Polinomios y series de Ehrhart

Abstract

The main goal of this dissertation is to prove the Eugèene Ehrhart’s theorem about integer-points enumerating functions over polytopes. It is, in fact, a generalization of Pick’s theorem in any finite-dimensional Euclidean space. We have structured this text in three parts. The introduction, where we focus our attention to hyperplanes, convex sets, polytopes and pointed cones, is the first part. The following chapters belong to the central part of the document. In Chapter 3, we will give some examples showing that the number of lattice points 2 in a (positive) integer dilate of a polytope $\mathcal {P} \subseteq \mathbb{R} ^{d}$ is a polynomial with the same degree as the dimension of P : this is the statement of Ehrhart’s theorem. After that, we will give a geometric proof of it, and finally (Chapter 4) we will demonstrate the Ehrhart-MacDonald reciprocity law and show the geometric meaning of some Ehrhart polynomial’s coefficients. The fifth Chapter is devoted to apply Ehrhart theory to solve some problems about combinatorics. The last part is the appendix which includes many propositions and lemmas that are used in the previous part.