Some functional equations and Picard constants of algebroid surfaces (Q1363219)

In order to compute Picard constants of algebroid surfaces, functional equations of type \[ \sum_{\mu=0}^ma_{\mu}(z)e^{\mu H(z)}=f(z)\sum_{\nu=0}^nb_\nu(z)e^{\nu L(z)}, \] where \(H\) and \(L\) are non-constant entire functions with \(H(0)=L(0)=0\), and \(a_\mu\), \(b_\nu\) and \(f\) are meromorphic functions, will be considered by using the Nevanlinna theory as a device. To give an example of results, normalize with \(a_m=b_n=1\) and assume that \(a_\mu\), \(b_\nu\) are small functions and that \(f\) has a few zeros and poles, specified in the Nevanlinna theory sense. Then either \(e^{mH(z)+nL(z)}=a_0(z)b_0(z)\) and \(f(z)=a_0(z)e^{-nL(z)}\) or \(e^{mH(z)-nL(z)}=a_0(z)/b_0(z)=f(z)\) and \(a_\mu\), \(b_\nu\) are to satisfy certain simple identities. Recall that a Picard constant \(P(R)\) of a Riemann surface \(R\) means the supremum of the number of values \(P(R)\) omitted by \(f\), taken over all non-constant meromorphic functions on \(R\). As an application of the above considerations, \(P(R)\) will be defined for certain four-sheeted algebroid surfaces defined by \(y^4+S_1(z)y^3+S_2(z)y^2+S_3(z)y+S_4(z)=0\), where \(S_1,\dots,S_4\) are entire functions and at least one of them is transcendental.