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We define the root polytope of a regular oriented matroid, and show that thegreedoid polynomial of an Eulerian branching greedoid rooted at vertex $v_0$ isequivalent to the $h^*$-polynomial of the root polytope of the dual of thegraphic 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 greedoidpolynomial for Eulerian branching greedoids, a fact which was first proved bySwee Hong Chan, and K\'evin Perrot and Trung Van Pham using sandpile models. Wealso obtain that the greedoid polynomial does not change if we reverse eachedge of an Eulerian digraph.

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