A Complexity Bound on Faces of the Hull Complex

Given a monomial kx₁,. . . ,x_n]-module M in the Laurent polynomial ring kx₁^±1, . . . , x_n^±1], the hull complex is defined to be the set of bounded faces of the convex hull of the points {t^a| x^a M} for sufficiently large t. Bayer and Sturmfels conjectured that the faces of this polyhedron are of bounded complexity in the sense that every such face is affinely isomorphic to a subpolytope of the (n – 1)-dimensional permutohedron, which in particular would imply that these faces have at most n! vertices. In this paper we prove that the latter statement is true, and give a counterexample to the stronger conjecture.