A polynomial that commutes with squaring

Unsolved Hard Archive

archive posted 3 weeks ago

Find all polynomials $P$ with real coefficients such that $P (x^{2}) = P (x)^{2}$ for every real $x$ .

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$P = 0$ , and $P (x) = x^{n}$ for $n = 0, 1, 2, \dots$

History and references

Comparing leading coefficients gives $a = a^{2}$ , so a nonzero $P$ is monic. If $P$ had a second nonzero term $c x^{k}$ below $x^{n}$ , the square $P (x)^{2}$ would contain $2 c x^{n + k}$ , while $P (x^{2})$ has no term of that degree. So $P (x) = x^{n}$ .

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