Given a discrete-time Markov chain with finite state space and a stationary transition matrix, a system of "local" Poisson equations characterizing the (exponential) Varadhan's functional J((.)) is given. The main results, which are derived for an arbitrary transition structure so that J((.)) may be nonconstant, are as follows: (i) Any solution to the local Poisson equations immediately renders Varadhan's functional, and (ii) a solution of the system always exist. The proof of this latter result is constructive and suggests a method to solve the local Poisson equations.