Factorization of the quadratic Misiurewicz-Thurston polynomials

TL;DR

This paper completely factors the Misiurewicz-Thurston polynomial related to the Mandelbrot set, classifying roots into hyperbolic and pre-periodic points with explicit multiplicities.

Contribution

It provides the full factorization of the polynomial over complex numbers, detailing the roots' classification and multiplicities, which was previously unknown.

Findings

Roots are classified into hyperbolic and pre-periodic points.

Hyperbolic roots have multiplicities depending on divisors of n.

Pre-periodic roots are simple.

Abstract

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