Summary: TROPICAL HYPERPLANE ARRANGEMENTS AND ORIENTED MATROIDS
FEDERICO ARDILA AND MIKE DEVELIN
Abstract. We study the combinatorial properties of a tropical hyperplane arrangement. We define
tropical oriented matroids, and prove that they share many of the properties of ordinary oriented
matroids. We show that a tropical oriented matroid determines a subdivision of a product of two
simplices, and conjecture that this correspondence is a bijection.
Tropical mathematics is the study of the tropical semiring consisting of the real numbers with
the operations of + and max. This semiring can be thought of as the image of a power series
ring under the degree map which sends a power series to its leading exponent. This semiring has
received great attention recently in several areas of mathematics, due to the discovery that there
are often strong relationships between a classical question and its tropical counterpart. One can
then translate geometric questions about algebraic varieties into combinatorial questions about
polyhedral fans. This point of view has been fruitful in algebraic geometry, combinatorics, and
phylogenetics, among others [2, 3, 6, 7, 11, 16, 22].
The triangulations of a product of two simplices are ubiquitous and useful objects. They are
of independent interest [4, 5, 9, 13], and have been used as a building block for finding efficient
triangulations of high dimensional cubes [10, 12] and disconnected flip-graphs [18, 19]. They also
arise very naturally in connection with the Schubert calculus , Hom-complexes , growth series
of root lattices , transportation problems and Segre embeddings , among others.