Coefficient functions of Ehrhart quasi-polynomials

Tyrrell McAllister

Given a rational polytope P, the Ehrhart quasi-polynomial of P is the function giving the number of lattice points in positive integer dilations of P. Such a function LP has the form LP(k) = cd(k) kd + . . . + c1(k) k + c0(k), where each "coefficient" cj is a periodic function from Z into Q. The connection between the geometry of the polytope P and the periods of the coefficient functions of LP is not well understood. McMullen has given bounds on these periods in terms of the so-called j-index of the polytope. McMullen's bounds satisfy nice relations, and one might hope to find similar relations for the periods themselves. Sadly, we can show with explicit constructions that no such relations exist in dimension 2. It is hoped that this talk will serve as an invitation to explore the behavior of these coefficient functions in higher dimensions.

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