# Tropical ideals do not realise all Bergman fans

## TL;DR

This paper demonstrates that not all Bergman fans, specifically the one from the direct sum of the Vámos matroid and a uniform matroid, can be realized as tropical varieties of tropical ideals with all maximal cones weighted equally.

## Contribution

It proves a specific non-realisability result in tropical geometry using matroid theory and tensor product non-existence results.

## Key findings

- Certain Bergman fans cannot be realized as tropical varieties of tropical ideals.
- The Vámos matroid combined with a uniform matroid produces a non-realisable Bergman fan.
- Not all weighted polyhedral complexes arise from tropical ideals.

## Abstract

Every tropical ideal in the sense of Maclagan-Rinc\'on has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the V\'amos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have weight one.