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For a natural number $n$, what are all the possible number of distinct real roots of $P(P(\dots(P(x))\dots))$, where $P$ is a polynomial of degree $d$ and there are $n$ copies of $P$'s?

The iterated polynomial has degree $d^n$, so the number of roots cannot be more than that, but can it be any number less than or equal to it? (At least if $d$ is odd, there should always be at least one real root.)

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  • $\begingroup$ If $P(x)$ has no real roots then neither does $P\circ P(x)$. After all, if $P\circ P(x_0)=0$ then $P(P(x_0))=0$. $\endgroup$
    – lulu
    Oct 26, 2020 at 13:31

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You're right with the odd part. If $d$ is odd, then $d^n$ is always odd, so there is always at least one real root.

I can provide a simple example that the number of roots can be less than $n$ when $d \equiv 0\pmod{2}.$ If $$P(x)=(x+1)^2 \text{ and } n=2,$$

then we get $((x+1)^2+1)^2=P(P(x)$. The LHS factors as $x^4+4x^3+8x^2+8x+4$ which has no real roots.

If you have a function that never crosses the x-axis, then raising it to the $n^\text{th}$ power when $n$ is even is going to keep it above the x-axis. Then, any shift up will keep it, again, above the x-axis.

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