# Ehrhart Polynomials of Dilated Polytopes

Katharina Victoria Jochemko (KTH Royal Institute of Technology)

The number of lattice points in integer dilates of a lattice polytope is given by a polynomial --- the Ehrhart polynomial of the polytope. Ehrhart polynomials encode fundamental properties such as the volume and the dimension of the polytope and arise in various guises in many areas, for example, commutative algebra, representation theory and optimization. A central question that is widely open is to characterize Ehrhart polynomials. An important tool is the h*-polynomial which encodes the Ehrhart polynomial in a particular basis. Beck and Stapledon conjectured that the h*-polynomial of any integer dilate of a lattice polytope is real-rooted whenever the dilation factor exceeds the dimension of the polytope. In this talk I will introduce the basic concepts of Ehrhart theory, highlight current research questions and show how interlacing polynomials can be used to prove the conjecture.