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Applications of the \(p\)-adic Nevanlinna theory to functional equations. (English) Zbl 1063.30043

Summary: Let \(K\) be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the \(p\)-adic Nevanlinna theory to functional equations of the form \(g= R\circ f\), where \(R\in K(x)\), \(f, g\) are meromorphic functions in \(K\), or in an “open disk”, \(g\) satisfying conditions on the order of its zeros and poles. In various cases we show that \(f\) and \(g\) must be constant when they are meromorphic in all \(K\), or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular, we have an easy way to obtain again Picard-Berkovich’s theorem for curves of genus \(1\) and \(2\). These results apply to equations \(f^m+g^n=1\), when \(f,\;g\) are meromorphic functions, or entire functions in \(K\) or analytic functions in an “open disk”. We finally apply the method to Yoshida’s equation \({y'}^m= F(y)\), when \(F\in K(X)\), and we describe the only case where solutions exist: \(F\) must be a polynomial of the form \(A(y-a)^d\) where \(m-d \) divides \(m\), and then the solutions are the functions of the form \(f(x)=a+\lambda(x-\alpha)^{m\over m-d}\), with \(\lambda^{m-d}({m\over m-d})^m=A\).

MSC:

30G06 Non-Archimedean function theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B99 Functional equations and inequalities
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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References:

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