Shared values for meromorphic functions
The value sharing problems of meromorphic functions of one variable have gamed significant attention in the value distribution theory during the last two decades. Recently attempts have been made to extend the results or studies to meromorphic functions of several complex variables. Let \(f\) denote a nonconstant meromorphic function in \(\mathbb{C}^n\). The main result of the paper is the following: If \(f^{-1} (\alpha_j) =(D_uf)^{-1} (\alpha_j)\) (counted with multiplicities) for three distinct polynomials \(\alpha_j\) or \(\infty\), \(j=1,2,3\), then \(f \equiv D_uf\), where \(D_uf\) is the directional derivative of \(f\) along a direction \(u\in S^{2n-j}\). The proof utilizes the \(C^n\) version of Nevanlinna's value distribution theory. Also presented is an elementary but tedious proof of the one complex variable case.
scientific article; zbMATH DE number 848986