Summer@icerm 2014 Description of Problems

Periods of rational polygons. A rational polygon P is a polygon in the plane, all of whose vertices have rational coordinates. If we dilate the polygon P by an integer k, the dilated polygon is written as kP. It is a theorem that the number of integer points in the plane (a, b) ∈ Z2, which are contained in kP, is given by a quasi-polynomial function of k. That is, |Z2 ∩ kP| = area(P)k2 + a1(k)k + a0(k), where a1(k) and a0(k) are periodic functions of k.