Periods of Ehrhart coefficients of rational polytopes

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ – that is, a “polynomial” in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.

💡 Research Summary

The paper investigates the periodic behavior of the coefficient functions in Ehrhart quasi‑polynomials associated with rational polytopes. For a rational polytope P ⊂ ℝⁿ, the number of integer points in its k‑th dilate, |kP ∩ ℤⁿ|, is given by a quasi‑polynomial \