Corrales Rodrigáñez, Capi On a characterization of the analytic and meromorphic functions defined on some rigid domains. (English) Zbl 0815.32017 J. Théor. Nombres Bordx. 6, No. 1, 117-125 (1994). Let \({\mathcal C}\) be a smooth irreducible curve over a nonarchimedean complete algebraically closed field \(k\) of characteristic zero. Let \(X\) be the subspace of the rigid analytification of \({\mathcal C}\) obtained by deleting either \(s\) points or \(s\) disks. Let \(u\) and \(v\) be holomorphic (resp. meromorphic) functions on \(X\) such that \(u^{-1} (\{\alpha_ i\}) = v^{-1} (\{\alpha_ i\})\) for distinct \(\alpha_ i, \dots, \alpha_ r\) in \(k\). Question: to find the smallest \(r\) such that \(u = v\)? The author proves \(r \leq 3\) (resp. \(r \leq 5)\). In the algebraic case, that is if \(u\) and \(v\) are nonconstant morphisms from \({\mathcal C}\) to \(\mathbb{P}^ 1_ k\), the author proves \(r \leq 2g + 3\) where \(g\) is the genus of \({\mathcal C}\) (and \(r = 3\) if \(g = 0)\). Reviewer: M.Reversat (Toulouse) MSC: 32P05 Non-Archimedean analysis 32A10 Holomorphic functions of several complex variables 32A20 Meromorphic functions of several complex variables 14G20 Local ground fields in algebraic geometry Keywords:nonarchimedean field; holomorphic function; meromorphic function PDFBibTeX XMLCite \textit{C. Corrales Rodrigáñez}, J. Théor. Nombres Bordx. 6, No. 1, 117--125 (1994; Zbl 0815.32017) Full Text: DOI Numdam EuDML EMIS References: [1] Adams, W.W., Straus, E.G., Non-archimedean analytic functions taking the same values at the same points, Illinois J. of Math15 (1971), 418-424. · Zbl 0215.13202 [2] Bourbaki, N., Eléments de mathématiques. Espaces vectoriels topologiques, Chap. I, Hermann, Paris, 1953. · Zbl 0050.38903 [3] Bosch, S., Dwork, B., Robba, P., Un théorème de prolongement pour des fonctions anatytiques, Math. Ann.252 (1980), 165-173. · Zbl 0446.32005 [4] Bosch, S., Gfjntzer, U., Remmert, R., Non archimedean analysis, Grund. der Math262, Springer Verlag, 1984. · Zbl 0539.14017 [5] Fresnel, J., Géométrie anlytique rigide, Université Bordeaux I, 1984. [6] Fresnel, J., Matignon, M., Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique, Annali di matematica pura ed applicata (IV) CXLV (1986), 159-210. · Zbl 0623.32020 [7] Fresnel, J., van der PUT, M., Géométrie analytique rigide et applications, Progress in Math.18, Birkhäuser, 1981. · Zbl 0479.14015 [8] Kiehl, R., Theorem A und Theorem B in der nichtarchimedischen Funktionertheorie, Invent. Math.2 (1967), 256-273. · Zbl 0202.20201 [9] Serre, J.-P., Géométrie algebrique et géométrie analytique, Ann. Inst. Fourier6 (1956), 1-42. · Zbl 0075.30401 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.