Ehrhart polynomial of coincides with the Hilbert polynomial of this line bundle. Ehrhart polynomials can be studied for their own sake. For instance, one could ask questions related to the roots of an Ehrhart polynomial. Furthermore, some authors have pursued the question of how these polynomials could be classified. Generalizations It is possible to study the number of integer points in a polytope if we dilate some facets of but not others. In other words, one would like to know the number of integer points in semi-dilated polytopes. It turns out that such a counting function will be what is called a multivariate quasi-polynomial. An Ehrhart-type reciprocity theorem will also hold for such a counting function. Counting the number of integer points in semi-dilations of polytopes has applications in enumerating the number of different dissections of regular polygons and the number of non-isomorphic unrestricted codes, a particular kind of code in the field of coding theory. See also Quasi-polynomial Stanley's reciprocity theorem Notes References . . . . Introduces the Fourier analysis approach and gives references to other related articles. . Definition and first properties. . Figurate numbers Polynomials Lattice points Polytopes