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 $\mathcal{M}$ such that its corresponding toric ideal $I_{\mathcal{M}}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$fromaminimalsetofbinomialgeneratorsof$I_{\mathcal M}$. 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 $d\ge 2$. 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.