The minimum period of the Ehrhart quasi-polynomial of a rational polytope

  • Authors:

  • Tyrrell B. McAllister;Kevin M. Woods

  • Affiliations:

  • Department of Mathematics, University of California, One Shields Avenue, Davis, CA;University of California, Berkeley

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2005

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Abstract

If P ⊆ Rd is a rational polytope, then ip(n) := #(nP ∩ Zd) is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P. The minimum period of ip(n) must divide D(P) = min{n ∈ Z 0:nP is an integral polytope}. Few examples are known where the minimum period is not exactly D(P). We show that for any D, there is a 2-dimensional triangle P such that D(P) = D but such that the minimum period of ip(n) is 1, that is, ip(n) is a polynomial in n. We also characterize all polygons P such that ip(n) is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.