Matroid toric ideals: complete intersection, minors and minimal systems of generators

TL;DR

This paper characterizes matroid toric ideals that are complete intersections, explores how to detect minors from generators, and identifies matroids with unique minimal generating sets, advancing understanding of their algebraic and combinatorial properties.

Contribution

It provides a complete classification of matroids with complete intersection toric ideals, criteria for detecting minors from generators, and characterizes matroids with unique minimal generating sets.

Findings

Classified all matroids with complete intersection toric ideals.

Developed criteria for detecting minors from minimal generators.

Characterized matroids with unique minimal generating sets.

Abstract

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 $\mathcal M$ 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.