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The S-unit equation over function fields. (English) Zbl 0533.10013
A simple alternative proof is given for the following result due to R. C. Mason [ibid. 93, 219-230 (1983; Zbl 0513.10016)]. Let k be an algebraically closed field of characteristic $$p\geq 0$$, C/k a smooth projective curve of genus g, S a finite set of points of C(k). If u and v are S-units satisfying $$u+v=1$$, then either $$u,v\in k^*$$ or $$\deg_ s(u)\leq 2g-2+| S|$$, where $$\deg_ s$$ denotes the separable degree and $$| |$$ the cardinality. The proof only relies on elementary algebraic geometry, principally the Riemann-Hurwitz formula.
Reviewer: S.Uchiyama

##### MSC:
 11D99 Diophantine equations 11R58 Arithmetic theory of algebraic function fields 11D88 $$p$$-adic and power series fields 11D41 Higher degree equations; Fermat’s equation
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##### References:
 [1] Hartshorne, Algebraic Geometry (1977) · doi:10.1007/978-1-4757-3849-0 [2] Mason, Math. Proc. Cambridge Philos. Soc. 93 pp 219– (1983)
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