Algebra Seminar: Polynomials, generating functions, and lattice point counting
1:10pm; Neill Hall, Room 106W
Abstract: The number of integral points (i.e., points with integer entries) in a convex polytope P in R^d gives its DISCRETE VOLUME. This quantity is the number of elements in the set P n Z^d ('n' represents intersection). We get a better approximation to the usual continuous volume of P if we shrink the lattice Z^d by an integer factor k and count the number of elements in the set P n (1/k)Z^d. This quantity is a polynomial in k and is called the Ehrhart polynomial of P. Discrete volumes of polytopes appear in several areas such as number theory, discrete optimization, algebraic geometry and others. In this talk, I will present a few simple examples of polytopes, and derive expressions for their discrete volumes by analyzing their generating functions.