Matroid Toric Ideals: Complete Intersection, Minors, and Minimal Systems of Generators

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids M such that its corresponding toric ideal IM is a complete intersection. Secondly, we handle the problem of detecting minors of a matroidM from a minimal set of binomial generators of IM. In particular, given a minimal set of binomial generators of IM we provide a necessary condition for M to have a minor isomorphic to Ud,2d for d ≥ 2. This condition is proved to be sufficient for d = 2 (leading to a criterion for determining whether M is binary) and for d = 3. Finally, we characterize all matroidsM such that IM has a unique minimal set of binomial generators.