Some further results on the unique range sets of meromorphic functions (Q1911164)

Let \(S\) be a subset of the complex plane \(\mathbb{C}\). For a non-constant meromorphic function \(f\) let \(E_f(S)= \bigcup_{n\in S} \{(z, p)\in \mathbb{C}\times N\mid f(z)= a\) with multiplicity \(p\}\). \(S\) is called a unique range set for entire functions (URSE) if for any pair of non-constant entire functions \(f\) and \(g\) the condition \(E_f(S)= E_g(S)\) implies \(f= g\). Unique range sets for meromorphic functions (URSM) are defined in a similar way. The existence of URSE with finitely many elements has been exhibited by \textit{H.-X. Yi} [Sci. China, Ser. A 38, No. 1, 8-16 (1995; Zbl 0819.30017)], thereby solving a question of \textit{F. Gross} [Complex analysis, Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51-67 (1977; Zbl 0357.30007)]. In the paper under review, the authors show that there exist URSE with 7 elements and URSM with 15 elements. The proof uses Nevanlinna's theorem on the so-called ``Borel-identities'' and a very careful analysis of the zeros and poles of the Wronskian determinant occurring in this theorem. Lower bounds for the cardinality of unique range sets are also given. To determine the exact value of the minimum cardinality of unique range sets is an interesting but still unsolved problem in value distribution theory.