# A Tropical F5 algorithm

## TL;DR

This paper adapts the efficient F5 algorithm for computing Gröbner bases to the tropical setting over valued fields, proving its correctness and termination, with potential applications in p-adic stability and computational algebra.

## Contribution

It introduces a tropical version of the F5 algorithm for homogeneous ideals, establishing its correctness and termination in the tropical context.

## Key findings

- Proves the tropical F5 algorithm terminates and is correct.
- Demonstrates the algorithm's efficiency and stability through numerical examples.
- Highlights potential for stable computations over p-adic fields.

## Abstract

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{\"o}b-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.