Toric Ideals of Flow Polytopes

TL;DR

This paper proves that toric ideals of flow polytopes are generated in degree 3, confirming a conjecture for the Birkhoff polytope and analyzing Gr"obner bases degrees for transportation polytopes.

Contribution

It establishes the degree 3 generation of toric ideals of flow polytopes and provides bounds on Gr"obner basis degrees, using a hyperplane subdivision method.

Findings

Toric ideals of flow polytopes are generated in degree 3.

Gr"obner bases of Birkhoff polytope have degree at most n, sharp for some orders.

Transportation polytopes have Gr"obner bases of degree at most mn/2, with examples showing sharpness.

Abstract

A referee found an error in the proof of the Main Theorem ("toric ideals of flow polytopes are generated in degree 3") that we could not fix. More precisely, the proof of Lemma 4.2.(ii) is incorrect. The results on Gr\"obner bases are untouched by this. ----- 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 Gr\"obner 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 Gr\"obner bases of at most degree $\lfloor mn/2\rfloor$. We construct a family of examples, where this bound is sharp.