The Ehrhart theorem

The Ehrhart theorem relates to counting how many integer points (points with whole number coordinates) lie within a scaled-up version of a polygon or polytope. Specifically, if you take a polygon and enlarge it by a factor of \( k \), the theorem provides a way to calculate the number of integer points inside this larger shape. It shows that this count can be expressed as a polynomial in \( k \), meaning there’s a systematic relationship between the size of the scaling and the number of integer points, making it a powerful tool in geometry and combinatorics.