Some functional equations and Picard constants of algebroid surfaces Dedicated to Professor Mitsuru Nakai on his 60th birthday

概要

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The authors study the functional equation $(\ast) \sum^m_{µ=0}a_µ(z)e^{µH(z)}=f(z)\sum^n_{\nu=0}b_\nu(z)e^{\nu L(z)}$ and give an application. Let $R$ be a Riemann surface, $M(R)$ the family of non-constant meromorphic functions on $R, P(f)$ the number of values which are not taken by an element $f$ of $M(R)$. The Picard constant $P(R)$ of $R$ is defined by $P(R)=\sup(P(f)| f\in M(R)). P(R)$ is conformally invariant, $P(R)\geq2$ if $R$ is open, and $P(R)\leq2n$ if $R$ is an $n$-sheeted algebroid surface. They apply a result on $(\ast)$ to obtain a result on $P(R)$ for a four-sheeted algebroid surface $R$, which is an improvement of a result obtained earlier by M. Ozawa and K. Sawada [Kodai Math. J. 17 (1994), no. 1, 101--124; MR1262956 (95g:30039); Kodai Math. J. 18 (1995), no. 2, 199--233; MR1346901 (96m:30043)].
1996-10-01