# Computing zero-dimensional tropical varieties via projections

## TL;DR

This paper introduces an efficient algorithm for computing zero-dimensional tropical varieties using projections and unimodular transforms of Gr"obner bases, with proven polynomial complexity and favorable implementation performance.

## Contribution

It presents a novel algorithm leveraging projections and unimodular transforms for zero-dimensional tropical varieties, improving computational efficiency and complexity analysis.

## Key findings

- Algorithm requires polynomial arithmetic operations given a Gr"obner basis.
- Implementation outperforms existing methods in speed and efficiency.
- Complexity for tropical links dominated by Gr"obner walk complexity.

## Abstract

We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast unimodular transforms of lexicographical Gr\"obner bases. We prove that our algorithm requires only a polynomial number of arithmetic operations if given a Gr\"obner basis, and we demonstrate that our implementation compares favourably to other existing implementations. Applying it to the computation of general positive-dimensional tropical varieties, we argue that the complexity for calculating tropical links is dominated by the complexity of the Gr\"obner walk.