Question

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 ?

Answer 1

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 non-constant, 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.