Tropical ideals do not realise all Bergman fans

Draisma, Jan; Rincón, Felipe (2019). Tropical ideals do not realise all Bergman fans. Séminaire Lotharingien de Combinatoire, 82B(Art. 69) Fakultät für Mathematik, Universität Wien

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Tropical ideals are combinatorial objects that abstract the possible collections of subsets arising as the supports of all polynomials in an ideal. Every tropical
ideal 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.

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