turkmath.org

The Ehrhart Polynomial counts the number of lattice points in integer dilates of a lattice polytope and is of fundamental importance in polyhedral combinatorics and toric geometry. The question of which polytopes have Ehrhart polynomials with non-negative coefficients is a well studied albeit poorly understood problem. I’ll talk about a recent result of mine, joint with Katharina Jochemko which shows that the linear coefficient of the Ehrhart polynomial of generalized permutohedra, an important class of polytopes introduced by Postnikov are indeed positive. We are also able to characterize all Minkowski additive functions on this class of polytopes, which includes several important classes of polytopes such as matroid and independent set polytopes. I’ll end with several open problems, some of which might be of interest to graduate or even undergraduate students.