A few remarks on a manin-mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates

We present here the natural extension of our Pila-Wilkie type estimates on the number of rational points of the trascendent part of a compact analytic subset of F_q(( 1 /T))_n(see [D1) to analogous subsets of Kⁿ, where K is a general local field of any characteristic. That would integrate the analogous estimate provided by F. Loeser, G. Comte and R. Cluckers in [CCL, Theorem 4.1.61. We remind in the first two sections the main ideas of our construction by correcting two minor mistakes we made in [Dl]. We then generalize the strategy to any local field.