Quasi-period collapse occurs when the Ehrhart Quasi-polynomial
of a rational polytope has a quasi-period less than the denominator of that
polytope. This phenomenon is poorly understood, and all known cases in
which it occurs have been proven with ad hoc methods. In this note, we
present a conjectural explanation for quasi-period collapse in rational polytopes.
We show that this explanation applies to some previous cases appearing
in the literature. We also exhibit examples of Ehrhart polynomials of rational
polytopes that are not the Ehrhart polynomials of any integral polytope.
Our approach depends on the invariance of the Ehrhart quasi-polynomial
under the action of affine unimodular transformations. Motivated by the similarity
of this idea to the scissors congruence problem, we explore the development
of a Dehn-like invariant for rational polytopes in the lattice setting.
|Title of host publication||Proceedings AMS-IMS-SIAM Summer Research Conference on Integer Points in Polyhedra (Snowbird UT, USA, June 11-15, 2006)|
|Place of Publication||Providence RI|
|Publisher||American Mathematical Society|
|Publication status||Published - 2008|
|Name||Contemporary Mathematics Series|