Journal Article

On the equation X1 + X2 = 1 infinitely generated multiplicative groups in positive characteristic

Peter Koymans and Carlo Pagano

in The Quarterly Journal of Mathematics

Volume 68, issue 3, pages 923-934
Published in print September 2017 | ISSN: 0033-5606
Published online February 2017 | e-ISSN: 1464-3847 | DOI: http://dx.doi.org/10.1093/qmath/hax007
On the equation X1 + X2 = 1 infinitely generated multiplicative groups in positive characteristic

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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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