Let
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
be a polynomial of degree
![n > 1](/media/m/c/8/9/c8999d29e042cf52e485c7a7b7301b0a.png)
with integer coefficients and let
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be a positive integer. Consider the polynomial
![Q(x) = P(P(\ldots P(P(x)) \ldots ))](/media/m/2/9/1/291458ae1ad88258c3e766b588fa1692.png)
, where
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
occurs
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
times. Prove that there are at most
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
integers
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
such that
![Q(t) = t](/media/m/1/6/a/16a7a8ee60fffebcb3cca0dbdbe8e0be.png)
.
%V0
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.