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

TL;DR

This paper provides a geometric proof that the greedoid polynomial of Eulerian branching greedoids is independent of the root, linking it to the $h^*$-polynomial of a root polytope, and shows it remains unchanged under edge reversal.

Contribution

It introduces a geometric approach to prove root-independence of the greedoid polynomial for Eulerian branching greedoids, connecting it to the $h^*$-polynomial of a root polytope.

Findings

The greedoid polynomial equals the $h^*$-polynomial of the root polytope of the dual graphic matroid.

The polynomial is independent of the choice of root vertex.

Reversing all edges in an Eulerian digraph does not change the greedoid polynomial.

Abstract

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, K\'evin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.