# the degree of the multiplier of Misiurewicz points

Define the iterated complex quadratic polynomial \begin{aligned}f^{0\phantom{+1}}_c(z) &= z \\ f_c^{n + 1}(z) &= (f_c^n(z))^2+c\end{aligned}

A Misiurewicz point $M_{q,p}$ satisfies $$f_c^{q + p}(0) = f_c^q(0)$$ where $q > 0$ is the preperiod and $p > 0$ is the period. The equation also has roots with lower preperiod (including $0$) and/or period, these should be discounted. By construction each $M_{q,p}$ is an algebraic integer. The multiplier $m$ of $c = M_{q,p}$ is defined as $$m = \prod_{n = q}^{q + p - 1} 2 f_c^n(0)$$ and is also an algebraic integer.

Question: is the degree of the minimal polynomial of $m$ always equal to the degree of the minimal polynomial of $M_{q,p}$?

(I had previously tabulated an attempt at finding the polynomials, but I realize now that I didn't check for irreducibility properly, so I removed the misleading tables and code from here and put them on my blog instead.)

• – Claude Aug 5 '18 at 15:19
• – Claude Aug 8 '18 at 15:39