Discrete truncated powers and lattice points in rational polytope

Discrete truncate power is very useful for studying the number of nonnegative integer solutions of linear Diophantine equations. In this paper, some detail information about discrete truncated power is presented. To study the number of integer solutions of linear Diophantine inequations, the generalized truncated power and generalized discrete truncated power are defined and discussed, respectively. We use generalized discrete truncated powers and multivariate splines to investigate the lattice points in rational polytopes. In particular, we present the degree and period of multidimensional Ehrhart polynomial.