A Markov model for factorisation of iterated cubic polynomials

Motivated by prior work on post-critically finite cubic polynomials, this paper proposes a Markov model based on critical orbits to describe the factorization of iterated PCF cubic polynomials over $\mathbb{Q}$ and constructs groups $M_n$ that are conjectured to contain the Galois groups of these iterations.

The Gist

Imagine you have a magical machine that takes a number, does some math to it, and spits out a new number. You feed that new number back into the machine, over and over again. This creates a chain of numbers: $x_1, x_2, x_3, \dots$

Now, imagine this machine is a bit of a trickster. Sometimes, if you look at the numbers it produces through a specific "lens" (mathematical primes), the machine seems to break the numbers into smaller, simpler pieces (factors). Sometimes it breaks them into three tiny pieces, sometimes into one big piece and one medium piece, and sometimes it doesn't break them at all.

The question mathematicians have been asking for years is: Can we predict exactly how often the machine will break the numbers, and what the "shape" of those breaks will be?

This paper, written by Javier San Martín Martínez, proposes a clever way to answer that question using a Markov Model. Here is the breakdown using everyday analogies.

1. The Tree of Roots

Imagine the numbers produced by the machine as leaves on a giant, upside-down tree.

• The top of the tree is the starting number.
• The first branch splits into three new numbers (because the machine is a "cubic" polynomial, meaning it deals with $x^3$).
• Each of those splits into three more, and so on.

This creates a massive, branching tree. The "Galois Group" is essentially the set of all possible ways you can shuffle the leaves of this tree without breaking the rules of the machine. If you know the rules of the shuffle, you know the secrets of the number machine.

2. The Problem: The Shuffle is Hard to See

Usually, figuring out the shuffle rules for these trees is incredibly difficult, like trying to guess the rules of a card game just by watching someone play for a few minutes. The patterns are hidden deep inside the math.

However, the author focuses on a special type of machine called a Post-Critically Finite (PCF) polynomial. Think of these as machines where the "critical points" (the most sensitive parts of the machine) eventually get stuck in a loop. They don't wander off forever; they bounce between a few specific spots. This makes the machine more predictable.

3. The Solution: A "Weather Forecast" for Numbers

The author proposes a Markov Model. In simple terms, a Markov model is like a weather forecast.

• If it's raining today (State A), there is a 70% chance it will rain tomorrow, and a 30% chance it will be sunny.
• If it's sunny today (State B), there is a 50% chance it will be sunny tomorrow, and a 50% chance it will rain.

The author says: "Let's treat the factorization of these numbers like the weather."

• The "Weather": Whether the number breaks into 3 pieces, 2 pieces, or 1 piece.
• The "Forecast": By looking at the "critical orbit" (the loop the machine gets stuck in), we can calculate the exact probabilities of what happens next.

The author defines "types" of numbers (like "Sunny" or "Rainy") based on whether certain mathematical values are "squares" or not. Then, they create a set of rules (a transition matrix) that tells us: If we have a "Sunny" number today, what are the odds we get a "Rainy" number tomorrow?

4. Building the "Shadow" Group

Here is the coolest part. The author doesn't just predict the weather; they build a fake group that follows these exact weather rules.

• Imagine you build a toy version of the tree-shuffling machine.
• You program it so that it shuffles the leaves in a way that perfectly matches the probabilities predicted by your Markov model.
• This toy machine is called the Markov Group.

The author proves that for specific types of cubic polynomials (those with loops of length 1 or 2), these toy machines can be built mathematically. They even calculated the "size" of these groups (using something called Hausdorff dimension), showing they are huge but still have a specific, measurable structure.

5. The Big Guess (The Conjecture)

The paper ends with a bold bet, or Conjecture:

"We believe that the real Galois group (the actual secret shuffler of the universe) is always hidden inside our Markov Group (the toy shuffler we built)."

Think of it like this:

• The Markov Group is a giant, slightly messy net.
• The Real Galois Group is a specific, perfect fish swimming inside that net.
• The author is saying: "We built a net based on the patterns we see. We are 99% sure the real fish is in there. We just haven't caught it yet."

Why Does This Matter?

If this conjecture is true, it changes the game. Instead of trying to solve the impossible puzzle of the real Galois group directly, mathematicians can study the "Markov Group" (which is easier to understand) and know that they are studying the right thing. It connects the chaotic world of number theory with the structured world of probability and group theory.

In a nutshell:
The author built a "probability simulator" for how cubic equations break apart. They proved that for certain special equations, this simulator creates a mathematical structure that likely contains the true answer to how these equations behave. It's like building a map of a city based on traffic patterns and betting that the actual city layout fits perfectly inside that map.

Technical Summary

Here is a detailed technical summary of the paper "A Markov model for factorisation of iterated cubic polynomials" by Javier San Martín Martínez.

1. Problem Statement

The paper addresses the challenge of describing the Galois groups of the splitting fields of iterated polynomials, specifically post-critically finite (PCF) cubic polynomials over the field $\mathbb{Q}(\sqrt{-3})$.

• Context: For a polynomial $f \in K[x]$, the Galois group of its $n$-th iterate, $\text{Gal}(f^n)$, acts on the roots of $f^n$, which form a regular $d$-ary tree ($d=3$ for cubics). These groups are generally difficult to characterize.
• Specific Gap: While Markov models have been successfully applied to quadratic polynomials (by Boston, Jones, and Goksel) to predict factorization patterns and bound Galois groups, a rigorous framework for cubic polynomials was lacking.
• Goal: To construct a probabilistic Markov model that predicts the factorization of iterates of PCF cubic polynomials and to define a sequence of groups $M_n$ that follow this model, conjecturing that these groups contain the actual Galois groups $\text{Gal}(f^n)$.

2. Methodology

The author employs a combination of arithmetic geometry, group theory (specifically wreath products and tree automorphisms), and probabilistic modeling.

A. The Markov Model for Factorization

The core of the methodology is a Markov process that tracks the "type" of irreducible factors as the polynomial is iterated.

• Critical Orbits: The model relies on the combined critical orbit of the polynomial $f$, defined by the set of values $\{f^k(\gamma_1), f^k(\gamma_2)\}$ where $\gamma_1, \gamma_2$ are the critical points. The paper focuses on orbits of length 1 and length 2.
• Types ($t(g)$): A polynomial $g$ is assigned a "type" based on whether specific values derived from the critical orbit are quadratic residues (squares, denoted 's') or non-residues (denoted 'n') in the finite field $\mathbb{F}_p$.
• Transition Rules: Using Lemma 2.3 (derived from the discriminant of $f-\beta$), the author establishes transition probabilities:
  • If the discriminant condition is a square ('s'), the polynomial either remains irreducible or splits into three factors of equal degree.
  • If the condition is a non-square ('n'), it splits into one factor of the original degree and one of double the degree.
• State Space: The states are sequences of 's' and 'n' corresponding to the critical orbit. The model defines transition probabilities between these states based on the composition $g \circ f$.

B. Construction of Markov Groups ($M_n$)

To realize these probabilities algebraically, the author constructs subgroups of the automorphism group of the ternary rooted tree, $\text{Aut}(T_n)$.

• Tree Automorphisms: The group $\text{Aut}(T_n)$ is isomorphic to the wreath product $\text{Aut}(T_{n-1}) \wr S_3$.
• Generators: The groups are generated by specific elements ($x_n, y_n, z_n$, etc.) that mimic the splitting behavior predicted by the Markov model.
  • Tripling ($t$): Corresponds to the 's' state (splitting into 3).
  • Doubling ($d$): Corresponds to the 'n' state (splitting into 2+1).
• Recursive Definition: The groups $M_n$ and their subgroups ($L_n, K_n, H_n$) are defined recursively using the structure of the tree and the specific cycle data required to match the Markov transition probabilities.

3. Key Contributions

A. Classification of PCF Cubics

The paper utilizes the classification of PCF cubic polynomials over $\mathbb{Q}$ by Anderson et al. [3]. It specifically analyzes two cases:

1. Combined critical orbit of length 1: (e.g., $-2z^3 + 3z^2$).
2. Combined critical orbit of length 2: (e.g., $2(z+a)^3 - 3(z+a)^2 + \dots$).

B. Construction of Specific Groups

The author explicitly constructs groups $M_n$ for both orbit lengths:

• Length 1: Defines $M_n = \langle x_n, y_n, z_n, L_{n-1} \rangle$. Proves that $L_n \trianglelefteq M_n$ with index 2, and $H_n \trianglelefteq L_n$ with index 12 (quotient $A_4$).
• Length 2: Defines $M_n = \langle L_n, y_n, l_n \rangle$. Proves that $L_n \trianglelefteq M_n$ with index 4 (quotient $V_4$).

C. Verification of Cycle Data

The paper rigorously proves that the constructed groups satisfy the cycle data predicted by the Markov model.

• Theorem 3.16 & 4.8: These theorems demonstrate that the cycle data (partition of degrees of irreducible factors) of the cosets of the constructed groups exactly matches the probabilities derived from the Markov process (e.g., $A^{(n)}_i = \text{CD}(\dots)$).
• Chebotarev Density: The construction ensures that the frequency of cycle types in the group matches the frequency of factorization types over $\mathbb{F}_p$ as predicted by the model.

D. Hausdorff Dimension

The paper computes the Hausdorff dimension of these groups within the profinite group $\text{Aut}(T)$.

• Result: For both orbit lengths, the dimension is calculated as:
  $$\text{hdim}(M_n) = 1 - \frac{1}{3}\log_2(6) \approx 0.871$$
  (Note: For length 2, the calculation involves a slightly different exponent in the group size, but the limit converges to the same value).

4. Results

1. Existence of Models: The paper successfully constructs groups $M_n$ for all PCF cubic polynomials with combined critical orbits of length 1 and 2 that strictly follow the proposed Markov factorization model.
2. Matching Frequencies: The cycle data of these groups matches the theoretical frequencies of factorizations over finite fields derived from the discriminant and critical orbit properties.
3. Connection to Known Cases: The constructed groups for the case of $-2z^3 + 3z^2$ (a Belyi map) are proven to coincide with the groups previously defined by Jones and others in the literature (specifically the groups $E_n$ defined by the kernel of the sign map).
4. Main Conjecture (Conjecture 0.1 & 5.1): The author conjectures that for any PCF cubic polynomial $f$, the constructed Markov group $M$ contains the actual Galois group $\text{Gal}(f^n)$ over $\mathbb{Q}(\sqrt{-3})$.
   • Support: This is supported by the fact that the groups contain all possible cycle data allowed by the arithmetic invariants, and by the rigidity of tree automorphisms (Conjecture 5.6 regarding local vs. global conjugation).

5. Significance

• Extension of Theory: This work extends the successful "Markov model" approach from quadratic polynomials to the more complex cubic case, opening the door to studying higher-degree iterated polynomials.
• Arithmetic-Group Bridge: It provides a concrete method to translate arithmetic properties (critical orbits, discriminants) into group-theoretic structures (subgroups of tree automorphisms).
• Predictive Power: The model offers a way to predict the density of primes for which the iterates of a cubic polynomial have specific factorization patterns, a problem central to arithmetic dynamics.
• Foundation for Future Work: By formulating the problem in terms of "local vs. global conjugation" (Conjecture 5.6), the paper sets a clear path for future research to prove that these constructed groups are indeed the maximal possible Galois groups for these polynomials.

In summary, San Martín Martínez provides a robust framework linking the arithmetic of PCF cubic polynomials to the geometry of tree automorphisms, constructing explicit groups that model the factorization behavior of these polynomials and conjecturing their role as the true Galois groups.