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978-3-8439-0659-3, Reihe Mathematik
Tropical Oriented Matroids and Cubical Complexes
189 Seiten, Dissertation Technische Universität Darmstadt (2012), Softcover, B5
Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes - in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes.
Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of pseudohyperplanes.
It was shown by Ardila and Develin that tropical oriented matroids can be represented subdivisions of products of two simplices and hence – by the Cayley Trick due to Huber, Rambau and Santos – mixed subdivisions of dilated simplices.
It also follows from a theorem by Develin and Sturmfels from 2004 that tropical hyperplane arrangements (and hence realisable tropical oriented matroids) are in bijection with regular subdivisions of products of two simplices.
In this thesis we define arrangements of tropical pseudohyperplanes in two different ways and prove Topological Representation Theorems for both of them. I.e., we show that any tropical oriented matroid can be realised as an arrangement of tropical pseudohyperplanes. These theorems provide tropical analogues to the famous Topological Representation Theorem by Folkman and Lawrence.
We then apply the Topological Representation Theorem for tropical oriented matroids to show that the cells in every mixed subdivision of a dilated simplex satisfy the tropical oriented matroid axioms. This completes the equivalence of the four concepts of tropical oriented matroids, mixed subdivisions of dilated simplices, subdivisions of products of two simplices and tropical pseudohyperplane arrangements.