Uniform bound for points of bounded degree in function fields of positive characteristic

I will present a bound for the number of F_q[t]-points of bounded degree in a variety defined over Z[t], uniform in q. This generalizes work by Sedunova for fixed q. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.