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Toric Ideals of Flow Polytopes

Abstract : We show that toric ideals of flow polytopes are generated in degree $3$. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m \times n)$-transportation polytopes, a similar result holds: they have Gröbner bases of at most degree $\lfloor mn/2 \rfloor$. We construct a family of examples, where this bound is sharp.
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Matthias Lenz. Toric Ideals of Flow Polytopes. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.889-896, ⟨10.46298/dmtcs.2837⟩. ⟨hal-01186265⟩

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