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Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of agraph, based on spanning trees and activities just like the originaldefinition, but using a fixed ribbon structure to order the set of edges in adifferent way for each tree. The interior polynomial $I$ is a generalization of$T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using aribbon structure on the underlying bipartite graph $G$. Our formula worksbecause it is determined by the Ehrhart polynomial of the root polytope of $G$in the same way as $I$ is. To prove this we interpret the Bernardi process as away of dissecting the root polytope into simplices, along with a shellingorder. We also show that our generalized Bernardi process gives a commonextension of bijections (and their inverses) constructed by Baker and Wangbetween spanning trees and break divisors.

Hypergraph polynomials and the Bernardi process