Alladi Ramakrishnan Hall

Ehrhart Polynomials

Kamalakshya Mahatab

IMSc

For a polytope P and any positive integer k, let i_P(k) denote the number of integral points in the dilation kP of P by a factor k. Ehrhart proved that if vertices of P have integral coordinates, then the function i_P is a polynomial in k. This polynomial is called the Ehrhart polynomial of P. If P has vertices with rational coordinates then i_P is `almost' a polynomial. The aim of this talk is to give an exposition to Ehrhart polynomials through examples. We will also discuss some basic properties of i_P, such as Periodicity and Reciprocity. The theory of Ehrhart polynomials is a very helpful tool in many combinatorial problems; we will see some applications in this talk. This talk requires no prerequisites from the theory of polytopes.