Every tropical ideal in the sense of Maclagan–Rincón 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 Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

中文翻译：

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热带理想没有实现所有伯格曼粉丝

Maclagan-Rincón 意义上的每个热带理想都有一个相关的热带品种，一个有限的多面体复合体，在其最大单元上配备了正积分权重。这导致了热带几何中普遍存在的可实现性问题，其中加权多面体复合体以这种方式出现。使用 Las Vergnas 关于拟阵张量积不存在的工作，我们证明没有热带理想的多样性是 Vámos 拟阵的直接和的伯格曼扇和三元上的二阶一致拟阵的伯格曼范，并且在所有最大圆锥的权重为 1。