Uniqueness problem with truncated multiplicities for meromorphic functions on a non-Archimedean field (Q1879051)

Let \(f\) be a non-constant meromorphic function on a complete non-Archimedean algebraically closed field \(K\) of characteristic zero. For a set \(S\subset K\) denote by \(E_f^{m_0}(S)\) the set \[ \bigcup_{a\in S}\{ (z,m)\in K\times \mathbb N\mid f(z)=a\text{ with multiplicity \(n\) and }m=\min (n,m_0)\}. \] The authors find sufficient conditions upon \(S\) for the equality \(E_f^{m_0}(S)=E_g^{m_0}(S)\) to imply \(f=g\) for any non-constant meromorphic functions \(f,g\). This covers the results obtained for \(m_0=\infty\) by \textit{P.-C. Hu} and \textit{C.-C. Yang}, [Acta Math. Vietnam. 24, No. 1, 95--108 (1999; Zbl 0986.30025)], and for \(m_0=1\) by \textit{A. Boutabaa} and \textit{A. Escassut} [Rend. Circ. Mat. Palermo 49, 501--520 (2000)].