Periods of Ehrhart Coefficients of Rational Polytopes

  • Tyrrell B. McAllister
  • Hélène O. Rochais
Keywords: Rational polytopes, Ehrhart quasi-polynomials


Let $\mathcal{P} \subset \mathbb{R}^{n}$ 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.

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