Generalizing Pick’s theorem

Abstract

[en] The main goal of this project is to analyze the function that counts the number of integer points inside the dilates of a lattice polytope. One of the first result in this area is Pick's Theorem [1] proven by the mathematician Georg Alexander Pick in 1899, which gives the area of a polygon by counting the points on its border and its interior. In the 1960s, the French mathematician Eugène Ehrhart further explored this field and proved that for lattice $d$-polytopes, these functions are polynomials of degree $d$ [2]. For this reason, these polynomials are called Ehrhart polynomials. In the following years, with the work of Richard P. Stanley and Ian G. Macdonald, the fundations of this field, Ehrhart Theory, were stablished.

Our research into this topic started with [3], results from which are stated on Chapter 6.

In the first chapter, we present and prove Pick's Theorem. Moreover, we also show that no analogue to this formula that expresses the volume of a polyhedron as a function only of its numbers of interior and boundary points exists.

From the second to the fifth chapter, we will follow Ehrhart Theory as it was done in [4]. All of the proofs are inspired in the ones that appear in this book, with some changes to make them easier to understand and the exercises left to the reader solved. In particular, in the second chapter, we make the first definitions on which this field of mathematics revolves. In addition, we also enunciate Ehrhart's Theorem, the first grand theorem on the field. In the third chapter, we develop the necessary mathematical tools needed to prove the before mentioned theorem. Lastly, in chapters four and five, we analyze the properties of the coefficients of the Ehrhart polynomial without new tools and with the help of the Ehrhart-Macdonald reciprocity respectively.

In the sixth chapter, as we said before, we present the study done by Max Kölbl, explaining it and showing its results. Moreover, some proofs were added or reworked for their presentation in this work.

Finally, in the seventh chapter, we return to the title of this paper and prove an $\mathrm{n}$-dimensional generalization for Pick's Theorem that we have arrived to ourselves.