Problem 5

Let

P(x)

be a polynomial of degree

n>1

with integer coefficients, and let

k

be a positive integer. Define

Q(x)=P(P(\ldots P(x)\ldots))

, where

P

occurs

k

times. Prove that there are at most

n

integers

t

such that

Q(t)=t

Step 8 of 8: Count the roots and finish

Previous step Next step

\#\{t\in\mathbb Z:Q(t)=t\}\le\#\{x:F(x)=0\}\le n

Detailed analysis

By Step 4, every integer fixed point of $Q$ is a fixed point of $P^2$ , and Step 7 places all such points among the roots of the nonzero degree- $n$ polynomial $F$ . Therefore $Q(t)=t$ has at most $n$ integer solutions, as required.