A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids

We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex $v_0$ is equivalent to the $h^*$-polynomial of the root polytope of the dual of the graphic matroid. As the definition of the root polytope is independent of the vertex $v_0$, this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, and K\'evin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse each edge of an Eulerian digraph.