UC Berkeley

The h*-polynomial captures the enumeration of lattice points in dilates of rational polytopes. For various classes of polytopes, there are many potential properties of this polynomial--including nonnegativity, monotonicity, unimodality, real-rootedness, and palindromicity--that are of interest within geometric combinatorics. In this dissertation, we investigate these properties for three variations of the h*-polynomial: the boundary h*-polynomial, the weighted h*-polynomial, and the local h*-polynomial. We conclude with an application of h*-polynomials to enumerating proper vertex colorings of graphs.