[Paper Review] Computing Tropical Varieties in Macaulay2

Carlos Améndola, Kathlén Kohn+6 more|arXiv (Cornell University)|Oct 29, 2017

Polynomial and algebraic computation2 references3 citations

TL;DR

This paper introduces a user-friendly Macaulay2 package for computing tropical varieties, integrating Gfan and Polymake functionalities while abstracting away low-level complexities. It enables efficient computation of tropical prevarieties, varieties, and stable intersections, supports non-prime ideals with multiplicity computation, and allows users to toggle between min and max conventions without prior knowledge of underlying algorithms.

ABSTRACT

We introduce a package for doing tropical computations in Macaulay2. The package draws on the functionality of Gfan and Polymake while making the process as simple as possible for the end user. This provides a powerful and user friendly tool for computing tropical varieties requiring little prerequisite knowledge.

Motivation & Objective

• To provide a user-friendly interface for tropical geometry computations in Macaulay2 without requiring expertise in Gfan or Polymake.
• To abstract the complexity of Gröbner fan computations and tropicalization algorithms behind intuitive functions.
• To extend functionality beyond existing tools by supporting multiplicities for non-prime ideals and user-selectable min/max conventions.
• To unify and streamline tropical computations within Macaulay2 as a central package for future development.
• To enable accurate computation of tropical prevarieties and stable intersections, including verification of tropical bases.

Proposed method

• The package uses the gfanInterface to call Gfan for Gröbner fan and tropical variety computations, while leveraging Polymake for advanced polyhedral geometry and intersection theory.
• It automatically selects between two algorithms: tropical starting cone traversal (for prime ideals) and brute-force Gröbner fan computation (for non-prime ideals), with user-specified flags.
• The package pre-processes non-homogeneous inputs to make them compatible with Gfan, reducing user input complexity.
• It constructs tropical cycles from fans and attached multiplicities, and verifies balancing using the isBalanced function.
• For stable intersections, it defaults to Polymake’s atint extension if available, falling back to Gfan otherwise.
• Users can choose between min and max conventions via a configuration option at package loading.

Experimental results

Research questions

• RQ1 How can tropical geometry computations be made accessible to users without deep knowledge of Gfan or Polymake?
• RQ2 What strategies can efficiently compute tropical varieties for both prime and non-prime ideals?
• RQ3 Can a unified Macaulay2 package be built to handle tropical prevarieties, varieties, and stable intersections with multiplicity computation?
• RQ4 How can user-configurable conventions (min vs. max) be seamlessly integrated into a high-level interface?
• RQ5 To what extent can the package verify tropical bases and compute stable intersections reliably?

Key findings

• The package successfully computes the tropical variety of a prime ideal like ⟨x + y + 1⟩ in ℂ*² as a 1-dimensional fan with three rays at (−1,−1), (1,0), and (0,1), each with multiplicity 1.
• For non-prime ideals, the package uses the more expensive gfan_tropicalbruteforce command and correctly computes multiplicities, ensuring accuracy over heuristic methods.
• The tropical prevariety of two polynomials f = x + y + z and g = x² + y² + z² is two-dimensional, while the true tropical variety of ⟨f,g⟩ is one-dimensional, proving f and g are not a tropical basis.
• The stable intersection of a random line and a random quadric in ℙ² results in a single point at the origin with multiplicity 2, confirming tropical Bézout’s theorem.
• The package correctly identifies that the tropical cycle formed by three rays in ℝ² with equal multiplicities is balanced, validating the geometric consistency of the output.
• The implementation supports user choice of min or max convention via configuration, with no impact on correctness or performance.