Quasi-polynomials, linear Diophantine equations and semi-linear sets

We investigate the family of semi-linear sets of Nt and Zt. We study the growth function of semi-linear sets and we prove that such a function is a piecewise quasi-polynomial on a polyhedral partition of Nt. Moreover, we give a new proof of combinatorial character of a famous theorem by Dahmen and Micchelli on the partition function of a system of Diophantine linear equations.