Excluded Minors for Matroids Satisfying Kinser's Inequalities

Kinser developed a hierarchy of inequalities for subspaces. These inequalities can be applied to the rank function of a matroid, and a matroid which is representable must satisfy each one of them. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids. We strengthen a result by Mayhew, Newman, and Whittle by showing that for every real-representable matroid and every layer of the hierarchy, there is an excluded minor for that layer of the hierarchy that contains as a minor.