Ehrhart theory for real dilates of polytopes - Royer, Tiago (2018)
The Ehrhart function L_P(t) of a polytope P is defined to be the number of integer points in the dilated polytope tP. Classical Ehrhart theory is mainly concerned with integer values of t; in this master thesis, we focus on how the Ehrhart function behaves when the parameter t is allowed to be an arbitrary real number. There are three main results concerning this behavior in this thesis. Some rational polytopes (like the unit cube [0, 1]^d) only gain integer points when the dilation parameter t is an integer, so that computing L_P(t) yields the same integer point count than L_P(t). We call them semi-reflexive polytopes. The first result is a characterization of these polytopes in terms of the hyperplanes that bound them. The second result is related to the Ehrhart theorem. In the classical setting, the Ehrhart theorem states that L_P(t) will be a quasipolynomial whenever P is a rational polytope. This is also known to be true with real dilation parameters; we obtained a new proof of this fact starting from the chraracterization mentioned above. The third result is about how the real Ehrhart function behaves with respect to translation in this new setting. It is known that the classical Ehrhart function is invariant under integer translations. This is far from true for the real Ehrhart function: not only there are infinitely many different functions L_{P + w}(t) (for integer w), but under certain conditions the collection of these functions identifies P uniquely.