The Parikh counting functions of sparse context-free languages are quasi-polynomials

Let L be a sparse context-free language over an alphabet of t letters and let f_L:N^t→N be its Parikh counting function. We prove the following two results: 1. There exists a partition of N^t into a finite family of polyhedra such that the function f_L is a quasi-polynomial on each polyhedron of the partition. 2. There exists a partition of N^t into a finite family of rational subsets such that the function f_L is a polynomial on each set of the partition.