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Prime rational functions. (English) Zbl 1370.11043
Summary: Let \(f(x)\) be a complex rational function. We study conditions under which \(f(x)\) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that \(f(x)\) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.
MSC:
11C08 Polynomials in number theory
12E05 Polynomials in general fields (irreducibility, etc.)
26C15 Real rational functions
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[1] [1]M. Ayad, Critical points, critical values of a prime polynomial, Complex Variables Elliptic Equations 51 (2006), 143–160. · Zbl 1091.12001
[2] [2]A. F. Beardon, Composition factors of polynomials, Complex Variables Theory Appl. 43 (2001), 225–239. · Zbl 1032.12001
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