DVR-Matroids of Algebraic Extensions

A matroid is a finite set E along with a collection of subsets of E, called independent sets, that satisfy certain conditions. The most well-known matroids are linear matroids, which come from a finite subset of a vector space over a field K. In this case the independent sets are the subsets that are linearly independent over K. Algebraic matroids come from a finite set of elements in an extension of a field K. The independent sets are the subsets that are algebraically independent over K. Any linear matroid has a representation as an algebraic matroid, but the converse is not true [7]. One tool that helps us better understand algbraic matroids is the Lindström valuation which is defined on basis sets of a matroid. This valuation is explicitly defined in [3]. In Chapter 2, we will show that the Lindström valuated matroid can be further refined to a DVR-matroid, or matroid over a discrete valuation ring as defined in [5]. In Chapter 3, we focus on a class of examples of algebraic matroids that come from homomorphisms of algebraic groups. We show that the d-vectors for the corresponding DVR-matroid can be computed in two different ways.