The Ehrhart polynomial L(P, t) counts the number of integer lattice points inside the dilated polytope tP = {tx : x ∈ P}, as a polynomial in t. For a convex lattice polytope P, Ehrhart (1962) proved this count is always a polynomial in t. The leading coefficient is the volume of P, the constant term is 1, and the degree equals the dimension. Ehrhart reciprocity: L(P, -t) = (-1)^d · L(P°, t), where P° is the interior — negative dilations count interior points. For the unit simplex in ℝ^d: L(Δ_d, t) = C(t+d, d), the binomial coefficient. These polynomials connect combinatorics, geometry, and number theory through generating functions called Ehrhart series.