Title: Weighted Ehrhart Theory
Abstract: Since Ehrhart's original work in the late 1960's, Ehrhart theory has developed into a key topic at the intersection of polyhedral geometry, number theory, commutative algebra, algebraic geometry, enumerative combinatorics, and integer programming.Here we examine an approach where instead of counting lattice points in polytopes, we define a weight function and count the evaluations of this function at the lattice points instead. We will study which of the properties of the original h*-vector hold more generally in the weighted case for some nicely behaved functions and we will see some counterexamples as to why we need some hypothesis on the weights. This leads us to a generalization of Stanley’s celebrated theorem about the h*-polynomial of the Ehrhart Series having nonnegative coefficients for some specific families of weight functions.
Title: No short polynomials in determinantal ideals