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 4 of 8: A closed integer orbit alternates between two values

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P^2(a)=a

Detailed analysis

Choose a minimum value in the orbit. Since all successive differences have the same nonzero magnitude and the orbit returns to its start, the signs must reverse at that minimum. The orbit therefore alternates between two integers $a,b$ , with $P(a)=b$ and $P(b)=a$ ; in particular every fixed point of $Q$ satisfies $P^2(a)=a$ .