Picard sets admitting exceptionally ramified meromorphic functions (Q752227)

A meromorphic function having more than two Picard exceptional values at a singularity is exceptionally ramified there. There are no meromorphic functions being exceptionally ramified, consequently no meromorphic functions having more than two Picard exceptional values (Picard's theorem), at an isolated singularity. In this paper, we showed the existence of singularity sets E such that any function of \(M_ E\), the totality of meromorphic functions with E as the set of singularities, cannot have more than two Picard exceptional values at any singularity (we call such a set E a Picard set), while, contrary to the case of isolated singularities, there are functions of \(M_ E\) being exceptionally ramified at each singularity. As for singularity sets E for which any function of \(M_ E\) cannot be exceptionally ramified at any singularity, see \textit{K. Matsumoto} and \textit{T. Kurokawa}, Analytic function theory of one complex variable, Pitman Res. Notes Math. Ser. 212, 123-148 (1989; Zbl 0683.30027).