The functional equation \(P(f)=Q(g)\) in a \(p\)-adic field.
The authors investigate whether a meromorphic function may admit two different decompositions of the form \(P\circ f\) and \(Q\circ g\), where \(P\), \(Q\) are two nonlinear polynomials in \(K\), which is a complete ultrametric algebraically closed field of characteristic \(\pi\). This work was motivated by some previous works of the functional equation \[ P(f)= Q(g)\tag{1} \] in the complex plane \(\mathbb{C}\) [see, e.g. \textit{H. K. Ha} and \textit{C. C. Yang}, Value distribution theory, Marcel Dekker, New York, 2003]. For a given \(a\in K\) and \(r> 0\), they denote by \(d(a, r^-)\) the disk \(\{x\in K\mid| x-a|< r\}\). Let \({\mathcal M}(d(a, r^-))\) and \({\mathcal A}(d(a, r^-))\) be the field of meromorphic functions and the ring of analytic functions in \({\mathcal M}(d(a, r^-))\), respectively. In \({\mathcal M}(d(a, r^-))\), they denote by \({\mathcal M}_b(d(a, r^-))\) the sub-field of functions \(f\in{\mathcal M}(d(a, r^-))\) which are of the form \(\varphi/\psi\) with \(\varphi,\psi\in{\mathcal A}A(d(a, r^-))\), and by \({\mathcal M}_u(d(a, r^-))\) the set \({\mathcal M}(d(a, r^-))\setminus{\mathcal M}_b(d(a, r^-))\). The authors use the Nevanlinna's value distribution theory in the characteristic \(\pi\) [see, e.g., \textit{H. G. W. Begehr}, \textit{R. P. Gilbert} and \textit{Man Wah Wong} (eds.), Analysis and applications -- ISAAC 2001. Proceedings of the 3rd international congress, Berlin, Germany, August 20--25 (2001; Zbl 1031.35002)], for examples, \(\chi\) is the characteristic exponent, \(\nu(f)\) is the ramification index of \(f\), \(T(\rho, f)\) is the Nevanlinna characteristics function of \(f\), and \(\widetilde N(\rho, f)\) is a counting function of \(f\). One of the main results is the following. Let \(P\), \(Q\) be in \(K[x]\) with \(P'\) and \(Q'\) not identically zero and let \(p= \deg(P)\), \(q= \deg(Q)\) satisfy \(2\leq\min(p, q)\). Assume that there exist \(k\) distinct zeros \(c_1,\dots, c_k\) of \(P'\) such that \(P(c_i)\neq P(c_j)\) for \(i\neq j\), and \(P(c_i)\neq Q(d)\) for every zero of \(d\) of \(Q'\), \(i= 1,\dots, k\). Assume that there exist two functions \(f, g\in{\mathcal M}_u(d,r^-)\) which satisfy (1) and let \(t= \nu(f)\). Then \[ \widetilde N(\rho,f)\geq {T(\rho, f)(kq- p)\over \chi^t q}+ O(1). \] Moreover, suppose that \(2p/3< q\). Then \(k\leq 2\). Further, if \(\neq q\), then \(k= 1\), \(c_1\) is a simple zero of \(P'\) and either \(q< p\) or g.c.d. \((p, q) =q -p\).
scientific article; zbMATH DE number 2066767
scientific article