Factorization of the quadratic Misiurewicz-Thurston polynomials

This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\mathrm{\ell },n}=p_{\mathrm{\ell }+n}(z)-p_{\mathrm{\ell }}(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where $$p_0(z)=0,p_{n+1}(z)=p_n(z{)}^2+z\mathrm{.}$$ The roots can be classified into two categories. First, there are hyperbolic points $\mathrm{hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{\mathrm{\ell },n}$ with multiplicity $\lfloor \frac{\mathrm{\ell }-1}{k}\rfloor +2$. Next are the points $\mathrm{mis}(j,k)$ for $2\le j\le \mathrm{\ell }$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\mathrm{\ell },n}$.