Séminaire Lotharingien de Combinatoire, 82B.69 (2019), 11 pp.

Jan Draisma and Felipe Rincón

Tropical ideals do not realise all Bergman fans

Abstract. 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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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