Is there a complete characterization of those integer polynomials, that is $P\in{\mathbb Z}[X]$, such that $P(D)\subset D$, where $D$ is the unit disk ? At least, $P(X)=\pm X^k$ works, when $k\in\mathbb N$. But are there other ones, many other ones ?

$\begingroup$ I apologize for having posted that an easy question. $\endgroup$ – Denis Serre Apr 5 '12 at 9:08
Since $P\in \mathbb{Z}[X]$, We have $P(0)\in \mathbb{Z}$. Suppose that $P(0)\neq 0$, then $P(0)\geq 1$. In that case $P(D)\subset D$ is not satisfied unless $P$ is constant by open mapping theorem. So, if $P$ is nonconstant, then we must have $P(0)=0$.
Write $P(X)=XQ(X)$. Then $Q(X)=P(X)/X$. On a disk $D_r$ of radius $0< r<1$ centered at $0$,
We have by Maximum modulus theorem that
$$Q(X)\leq 1/r$$
since $P(X)\leq 1$. Letting $r\rightarrow 1$, we have also that $Q(X)\leq 1$ whenever $X\leq 1$. By repeating the same argument for $Q(X)$, we obtain that $P(X)=\pm X^k$, when $k\in \mathbb{N}$ are the only polynomials in $\mathbb{Z}[X]$ satisfying the property.

2

$\begingroup$ math.stackexchange.com/questions/2769600/… related .... with complex coefficients and monic. $\endgroup$ – user111524 May 6 '18 at 19:28