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**Abstract**

Let *K* be a field of characteristic [math] and let *G* be a subgroup of [math]with [math] finite. Then Voloch proved that the equation [math] for given [math] has at most [math] solutions [math], unless [math] for some [math]. Voloch also conjectured that this upper bound can be replaced by one depending only on *r*. Our main theorem answers this conjecture positively. We prove that there are at most [math] solutions (*x*, *y*) unless [math] for some [math] with (*n*, *p*) = 1. During the proof of our main theorem, we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods cannot be transferred to positive characteristic.

*Journal Article.*
*3105 words.*

*Subjects: *
Pure Mathematics

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