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 3 of 8: All nonzero successive differences have equal size

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|x_{i+1}-x_i|=|x_i-x_{i-1}|

Detailed analysis

In the remaining case, take a nontrivial orbit. By the lemma, $x_i-x_{i-1}$ divides $x_{i+1}-x_i$ for every $i$ , and the last difference divides back into the first because the orbit closes. Thus all nonzero successive differences have equal absolute value.