# Factorization of the quadratic Misiurewicz-Thurston
polynomials

Nicolae Mihalache, François Vigneron

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Nicolae Mihalache, François Vigneron. Factorization of the quadratic Misiurewicz-Thurston polynomials. 2025. hal-05124006

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Distributed under a Creative Commons Attribution 4.0 International License
Factorization of the quadratic
Misiurewicz-Thurston polynomials

Nicolae Mihalache¹ and François Vigneron²

June 21, 2025

Abstract

This note provides the complete factorization of the Misiurewicz-Thurston polyno-
mial $q_{\ell,n}$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set. The
roots can be classified into two categories. First, there are hyperbolic points $\text{Hyp}(k)$
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 $\lfloor \frac{\ell-1}{k} \rfloor + 2$. Next are the points $\text{Mis}(j,k)$
for $2 \le j \le \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact
pre-period $j$. Those are simple roots of $q_{\ell,n}$.

In this article, we are interested in the factorization of two important families of poly-
nomials. The first family is defined recursively by

$$
p_0(z) = 0, \quad p_{n+1}(z) = p_n(z)^2 + z. \tag{1}
$$

For $n \ge 1$, one has $\deg p_n = 2^{n-1}$. The second family is a two parameters family defined for $\ell, n \in \mathbb{N}$ by

$$
q_{\ell,n}(z) = p_{\ell+n}(z) - p_{\ell}(z). \tag{2}
$$

The factorisation of $p_n$ over $\mathbb{C}$ is well known [DH82] and is recalled first to fix essential
notations. The main result of this article is the proof of Theorem 2 below, which gives
the complete factorization of $q_{\ell,n}$ in terms of simple factors whose dynamical significance is
explained briefly in the first section. This result plays a role in the systematic computation
of hyperbolic centers and of pre-periodic parameters in the Mandelbrot set, up to period
$n = 41$ for $p_n$ and all types $\ell+n \le 35$ for $q_{\ell,n}$, as detailed in [MV25], yet remains of general
interest.

# 1 About the Mandelbrot set

The polynomials $p_n$ and $q_{\ell,n}$ are closely related to the *Mandelbrot set* $\mathcal{M}$ (Fig. 1).
This set is composed of the parameters $c \in \mathbb{C}$ for which the sequence $(p_n(c))_{n \in \mathbb{N}}$ remains
bounded. This sequence is the orbit by iterated compositions of the only critical point
$z = 0$ of the map $f_c(z) = z^2 + c$, i.e.

$$
\forall n \in \mathbb{N}, \quad f_c^n(0) = \underbrace{f_c \circ \dots \circ f_c}_{n \text{ times}}(0) = p_n(c). \qquad (3)
$$
For $c \in \mathbb{C}$, the dynamics of $f_c$ splits $\bar{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ in two complementary sets. The Fatou set $\mathcal{F}_c$ is the open subset composed by the points $z \in \bar{\mathbb{C}}$ in the neighborhood of which, the sequence $(f_c^n)_{n \in \mathbb{N}}$ is a normal family i.e. precompact in the topology of local uniform convergence. On the contrary, on the Julia set $\mathcal{J}_c = \bar{\mathbb{C}} \setminus \mathcal{F}_c$, the dynamics is chaotic. Both $\mathcal{F}_c$ and $\mathcal{J}_c$ are fully invariant (i.e. invariant sets of the forward and backwards dynamics) and that $c \in \mathcal{M}$ if and only if $\mathcal{J}_c$ is connected. For a review of the properties of Fatou and Julia sets, see e.g. [CG93], [Mil90] for polynomial and rational maps and [Ber93], [MPRW22] for entire and meromorphic functions.

<figure><img src="image_2.png" /><figcaption>Figure 1: The Mandelbrot set $\mathcal{M}$ with $\partial\mathcal{M}$ in black and in gray, the interior of $\mathcal{M}$.</figcaption></figure>

## 1.1 Hyperbolic centers

For $n \ge 1$, the roots of $p_n$ are parameters $c \in \mathcal{M}$, called hyperbolic centers, whose critical orbits are periodic of period $n$. The roots of $p_n$ are simple [DH82, Buf18] and the polynomial $p_n(z)$ is divisible by $p_k(z)$ for any divisor $k$ of $n$. A. Douady and J.H. Hubbard [DH82, DH85] have shown that the hyperbolic centers are interior points of $\mathcal{M}$, with at most one hyperbolic center per connected component of the interior. The set of all hyperbolic centers is also dense in the boundary of $\mathcal{M}$ in the sense that its closure in $\mathbb{C}$ contains $\partial\mathcal{M}$.

Let us define the set of hyperbolic centers of order $n \ge 1$ as the subset of $p_n^{-1}(0)$ whose minimal (or fundamental) period is exactly $n$; in other terms:

$$ \mathrm{Hyp}(n) = \{z \in p_n^{-1}(0) \mid \forall k \in \mathrm{Div}(n)^*, p_k(z) \neq 0\} \quad (4) $$

where $\mathrm{Div}(n) = \{k \in \mathbb{N}^*; \exists k' \in \mathbb{N}, kk' = n\}$ and $\mathrm{Div}(n)^* = \mathrm{Div}(n)\setminus\{n\}$ is the set of strict divisors of $n$. For example, $\mathrm{Hyp}(1) = \{0\}$ and $\mathrm{Hyp}(2) = \{-1\}$. The reduced
polynomial, also known as Gleason's polynomial, is:

$$h_n(z) = \prod_{r \in \text{Hyp}(n)} (z - r). \qquad (5)$$

The polynomials $p_n$ and $h_n$ have integer coefficients. While expected, the irreducibility of $h_n$ over $\mathbb{Z}[z]$ remains conjectural; see [HT15, last remark of §3] and [SS17, p.155].

**Theorem 1 ([DH82]).** The complete factorization of $p_n$ is

$$p_n(z) = \prod_{k|n} h_k(z). \qquad (6)$$

Moreover, the cardinal of Hyp($n$) is given by

$$|\text{Hyp}(n)| = \sum_{k|n} \mu(n/k) 2^{k-1} \qquad (7)$$

where $\mu$ is the Möbius function, i.e.

$$\mu(n) = \begin{cases} (-1)^\nu & \text{if } n \text{ is square free and has } \nu \text{ distinct prime factors,} \\ 0 & \text{if } n \text{ is not square free.} \end{cases}$$

The notation $k|n$ means $k \in \text{Div}(n)$ and $|S|$ denotes the cardinal of a finite set. The identity (6) is well known and is key in the count of hyperbolic centers. The Online Encyclopedia of Integer Sequences [OEIS] attributes (7) to Warren D. Smith and Robert Munafo (2000), sadly with no published reference. For a more general result that applies to iterated maps of the form $z^d + c$, see [HT15].

<figure><img src="image_3.png" /><figcaption>Figure 2: The hyperbolic points Hyp($n$) for $n \le 18$ in green and the Misiurewicz-Thurston parameters $M_{\ell,n} = \text{Mis}(\ell, n)$ with $\ell + n \le 16$ in red in different parts of the Mandelbrot set $M$, with $\Re c$ increasing from left to right between images.</figcaption></figure>
## 1.2 Pre-periodic or Misiurewicz-Thurston parameters

For integers $n \ge 1$ and $\ell \ge 2$, the roots of $q_{\ell,n}$ (defined in (2)) are parameters $c \in \partial\mathcal{M}$, called *pre-periodic parameters* or *Misiurewicz-Thurston points*, whose critical orbits are pre-periodic, that is it becomes periodic of period $n$ after the first $\ell$ steps. For $\ell \in \{0, 1\}$, one can check immediately that

$$q_{0,n}(z) = p_n(z) \quad \text{and} \quad q_{1,n}(z) = p_n^2(z). \qquad (8)$$

The dynamical reason for the second identity in (8) is that 0 is the only pre-image of $c$ under $f_c$ so a pre-periodic orbit of type $(1, n)$ starting at zero actually loops back to zero, so is periodic of period $n$. The polynomial $q_{\ell,n}(z)$ is divisible by $q_{\ell,k}(z)$ for any divisor $k$ of $n$ and by $q_{\ell',n}(z)$ for any $\ell' \le \ell$. Some of the roots of $q_{\ell,n}(z)$ have multiplicity; however, the polynomial

$$s_{\ell,n}(z) = \frac{q_{\ell,n}(z)}{q_{\ell-1,n}(z)} \in \mathbb{Z}[z] \qquad (9)$$

has simple roots [Buf18], [HT15, Lemma 3.1]. Douady-Hubbard [DH82, DH85] have shown that the set of all pre-periodic points is dense in the boundary of $\mathcal{M}$. Visually, those points are either branch tips, centers of spirals or points where branches meet (see Fig. 2).

By analogy with the hyperbolic case, let us define the set of Misiurewicz points of type $(\ell, n)$ as the subset of $q_{\ell,n}^{-1}(0)$ whose dynamical parameters are exactly $\ell$ and $n$; in other terms:

$$\mathrm{Mis}(\ell, n) = \left\{ z \in q_{\ell,n}^{-1}(0) \left| \begin{array}{l} q_{\ell-1,n}(z) \neq 0, \\ \forall k \in \mathrm{Div}(n)^*, q_{\ell,k}(z) \neq 0 \end{array} \right. \right\}. \qquad (10)$$

We call $\ell + n$ the order of $\mathrm{Mis}(\ell, n)$ because $q_{\ell,n}$ is a polynomial of degree $2^{\ell+n-1}$. The reduced polynomial whose roots are exactly $\mathrm{Mis}(\ell, n)$ is denoted by

$$m_{\ell,n}(z) = \prod_{r \in \mathrm{Mis}(\ell,n)} (z-r) \in \mathbb{Z}[z]. \qquad (11)$$

Note that $\mathrm{Mis}(0, n) = \mathrm{Hyp}(n)$ and $\mathrm{Mis}(1, n) = \emptyset$ because of (8). The count of the Misiurewicz points has been established by B. Hutz and A. Towsley [HT15, Cor. 3.3]:

$$|\mathrm{Mis}(\ell, n)| = \Phi(\ell, n) |\mathrm{Hyp}(n)| \qquad (12)$$

where

$$\Phi(\ell, n) = \begin{cases} 1 & \text{if } \ell = 0, \\ 2^{\ell-1} - 1 & \text{if } \ell \neq 0 \text{ and } n|\ell-1, \\ 2^{\ell-1} & \text{otherwise.} \end{cases}$$

To get a complete factorization of $q_{\ell,n}(z)$ in terms of (5) and (11), one needs to understand the multiplicity of the hyperbolic factors. We claim the following result.
**Theorem 2.** For $\ell \in \mathbb{N}$ and $n \in \mathbb{N}^*$, one has

$$ q_{\ell,n}(z) = \prod_{k|n} \left( h_k(z)^{\eta_{\ell}(k)} \prod_{j=2}^{\ell} m_{j,k}(z) \right) \qquad (13) $$

where the multiplicity $\eta_{\ell}(k)$ is given by<fnref n="1" />

$$ \eta_{\ell}(k) = \left\lfloor \frac{\ell - 1}{k} \right\rfloor + 2. \qquad (14) $$

In other words, the roots of $q_{\ell,n}$ are composed of the points $\text{Hyp}(k)$ for any divisor $k$ of $n$, which are roots with multiplicity $\eta_{\ell}(k)$, and of the points $\text{Mis}(j,k)$ for $2 \le j \le \ell$, which are simple roots.

This result appears to be new. We propose a direct proof in Section 2. In [MV25], we use this result to compare the performance of a new splitting algorithm in the case of simple roots, like for $p_n$, or in the presence of high-multiplicity roots, like for $q_{\ell,n}$. Knowing the exact multiplicity is crucial for the interpretation of the numerical benchmarks.

## 2 Proof of Theorem 2

Let us give here a direct proof of the factorization theorem.

*Proof.* As mentioned above, thanks to [HT15], only the multiplicity of the hyperbolic factors has to be established. For $\ell \in \{0,1\}$, the formula (13) boils down to (8) so we will suppose $\ell \ge 2$ from now on. In particular, $\eta_{\ell}(k) \ge 2$.

First, let us check that the multiplicities in (13) are consistent with $\text{deg} q_{\ell,n}$, i.e. :

$$ \sum_{k|n} \left( \eta_{\ell}(k) |\text{Hyp}(k)| + \sum_{j=2}^{\ell} |\text{Mis}(j,k)| \right) = 2^{\ell+n-1}. \qquad (15) $$

Indeed, using (7) and (12) and a geometric sum, the left-hand side equals

$$ \sum_{k|n} |\text{Hyp}(k)| \left( \eta_{\ell}(k) + \sum_{j=2}^{\ell} \Phi(j,k) \right) = \sum_{k|n} \left( \sum_{m|k} \mu(k/m) 2^{m-1} \right) \left( 2^{\ell} + \left\lfloor \frac{\ell-1}{k} \right\rfloor - \sum_{j=2}^{\ell} \delta_{k|j-1} \right) $$

where $\delta_{k|j-1} = 1$ if $k$ divides $j-1$ and 0 otherwise. For any integers $\lambda, k \in \mathbb{N}^*$, let us observe that

$$ \left\lfloor \frac{\lambda}{k} \right\rfloor = |\{j \in \mathbb{N}^*; j \text{ is an integer}\}| = \sum_{j=1}^{\lambda} \delta_{k|j}. \qquad (16) $$

<footnote n="1">¹In equation (14), the notation $\lfloor \cdot \rfloor$ represents the floor function, i.e. $\lfloor x \rfloor = n$ if $n \in \mathbb{Z}$ and $x \in [n, n + 1)$.</footnote>
The claim (15) thus follows from Möbius inversion formula [Möb32]-[Rot63]:

$$ \sum_{k|n} \left( \sum_{m|k} \mu(k/m) 2^m \right) = 2^n. \qquad (17) $$

The case of Hyp(1) = {0} is essentially based on the well known fact (proven by direct induction) that the trailing coefficients of $p_n(z)$ stabilize as $n$ grows; more precisely:

$$ \forall k \in \mathbb{N}, \quad p_n(z) \equiv p_{n+k}(z) \mod z^{n+1}, \qquad (18) $$

thus $q_{\ell,n}(z) \equiv 0 \mod z^{\ell+1}$. On the other hand, using (20) below and $p_k(z) = z \mod z^2$ for $k \ge 1$, one has by induction:

$$ q_{\ell,n}(z) \equiv 2^{\ell-1} z^{\ell+1} \mod z^{\ell+2}, \qquad (19) $$

so zero is exactly of multiplicity $\ell + 1 = \eta_\ell(1)$.

Let us now consider the case of Hyp($k$) for $k|n$ and $1 < k \le n$. Note that if $n$ is prime, there is only one hyperbolic factor left, namely $k = n$ and the factorization (13) follows for example from an argument of divisibility and the identity of degrees (15). We can however treat the general case in a unified way, regardless of whether $n$ is composite or not and prove (13) by recurrence on $\ell \ge 2$. One has

$$ \begin{aligned} q_{\ell,n}(z) &= p_{\ell+n}(z) - p_{\ell}(z) \\ &= p_{\ell+n-1}^2(z) - p_{\ell-1}^2(z) \\ &= q_{\ell-1,n}(z)(p_{\ell+n-1}(z) + p_{\ell-1}(z)) \\ &= q_{\ell-1,n}(z)(q_{\ell-1,n}(z) + 2p_{\ell-1}(z)). \end{aligned} \qquad (20) $$

The recurrence assumption reads

$$ q_{\ell-1,n} = \prod_{k|n} \left( h_k^{\eta_{\ell-1}(k)} \prod_{j=2}^{\ell-1} m_{j,k} \right) \qquad (21) $$

with $\eta_{\ell-1}(k) \ge 2$ and, from the hyperbolic case, we know that

$$ p_{\ell-1} = \prod_{j|\ell-1} h_j. $$

If $k$ divides $n$ but not $\ell - 1$, then $h_k$ divides $q_{\ell-1,n}(z)$ but is relatively prime to $p_{\ell-1}$ in $\mathbb{C}[z]$, so $h_k$ and $q_{\ell-1,n} + 2p_{\ell-1}$ are also relatively prime. Alternatively, if $k$ divides both $n$ and $\ell - 1$ then $h_k$ divides $q_{\ell-1,n} + 2p_{\ell-1}$. Finally, the polynomials $m_{\ell,k}$ for $k|n$ are known factors of $q_{\ell,n}$ that are relatively prime to all $h_j$ and to $q_{\ell-1,n}$, and as $q_{\ell,n}/q_{\ell-1,n}$ has simple roots, they are simple factors of $q_{\ell,n}$. There are no other Misiurewicz-type factors and all hyperbolic factors are relatively prime with one another. We have thus established that:

$$ \left( \prod_{k \mid \gcd(n, \ell-1)} h_k \right) \left( \prod_{k|n} m_{\ell,k} \right) q_{\ell-1,n} \text{ divides } q_{\ell,n}. \qquad (22) $$
Let us observe that (14) and (16) imply that, for $\ell, k \ge 2$:

$$ \eta_{\ell-1}(k) = \begin{cases} \eta_{\ell}(k) - 1 & \text{if } k|\ell-1, \\ \eta_{\ell}(k) & \text{otherwise.} \end{cases} \qquad (23) $$

Combining (21), (22) and (23), we get that

$$ \prod_{k|n} \left( h_k^{\eta_{\ell}(k)} \prod_{j=2}^{\ell} m_{j,k} \right) \quad \text{divides} \quad q_{\ell,n}. $$

Finally the identity of degrees (15) and the unitary assumption of $q_{\ell,n}$ ensure that (13) holds for the next pre-period. □

**Remark 1.** The previous proof also establishes the following identity regarding (9):

$$ s_{\ell,n} = \frac{q_{\ell,n}}{q_{\ell-1,n}} = p_{\ell+n-1} + p_{\ell-1} = \left( \prod_{k|\text{gcd}(n,\ell-1)} h_k \right) \left( \prod_{k|n} m_{\ell,k} \right). \qquad (24) $$

**Example 2.** To illustrate our method, the simplest case of a composite period is:

$$ q_{2,4}(z) = p_6(z) - p_2(z) = p_5^2(z) - p_1^2(z) = q_{1,4}(z)(p_5(z) + p_1(z)) = p_4^2(z)(p_4^2(z) + 2z). $$

As $p_4 = h_1h_2h_4$, the polynomial $q_{2,4}$ is divisible by $h_4^2$ but not by $h_4^3$. The exponent of Hyp(4) is therefore exactly 2 and we are left with a single hyperbolic factor, namely Hyp(2). Using the degree identity (15) thus ensures that

$$ q_{2,4}(z) = h_1^3(z)h_2^2(z)h_4^2(z)m_{2,1}(z)m_{2,2}(z)m_{2,4}(z) $$

and

$$ s_{2,4}(z) = p_4^2(z) + 2z = h_1(z)m_{2,1}(z)m_{2,2}(z)m_{2,4}(z). $$

## References

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<footnote n="1">¹ *Nicolae Mihalache*. Université Paris-Est Creteil, UMR 8050 CNRS, LAMA, F-94010 Creteil, France & Université Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France

*nicolae.mihalache@u-pec.fr*</footnote>

<footnote n="2">² *François Vigneron*. Université de Reims Champagne-Ardenne, Laboratoire de Mathématiques de Reims, UMR 9008 CNRS, Moulin de la Housse, BP 1039, F-51687 Reims

*francois.vigneron@univ-reims.fr*</footnote>