Local Ehrhart Theory and Gale Duality

The central object of this dissertation is the local h*-polynomial of a lattice polytope. This is an invariant that arises in the study of counting the lattice points inside polytopes and their dilations, the so-called Ehrhart theory. If a lattice polytope is spanning, it can be defined by the data of its Gale dual. In this thesis, we try to understand what we can say about the local h*-polynomial from a given Gale dual. The thesis comprises of three projects. In the first one, we introduce the shifted products of circuits and give arithmetic-flavoured expressions for the their (local) h*-polynomial, extending the known results for circuits. In the second project, we prove that there exists a geometric construction which preserves the coefficients of the local h*-polynomial up to an overall shift. We call it a Lawrence twist. This construction in particular helps to disprove a conjecture about polytopes with vanishing local h*-polynomial. In the third project, we obtain the complete classification of the four-dimensional simplices with vanishing local h*-polynomial. We show that any such simplex that is not a free join must belong either to a certain one-parameter family of simplices or it must be one of the six sporadic cases.