We investigate the existence of distinct polynomialsF, G having roots of prescribed multiplicities and deg(F −G) as small as predicted by Mason’sabc theorem. The case of characteristic zero has been treated completely in a previous paper, but those methods do not apply in positive characteristic. Here we study this problem through reduction; it turns out that what we require amounts to proving good reduction for certain covers of the projective line, unramified except above 0, 1, ∞. We shall give sufficient conditions for good reduction of those covers, which sometimes go beyond known criteria due to Grothendieck, Fulton and Beckmann. The methods are completely different from those used by such authors and rely on results by Dwork and Robba onp-adic analytic continuation of Puiseux series.