Hensley's problem for complex and non-Archimedean meromorphic functions
The authors consider Hensley's \(n\)-th power problem for complex meromorphic functions and for non-Archimedean meromorphic functions. In the complex case they use classical Nevanlinna theory, and in the non-Archimedean case they use \(p\)-adic Nevanlinna theory. One of the results in the complex case is the following: Suppose that \(f\), \(g\) and \(x_1,\dots, x_M\) are meromorphic functions and at least one of the \(x_i\) is not constant. Let \(n\) be an integer no less than \(2\) and \((c_1, \dots, c_M)\) be a sequence of distinct constants in \(\mathbb C\). Suppose that \[ x_i^n =(f+c_i)^n-g \quad \text{for }1\leq i\leq M. \] Then \(g=0\) if (i) \(M> \frac{7n^4-n^3-4n+3}{(n-1)^2}\), or (ii) \(f\) and \(g\) are entire functions and \(M>\frac{4n^4-n^3-4n+3}{n-1}\). When \(n\geq 9\), the size of \(M\) can be much smaller. The authors also obtain the corresponding result in the non-Archimedean case.
scientific article; zbMATH DE number 5909847
scientific article