[Paper Review] Enumerating Polytropes

Ngoc Mai Tran|arXiv (Cornell University)|Oct 8, 2013

Polynomial and algebraic computation15 references4 citations

TL;DR

This paper introduces a novel Gröbner fan-based approach to enumerate tropical types of full-dimensional and maximal polytropes in tropical projective space $\mathbb{TP}^{n-1}$, leveraging the refinement of the polytrope map fan and the bipartite binomial fan. It establishes a bijection between tropical types and compatible sets of bipartite and triangle binomials, enabling efficient computation: 1013 tropical types for $n=4$ and 27,248 maximal types for $n=5$, resolving a key open problem in tropical geometry.

ABSTRACT

Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in $\mathbb{TP}^{n-1}$ are in bijection with cones of a certain Gröbner fan $\mathcal{GF}_n$ in $\mathbb{R}^{n^2 - n}$ restricted to a small cone called the polytrope region. These in turn are indexed by compatible sets of bipartite and triangle binomials. Geometrically, on the polytrope region, $\mathcal{GF}_n$ is the refinement of two fans: the fan of linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite binomial fan. This gives two algorithms for enumerating tropical types of polytropes: one via a general Gröbner fan software such as extsf{gfan}, and another via checking compatibility of systems of bipartite and triangle binomials. We use these algorithms to compute types of full-dimensional polytropes for $n = 4$, and maximal polytropes for $n = 5$.

Motivation & Objective

• To provide a systematic method for enumerating tropical types of polytropes in $\mathbb{TP}^{n-1}$, which are both tropical and ordinary polytopes.
• To resolve the computational infeasibility of brute-force Gröbner fan enumeration by restricting to the polytrope region.
• To establish a bijection between tropical types of polytropes and compatible sets of bipartite and triangle binomials.
• To compute the number of combinatorial tropical types of full-dimensional polytropes in $\mathbb{TP}^3$ and maximal polytropes in $\mathbb{TP}^4$.
• To address the open problem of characterizing compatibility among bipartite monomials, which underlies the enumeration of tropical types.

Proposed method

• The tropical types of polytropes are shown to be in bijection with cones of the Gröbner fan $\mathcal{GF}_n$ restricted to the polytrope region $\mathcal{P}_n$.
• The restricted fan $\mathcal{GF}_n|_{\mathcal{P}_n}$ is proven to equal the refinement of two fans: the fan of linearity of the polytrope map $\mathcal{P}_n$ and the bipartite binomial fan $\mathcal{BB}_n$.
• The bipartite binomial fan $\mathcal{BB}_n$ is constructed as a refinement of fans isomorphic to the braid arrangement, with hyperplanes defined by bipartite binomials.
• Compatibility of sets of bipartite and triangle binomials is used as a combinatorial criterion to index tropical types, replacing full Gröbner fan computation.
• Two algorithms are proposed: one using general Gröbner fan software (e.g., gfan) restricted to the polytrope region, and another based on checking compatibility of binomial systems.
• The method leverages symmetry to reduce computation, and the results are verified via agreement between gfan outputs and combinatorial counts.

Experimental results

Research questions

• RQ1 How can tropical types of polytropes be systematically enumerated beyond small dimensions using algebraic geometry tools?
• RQ2 What is the precise combinatorial structure underlying the tropical types of polytropes in $\mathbb{TP}^{n-1}$?
• RQ3 How does the Gröbner fan $\mathcal{GF}_n$ restrict to the polytrope region, and what is its geometric and combinatorial structure there?
• RQ4 What conditions ensure compatibility of sets of bipartite and triangle binomials that correspond to non-empty cones in the restricted fan?
• RQ5 Can the number of tropical types of maximal polytropes be computed efficiently, and what is its value for $n=5$?

Key findings

• The tropical types of full-dimensional polytropes in $\mathbb{TP}^3$ are enumerated as 1013 distinct combinatorial types, confirmed via both gfan and combinatorial refinement.
• For $n=5$, there are 27,248 combinatorial tropical types of maximal polytropes in $\mathbb{TP}^4$, computed using the compatibility condition on binomial systems.
• The restricted Gröbner fan $\mathcal{GF}_n|_{\mathcal{P}_n}$ is exactly the refinement of the polytrope map fan $\mathcal{P}_n$ and the bipartite binomial fan $\mathcal{BB}_n$, providing a geometric decomposition.
• The open, full-dimensional cones in $\mathcal{GF}_n|_{\mathcal{P}_n}$ correspond bijectively to maximal polytropes, indexed by compatible sets of bipartite and triangle binomials.
• The method avoids full computation of $\mathcal{GF}_n$, which is infeasible for $n \geq 5$, by restricting to the polytrope region and using combinatorial compatibility checks.
• The paper corrects an error in prior work: there are six, not five, tropical types of full-dimensional polytropes in $\mathbb{TP}^3$ with maximal vertices, resolving a discrepancy in Joswig and Kulas' table.