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!

The result is Corollary 3.3 in this paper. 
