Pick’s Theorem and Applications to Restricted Integer Partition Functions

Daniel Gulbrandsen
University of Wisconsin-Milwaukee
PhD Graduate Student

“A point in R^n is called an integral point if all of its coordinates are integers. In R^2, Pick’s Theorem relates the area of a polygon P, whose vertices all have integer coordinates, to the number of integral points it contains with the formula A = i + B/2 – 1, where A is the area of the P, i is the number of integral points in the interior of P, and B is the number of integral points contained on the boundary of P.
Now imagine placing a polygon in R^3 so that all of its vertices have integer coordinates. Does there exist a formula (similar to Pick’s Theorem) that relates the area of a polygon in R^3 to the number of integral points it contains? In this talk we will derive such a formula, show how our formula is a generalization of Pick’s formula, and investigate applications of the formula to restricted integer partition functions”.