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Rational functions with linear relations. (English) Zbl 1142.39017

Let \(K\) be a field. Pick \(\alpha\), \(\beta\), \(\gamma\), \(\delta\in K\) with \(\alpha,\gamma\neq 0\) and let \(f\in K[x]\) have degree \(n> 0\). The authors determine when \(f(\alpha x+ \beta)= \gamma f(x)+\delta\).
Further, the authors solve the equation \(f\circ g= h\circ f\) in rational functions \(f,g,h\in K(x)\), \(K\) being a field of characteristic \(p\geq 0\). These contributions allow to the authors to evoke some results of P. Fatou, G. Julia and J. F. Ritt from the nineteentwentieths concerning commuting rational functions \(f,g\in C(x)\), \(f(g(x))= g(f(x))\). By applying the theorems of the authors one obtains in an easier way some results of C. Wells [ibid. 46, 347–350 (1974; Zbl 0298.12009)], G. L. Mullen [ibid. 84, 315–317 (1982; Zbl 0498.12018)] and H. G. Park [Bull. Korean Math. Soc. 29, No. 2, 277–283 (1992; Zbl 0769.11046)].

MSC:

39B12 Iteration theory, iterative and composite equations
12E05 Polynomials in general fields (irreducibility, etc.)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:

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