## Christopher Vander Wilt and Daniel Gulbrandsen

### Utah Valley University

Let nP denote the polytope obtained by expanding the convex integral polytope P⊂R^d by a factor of n in each dimension. Ehrhart [1] proved that the number of lattice (integer) points contained in nP is a rational polynomial of degree d in n. What happens if the polytope is expanded by not necessarily the same factor in each dimension? In this talk a partial answer to this question will be provided, using powers of n as different factors to expand the polytope. It will be shown that the number of lattice points contained in the polytope formed by expanding P by multiplying each vertex coordinate by such a factor is a quasi-polynomial in n. Quasi-polynomials are a generalization of polynomials, where the coefficients of the quasi-polynomials are periodic functions with integral period. Furthermore, particular cases where the number of such lattice points is a polynomial will be presented. In addition, the period of these quasi-polynomials as well as the Law of Reciprocity will be addressed. At the end, future work will be discussed. [1] E. Ehrhart, “Sur les Poly`edres Rationnels Homoth ´etiques `a n Dimensions,” C. R. Acad. Sci. Paris 254 (1962)