Ehrhart polynomial

The Ehrhart polynomial is a mathematical tool used in combinatorial geometry to count how many integer points (points with whole number coordinates) exist within a scaled version of a geometric shape, like a polygon. Specifically, if you take a shape and scale it by an integer factor, the Ehrhart polynomial gives you a formula to determine the number of those integer points inside the scaled shape. This polynomial encodes the relationship between the shape's area, its vertices, and how the points are distributed, making it useful in various fields such as optimization, number theory, and algebraic geometry.