A Markov model for factorisation of iterated cubic polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915847556038656 |
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| author | Martínez, Javier San Martín |
| author_facet | Martínez, Javier San Martín |
| contents | 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 $\mathrm{Gal}(f^n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Markov model for factorisation of iterated cubic polynomials Martínez, Javier San Martín Number Theory 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 $\mathrm{Gal}(f^n)$. |
| title | A Markov model for factorisation of iterated cubic polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.16202 |