Computing tropical points and tropical links

TL;DR

This paper introduces algorithms for computing points and links on tropical varieties, utilizing triangular decomposition and Newton polygons, and applies these to analyze the structure of tropical Grassmannians.

Contribution

It presents new algorithms for zero-dimensional tropical varieties and extends them to higher dimensions, demonstrating their application on complex tropical Grassmannians.

Findings

Tropical Grassmannians $ ext{G}_{3,8}$ and $ ext{G}_{4,8}$ are not simplicial.

Algorithms effectively compute points and links on tropical varieties.

New methods improve understanding of tropical geometric structures.

Abstract

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical varieties, using intersections with affine hyperplanes to reduce the dimension to zero. We use the algorithms to show that the tropical Grassmannians $\mathcal G_{3,8}$ and $\mathcal G_{4,8}$ are not simplicial.