On a characterization of the analytic and meromorphic functions defined on some rigid domains (Q1340680)

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 \(g = 0)\).