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 7 of 8: Put every point on one degree-

n

polynomial

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F(x)=C-x-P(x),\qquad C=a+b,\qquad F(\alpha)=0

Detailed analysis

Set $C=a+b$ . For every fixed point $\alpha$ of $P^2$ , with $\beta=P(\alpha)$ , Step 6 gives $\alpha+\beta=C$ . Thus $F(\alpha)=C-\alpha-P(\alpha)=0$ . The two points $a,b$ also satisfy this equation, and $F$ has degree $n$ because $P$ does.