IR-MO 2003 Problem 29

Let $c\in\mathbb C$ and $A_c = \{p\in \mathbb C[z]|p(z^2 + c) = p(z)^2 + c\}$ .

a) Prove that for each $c\in C$ , $A_c$ is infinite.

b) Prove that if $p\in A_1$ , and $p(z_0) = 0$ , then $|z_0| < 1.7$ .

c) Prove that each element of $A_c$ is odd or even.

Let $f_c = z^2 + c\in \mathbb C[z]$ . We see easily that $B_c: = \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $A_c$ . Prove that in the following cases $A_c = B_c$ .

d) $|c| > 2$ .

e) $c\in \mathbb Q\backslash\mathbb Z$ .

f) $c$ is a non-algebraic number

g) $c$ is a real number and $c\not\in [ - 2,\frac14]$ .