Silverman, Joseph H. The S-unit equation over function fields. (English) Zbl 0533.10013 Math. Proc. Camb. Philos. Soc. 95, 3-4 (1984). 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 Cited in 1 ReviewCited in 14 Documents 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 Keywords:S-unit equation over function fields; separable degree; Riemann-Hurwitz formula PDF BibTeX XML Cite \textit{J. H. Silverman}, Math. Proc. Camb. Philos. Soc. 95, 3--4 (1984; Zbl 0533.10013) Full Text: DOI 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.