minkowski Integral, Fiber Polytope
  • Example 2.1
  • Lemma 2.2: Theorem 10 in esterov2008
  • Remark 2.1
  • Definition 2.3: Mixed Fiber Polytope
  • Remark 2.2
  • Theorem 2.4: Definition 5, Theorem 2 and Theorem 7 in esterov2008
  • Remark 2.3
  • Definition 2.5: Face, Vertex, Normal Cone
  • ...and 19 more