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 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

Matroid Powers in Tropical Geometry

We examine three themes in the study of tropical ideals and further introduce a new cryptomorphism for valuated matroids, which we extend to a “cryptomorphism” for tropical ideals. Our first theme studies tropically non-realizable varieties, which are tropical varieties that cannot arise as the variety of a tropical ideal. We approach this question from the perspective of matroid tensor and symmetric powers, generalizing the techniques of Las Vergnas. We show that matroid tensor and symmetric powers are minor closed and conclude that there are both infinitely many tropical linear spaces that are tropically non-realizable and infinitely many forbidden minors to the class of tropically realizable matroids. The second theme that we study is non-realizable tropical ideals. We provide a substantial revision to the author’s previous work “Boolean Paving Tropical Ideals with Constant Hilbert Function.” We re-frame paving tropical ideals in the context of finitary matroids and minimal rank symmetric quasi powers, simplify many of the original statements, techniques, and proofs, we further expand upon the original results, and provide additional statements about the realizability of paving tropical ideals. The third theme explores realizable tropical varieties. We establish that the theory of large matroid symmetric powers is encoded in the question of tropical realizability. Using algebro-geometric techniques, we deduce a characterization of matroid symmetric powers in terms of the geometry of the underlying matroids. As a key lemma, we provide an elementary proof that tropical manifolds are connected through codimension one. We conclude this thesis by introducing a new cryptomorphism for valuated matroids in terms of closure operators. Valuated closure operators naturally extend to a “cryp- tomorphism” for tropical ideals, which expresses the tropical ideal as the map between the ambient polynomial semiring and the coordinate semiring of the tropical ideal. We also note some further connections to phylogenetics and tropical statistics