#### 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.

Done