Functional Equations in Complex Analysis and Number Theory
نویسنده
چکیده
We study the following questions: (1) What are all solutions to f ◦f̂ = g◦ĝ with f, g, f̂ , ĝ ∈ C(X) being complex rational functions? (2) For which rational functions f(X) and g(X) with rational coefficients does the equation f(a) = g(b) have infinitely many solutions with a, b ∈ Q? We utilize various algebraic, geometric and analytic results in order to resolve both (1) and a variant of (2) in case the numerator of f(X) − g(Y ) is an irreducible polynomial in C[X, Y ]. Our results have applications in various mathematical fields, such as complex analysis, number theory, and dynamical systems. Our work resolves a 1973 question of Fried, and makes significant progress on a 1924 question of Ritt and a 1997 question of Lyubich and Minsky. In addition, we prove a quantitative refinement of a 2015 conjecture of Cahn, Jones and Spear.
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