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arXiv · 2502.16202

A Markov model for factorisation of iterated cubic polynomials

Javier San Martín Martínez
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Abstract

Motivated by the work of Boston, Jones and Goksel, we propose a Markov model for the factorisation of post-critically finite (PCF) cubic polynomials f. Using the information encoded in the critical orbits, we define a Markov model for PCF cubic polynomials with combined critical orbits of lengths one and two. Thanks to the work of Anderson et al., a complete list of PCF cubic polynomials over $\mathbb{Q}$ is available. Some of these polynomials have already been studied, such as those with colliding critical orbits analysed by Benedetto et al., which align with our model. We construct groups $M_n$ and prove that they follow our Markov model. These groups $M_n$ are conjectured to contain $\mathrm{Gal}(f^n)$.

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Javier San Martín Martínez. 2025-02-22. A Markov model for factorisation of iterated cubic polynomials. https://arxiv.org/abs/2502.16202

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