The Ehrhart polynomial of a lattice polytope counts integer points in integral dilates of the polytope. The coefficients of these polynomials are, for the most part, a complete mystery. We have established linear inequalities between the coefficients of an Ehrhart polynomial, depending only on the dimension of the polytope. These relations imply, in particular, that in a fixed dimension the roots of any Ehrhart polynomial are bounded. Our result can be generalized slightly, to Poincaré series of a certain type.

Furthermore, we give partially tight bounds for the real roots of an Ehrhart polynomial.

Finally, I will report on studies of special classes of polytopes whose Ehrhart polynomials exhibit remarkable behavior.

This is joint work with Mike Develin (Berkeley), Jesus DeLoera (Davis), and Julian Pfeifle (Barcelona).