Abstract
AbstractLetKbe a number field or a function field of characteristic 0, letÏâK(z) with deg(Ï) â©Ÿ 2, and letαâ â1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereÏhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(Ï,α) of â€â©Ÿ0Ă â€>0and show that for all but finitely many (m,n) âA(Ï,α) there is a prime đ âSsuch that ordđ(Ïm+n(α)âÏm(α)) = 1 andαhas portrait (m,n) under the action ofÏmodulo đ. This latter condition implies ordđ(Ïu+v(α)âÏu(α)) â©œ 0 for (u,v) â â€â©Ÿ0Ă â€>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for â1Ă â1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of IngramâSilverman, FaberâGranville and the authors.
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