Let $f$ and $g$ be two polynomials. What can we conclude if $$f(g(x))=g(f(x)).$$ I can only check some trivial cases like $f(x)=x^n$ and $g(x)=x^m$.
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1There are non-trivial examples...The Chebyshev polynomials all commute, for instance. see this question – lulu Dec 09 '23 at 11:39
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According to this answer there is a classic result by Ritt which classifies all polynomials that commute. – Kurt G. Dec 09 '23 at 12:06
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2Does this answer your question? When functions commute under composition – Kurt G. Dec 09 '23 at 12:08
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Thank you for your answers. I may read the materials first. – HHe Dec 09 '23 at 12:49