Find all polynomials $P(x)$ which have the property $$P[F(x)]=F[P(x)], \quad P(0) = 0$$ where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$.
This is an exercise from my homework. I would appreciate any kind explanations. Thanks!
Denote $x_0=0$ and $x_{n+1}=F(x_n)$ for every $n\ge 0$. Since $F(x)>x$ for every $x\ge 0$, $x_{n+1}>x_n$ for every $n\ge 0$. Note that $P(x_0)=x_0$ and $$P(x_{n+1})=P(F(x_n))=F(P(x_n)),\quad\forall n\ge 0.$$ Then by induction, it is easy to see that $$P(x_n)=x_n,\quad\forall n\ge 0.\tag{1}$$ $(1)$ implies that the polynomial $P(x)-x$ has infinitely many roots, i.e. $P(x)\equiv x$.