# Qualifying Exam

Abstract:

The Ehrhart polynomial of a lattice polytope encodes information about the number of integer lattice points in positive integral dilates of . The -polynomial of is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is the Minkowski sum of line segments. In this talk I will present two theorems of Matthias Beck, Katharina Jochemko, and Emily McCullough from their paper, titled, "** -polynomials of zonotopes**". They give a combinatorial description of the -polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the -polynomial of every lattice zonotope has only real roots. Furthermore, I will outline my current and future research directions and questions.