Picard constants of \(n\)-sheeted algebroid surfaces (Q1861085)

Denote by \({\mathfrak M}(R)\) the family of non-constant meromorphic functions on a Riemann surface \(R\), and let \(p(f)\) be the number of Picard exceptional values of \(f\in{\mathfrak M}(R)\). Then \({\mathcal P}(R)= \sup_{f\in{\mathfrak M} (R)}p(f)\) is called the Picard constant of \(R\). It is well known that \({\mathcal P} (R)\geq 2\) if \(R\) is open and \({\mathcal P}(R)=0\) if \(R\) is compact. An \(n\)-sheeted algebroid surface is the domain of existence of an \(n\)-valued algebroid function \(y\), defined by an irreducible equation \[ F(zy)=S_0(z)y^n- S_1(z)y^{n-1}+ \cdots + (-1)^{n-1}s_{n-1} (z)y+(-1) S_n(z)=0, \] where the \(S_j\) are entire functions with no common zeros. If \(R\) is an \(n\)-sheeted algebroid surface, then \({\mathcal P} (R)\leq 2n\) by the Selberg theory of algebroid functions. As for recent investigations on \(p(f)\) and \({\mathcal P}(R)\) see [\textit{K. Niinō} and \textit{K. Tohge}, J. Math. Soc. Japan 48, 649-665 (1996; Zbl 0877.30013)], \textit{M. Ozawa} and \textit{K. Sawada}, Kodai Math. J. 17, 101-124 (1994; Zbl 0913.30023)], \textit{M. Ozawa} and \textit{K. Sawada}, Kodai Math. J. 18, 99-141 (1995; Zbl 0836.30019)] \textit{K. Sawada} and \textit{K. Tohge}, Kodai Math. J. 18, 142-155 (1995; Zbl 0835.30023)]. The present paper is devoted to constructing all \(a\)-sheeted entire \((S_0(z)\equiv 1)\) algebroid Riemann surfaces \(R\) such that \({\mathcal P}(R)\geq n+1\). This characterization is too complicated to be repeated here. However, the determining equation \(F(z,y)=0\) takes the form \[ F(z,y)=P(y)+ \sum_{j=1}^l Q_j(y) e^{H_j^* (z)}=0, \] where the \(H^*_j\) are non-constant entire functions with \(H_j^*(0)=0\), \(P(y)\) is a certain polynomial and the \(Q_j\) are certain rational functions. In particular, if \(l= 1\), then \(F(z,y) =P(y)+Q(y)e^{H(z)}\), where \[ \begin{aligned} P(y)& =\prod_{k=1}^m(y-b_k)^{n_k}, \quad n_1+ \cdots +n_m=n,\\ \text{and}\\ Q(y)& =a \prod_{k =1}^{p-m} (y-a_k)^{s_k}, \quad s_1+\dots +s_{p-m}\leq n-1,\quad a\neq 0. \end{aligned} \] Among the consequences of this characterization we observe that if \(p(y)>3 n/2\) is the number of exceptional values of an \(n\)-valued entire algebroid function \(y\), then \(l=1\) follows. In addition to the characterization described above, the case \(l=1\) is investigated extensively in this paper. As an example of the results let us mention the following theorem: Let \(R\) be an \(n\)-sheeted algebroid surface defined by \(P(y)+Q(y) e^{H(z)}=0\) and let \(f\) be an entire function on \(R\). Then \(f\) has a representation \(f=f_0+ f_1y +\cdots +f_{n -1} y^{n-1}\), where the \(f_j\) are meromorphic in the complex plane, regular at all points \(z\) such that \(H'(z)\neq 0\). Moreover, the Picard constants are investigated for \(l=1\) in the final section. Some open problems are also offered for future studies.