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. Firstly, we list all matroids M such that its corresponding toric ideal I M is a complete intersection. Secondly, we handle the problem of detecting minors of a matroid M from a minimal set of binomial generators of I M. In particular, given a minimal set of binomial generators of I M we provide a necessary condition for M to have a minor isomorphic to U d,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 matroids M such that I M has a unique minimal set of binomial generators.
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Ignacio García-Marco, Jorge Luis Ramírez Alfonsín. Matroid Toric Ideals: Complete Intersection, Minors, and Minimal Systems of Generators. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2015, 29 (4), pp.2267-2276. ⟨10.1137/140986608⟩. ⟨hal-02049103⟩

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