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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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[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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