The magic of the h-vector

Abstract

Throughout the history of mathematics we have been able to appreciate the existence of diverse and wonderful connections between dierent areas, resulting in the interest of studying topics involving ideas from dierent elds of mathematics. Such is the case of polyhedral combinatorics. One of the most important results in this eld is the Upper Bound Theorem. With the proof of this theorem, a wonderful connection between geometry and combinatorial polytopes was discovered. The purpose of this text is to introduce some denitions and ideas that are developed in the proof of the Upper Bound Theorem. The main character of our brief text will be the h-vector.