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

Abstract

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. First, we list all matroids $\mathcal{M}$ such that its corresponding toric ideal $I_{\mathcal{M}}$ is a complete intersection. Second, we handle the problem of detecting minors of a matroid $\mathcal{M}$ from a minimal set of binomial generators of $I_{\mathcal{M}}$. In particular, given a minimal set of binomial generators of $I_{\mathcal{M}}$ we provide a necessary condition for $\mathcal{M}$ to have a minor isomorphic to $\mathcal{U}_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal{M}$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal{M}$ such that $I_{\mathcal{M}}$ has a unique minimal set of binomial generators.