Ehrhart Theory on Arrangements of Hyperplanes

Hyperplanes can be thought of as codimension 1 affine subspaces of a given Euclidean space. Given an arrangement A of hyperplanes in R^d, the set of intersections of hyperplanes define a partially ordered set called intersection poset. The characteristic polynomial of A can then be defined using Mobius function of the intersection poset. One can recover some topological information about the complement of hyperplane arrangement from its characteristic polynomial. Some family of arrangements of interest are Catalan arrangements, Shi arrangements and Linial arrangements. Another way of recovering characteristic polynomial is finite field method, which involves counting points in the complement of good reduction of A. Recently, Yoshinaga used Ehrhart theoretic methods to give a uniform proof of Postnikov-Stanley conjectures on m-Linial arrangements. In this project, we would like to extend these methods to other problems of interest. We would also like to explore further into relation of characteristic polynomial and rook factorial polynomial.

Faculty Supervisor:

Graham Denham

Student:

Partner:

Hokkaido University

Discipline:

Mathematics

Sector:

Agriculture; Education

University:

Western University

Program: