Suppose there is a polynomial $p(x)$ with degree $n.$ The problem is to tell the nature of the roots of the equation $p(p(x)) = 0$ if the nature of roots of the equation $p(x) = 0$ is given.
Particularly, what will be the nature of the roots of the equation $p(p(x)) = 0,$ if:
i) All roots of the equation $p(x) = 0$ are real
ii) All roots of the equation $p(x) = 0$ are complex
I tried as much as I can but I am not able to find a single general feature which allows me to move forward towards the solution. What should I look for in this problem?