Valuated matroid polytopes and linking system composition.

PhD Theses.Valuated matroids are a generalisation of matroids; matroids themselves being an abstraction of the notion of independence. Valuated matroids have many equivalent de nitions including via independent sets and circuits, and in this thesis we show that a valuated matroid has an equivalent de nition in terms of a rank function which we construct by analogy with the matroid rank function by looking at matroid and valuated matroid polytopes. We separately construct a hyperoperation which is an extension of a previously studied operation of composing valuated matroids, this being the composition of valuated linking systems. The composition of valuated linking systems can be seen as a generalisation of matrix multiplication to tropical linear spaces. In particular, the hyperoperation we introduce has been in uenced by viewing matrices as representing linear spaces, which we can do by looking at their row space, and consequently by how these relate to Pl ucker coordinates. Working tropically, since tropical linear spaces are equivalent to valuated matroids, which are also known as tropical Pl ucker vectors, we create the hyperoperation by using the parallels with matrices representing linear spaces over a eld. We describe the hyperproduct completely for small rank, where this operation forms a hypergroup. In higher rank we investigate what known matroid subdivisions it contains, as well as also showing that it does not form a fan, and nor is it convex in general. We also conjecture this hyperoperation forms a hypergroup for higher rank, and present some investigation towards this