Tropical ideals do not realise all Bergman fans

Tropical ideals do not realise all Bergman fans

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 i...

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Veröffentlicht in:Research in the mathematical sciences 2021, Vol.8 (3), p.44-44, Article 44
Hauptverfasser: Draisma, Jan, Rincón, Felipe
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Applications of Mathematics
Computational Mathematics and Numerical Analysis
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Mathematics and Statistics
Tensors
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description 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.
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Computational Mathematics and Numerical Analysis
Cones
Mathematics
Mathematics and Statistics
Tensors
title Tropical ideals do not realise all Bergman fans
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