A Markov model for factorization of iterated cubic polynomials

2025/02/22 by Martínez, Javier San Martín
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.16202

Abstract

Motivated by Boston and Jones and Goksel, we propose a Markov model for the factorization of PCF cubic polynomials f. Using the information encoded in the critical orbits of cubic polynomials, we define a Markov model for PCF cubic polynomials with combined critical orbits of lengths one and two. A complete list of PCF cubic polynomials over ℚ is available thanks to the work of Anderson et al. Some of these polynomials have been previously studied -- for example, those with colliding critical orbits analyzed by Benedetto et al.; the results from these studies 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 Gal(fⁿ).

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