The functional equation \(P(f)=Q(g)\) in a \(p\)-adic field.
type: Item
instance of: scholarly article
title: The functional equation \(P(f)=Q(g)\) in a \(p\)-adic field.
zbMATH Open document ID: 1054.30043
author: Alain Escassut
author: Chungchun Yang
published in: Journal of Number Theory
publication date: 2004-05-27T00:00:00Z
review text: 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)\…
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30G06
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 2066767
zbMATH Keywords: \(p\)-adic Nevanlinna theory
zbMATH Keywords: functional equation
MaRDI profile type: Publication
full work available at URL: j.jnt.2003.11.005
OpenAlex ID: W2084304436
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DOI: 10.1016/J.JNT.2003.11.005
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description: scientific article; zbMATH DE number 2066767