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 2 of 8: Follow the orbit of a fixed point of

Q

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x_{i+1}=P(x_i),\qquad x_k=x_0

Detailed analysis

Let $Q(x_0)=x_0$ and define $x_{i+1}=P(x_i)$ . Then $x_k=x_0$ . If every such integer $x_0$ also satisfies $P(x_0)=x_0$ , the roots of $P(x)-x$ already give the desired bound because this polynomial has degree $n$ .