  We study the relations between the finite generation of Cox ring, the
rationality of Euler-Chow series and Poincar\'e series and Zariski's conjecture
on dimensions of linear systems. We prove that if the Cox ring of a smooth
projective variety is finitely generated, then all Poincar\'e series of the
variety are rational. We also prove that the multi-variable Poincar\'e series
associated to big divisors on a smooth projective surface are rational,
assuming the rationality of multi-variable Poincare series on curves.
