Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series [Formula: see text] over a field of positive characteristic with residue field [Formula: see text], we prove effective bounds for the number of possible roots of unity in terms of the number of zeros of the auxilliary function [Formula: see text] on the unit circle. In characteristic 0 our bound is still effective but also depends on the ramification degree of the base field over [Formula: see text] as well as the size of the coefficients of [Formula: see text]. This has applications to the Manin-Mumford conjecture in [Formula: see text]. In characteristic [Formula: see text], this work builds upon a pigeon-hole based method by Schmidt.