# When polynomial is power

$P(x)$ ia a polynomial with real coefficients, and $k>1$ is an integer. For any $n\in\Bbb Z$, we have $P(n)=m^k$ for some $m\in\Bbb Z$. Show that there exists a real coefficients polynomial $H(x)$ such that $P(x)=(H(x))^k$, and $\forall n\in\Bbb Z,$ $H(n)$ is an integer.

This is an old question, but I never saw a complete proof. Thanks a lot!

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mast.queensu.ca/~murty/poly2.pdf proves this for $k=2$ but for polynomials in any number of variables. It mentions the result you want, but gives no references. – Gerry Myerson Aug 11 '13 at 10:17

The result is Corollary 3.3 in this paper.

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You can skip the nonelementary stuff in §2: To arrive at corollary 3.3, you need only read up to lemma 1.5 and then §3. – Hagen von Eitzen Aug 11 '13 at 16:42