Hyperplane Arrangements over the Ring of Integers Modulo N.

Let V be a finite vector space of dimension n over the field K. A hyperplane in V is an n − 1 dimensional subspace of V defined by an equation of the form∑ni=1 aixi = 0, ai ∈ K, (x1, ..., xn) ∈ V . A hyperplane arrangement is any finite collectionof hyperplanes. This work focuses on some hyperplanes arrangements such as the braid and the graphical arrangement {xi − xj = 0 : 1 ≤ i \u3c j ≤ n}, the shi arrangement {xi − xj = 1 : 1 ≤ i \u3c j ≤ n}, the threshold arrangement {xi + xj = 0 : 1 ≤ i \u3c j ≤ n}, and the complete arrangement {∑i∈S xi = 0 : S ⊆ [n]}. The graphical arrangement is a subarrangement of the threshold arrangement. Chapter 1 gives an introductory overview of the entire work, the significance of hyperplane arrangements, and their interconnections.Chapter 2 presents some of the important definitions and the general theory underlying hyperplane arrangements. Chapter 2 also introduces what we would call the most important concept in the study of hyperplane arrangement, which is the characteristic polynomial. Every hyperplane arrangement(or simply arrangement) can be identified by a unique characteristic polynomial. Chapter 3 covers the braid, graphical, and Shi arrangement, and application to graph coloring, while Chapter 4 explores the threshold arrangement, and how it relates to threshold graphs. There is a bijection between the set of threshold graphs on n vertices and the set of regions of the boxed threshold arrangement. Finally, chapter 5 gives an introduction to the complete arrangement. KEYWORDS: Threshold Arrangement, Characteristic Polynomial, Finite Field Method