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Is there an elementary real function $F$ such that $F(1+F^{-1}(x))$ is a polynomial of degree at least 2 without real fixpoints.

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I don't know. Is there any use for such a function? Do you know an $F$ such that you get a polynomial of degree at least 2 with a real fixed point? – Gerry Myerson Nov 7 '10 at 4:10
Ya one gets the Chebychev polynomials $T_n$ of degree $n$ with $F(x)=\cos(n^x)$. For example $\cos(2^{1+log_2(\arccos(x))})=\cos(2 \arccos(x))=2 x^2-1 = T_2(x)$. – bo198214 Nov 7 '10 at 4:43
And I forgot to mention: All Chebychev polynomials have fixpoint 1. – bo198214 Nov 7 '10 at 12:07

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