Anton Hilding: Ehrhart polynomials of alcoved polytopes

Abstract.

The Ehrhart counting function enumerates the number of integer lattice points in positive integer dilates of a lattice polytope. Ehrhart proved in 1962 that this function is a polynomial of degree d for a d-dimensional polytope. In 1976 Scott characterized which polynomials may be Ehrhart polynomials for two-dimensional polytopes, by determining a series of inequalities that the coefficients satisfy.

In this talk, we discuss the Ehrhart polynomials of alcoved polytopes, a subclass of polytopes given as the intersection of certain half-spaces. Using number-theoretic, analytic and geometric methods, we state and prove necessary conditions further restricting which polynomials may be Ehrhart polynomials for two-dimensional alcoved polytopes, akin to what Scott did. Furthermore, we conjecture a generalization of these results to hold in higher dimension, proving partial positive results that support the conjecture.