  We give an upper bound for the number of functionally independent meromorphic
first integrals that a discrete dynamical system generated by an analytic map
$f$ can have in a neighborhood of one of its fixed points. This bound is
obtained in terms of the resonances among the eigenvalues of the differential
of $f$ at this point. Our approach is inspired on similar Poincar\'e type
results for ordinary differential equations. We also apply our results to
several examples, some of them motivated by the study of several difference
equations.
