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

If P ⊂ R d is a rational polytope, then i P ( n ) ≔ # ( nP ∩ Z d ) is a quasi-polynomial in n , called the Ehrhart quasi-polynomial of P . The minimum period of i P ( n ) must divide D ( P ) = min { n ∈ Z > 0 : nP is an integral polytope } . Few examples are known where the minimum period is not...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2005-02, Vol.109 (2), p.345-352
Main Authors: McAllister, Tyrrell B., Woods, Kevin M.
Format: Article
Language:English
Subjects:
Convex bodies
Ehrhart polynomials
Hilbert series
Lattice points
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Summary:If P ⊂ R d is a rational polytope, then i P ( n ) ≔ # ( nP ∩ Z d ) is a quasi-polynomial in n , called the Ehrhart quasi-polynomial of P . The minimum period of i P ( 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 i P ( n ) is 1, that is, i P ( n ) is a polynomial in n . We also characterize all polygons P such that i P ( 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.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2004.08.006