A note on the factorization of iterated quadratics over finite fields
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 163-178

Let $ f $ be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of $ f $ and conjectured that its limiting distribution explains the factorization of large iterates of $ f $. Later, Xia, Boston, and the author performed extensive Magma computations and found some exceptional families of quadratics that do not seem to follow the original Markov model conjectured by Boston and Jones. They discovered this by empirically observing that certain factorization patterns predicted by the Boston–Jones model never seem to occur for these polynomials, and they suggested a multi-step Markov model that accounts for these missing factorization patterns. In this note, we provide proofs for all these missing factorization patterns. These are the first results that explain why the original conjecture of Boston and Jones does not hold for all monic quadratic polynomials.

Soit $f$ un polynôme quadratique unitaire sur un corps fini de caractéristique impaire. En 2012, Boston et Jones ont construit un processus de Markov basé sur l’orbite post-critique de $f$ et ont conjecturé que sa distribution limite explique la factorisation des grands itérés de $f$. Plus tard, Xia, Boston et l’auteur ont effectué de nombreux calculs avec Magma et ont trouvé certaines familles exceptionnelles de quadratiques qui ne semblent pas suivre le modèle de Markov initial conjecturé par Boston et Jones. Ils ont découvert cela en observant empiriquement que certains schémas de factorisation prédits par le modèle de Boston–Jones ne semblent jamais se produire pour ces polynômes, et ils ont proposé un modèle de Markov à plusieurs étapes qui rend compte de ces schémas de factorisation manquants. Dans cette note, nous fournissons des démonstrations pour tous ces schémas de factorisation manquants. Ce sont les premiers résultats qui expliquent pourquoi la conjecture originale de Boston et Jones ne vaut pas pour tous les polynômes quadratiques unitaires.

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DOI : 10.5802/jtnb.1358
Classification : 11T55, 37P25, 12E05, 20E08
Keywords: finite fields, iteration of polynomials, Markov processes, arithmetic dynamics
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Vefa Goksel. A note on the factorization of iterated quadratics over finite fields. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 163-178. doi: 10.5802/jtnb.1358
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[1] Faseeh Ahmad; Robert Benedetto; Jennifer Cain; Gregory Carroll; Lily Fang The arithmetic basilica: a quadratic PCF arboreal Galois group, J. Number Theory, Volume 238 (2022), pp. 842-868 | DOI | MR | Zbl

[2] Robert Benedetto; Patrick Ingram; Rafe Jones; Michelle Manes; Joseph H. Silverman; Thomas J. Tucker Current trends and open problems in arithmetic dynamics, Bull. Am. Math. Soc., Volume 56 (2019) no. 4, pp. 611-685 | DOI | MR | Zbl

[3] Andrew Bridy; John R. Doyle; Dragos Ghioca; Liang-Chung Hsia; Thomas J. Tucker A question for iterated Galois groups in arithmetic dynamics, Can. Math. Bull., Volume 64 (2021) no. 2, pp. 401-417 | MR | DOI | Zbl

[4] Andrew Bridy; Thomas J. Tucker Finite index theorems for iterated Galois groups of cubic polynomials, Math. Ann., Volume 373 (2019) no. 1-2, pp. 37-72 | DOI | MR | Zbl

[5] Andrea Ferraguti; Giacomo Micheli An equivariant isomorphism theorem for mod p reductions of arboreal Galois representations, Trans. Am. Math. Soc., Volume 373 (2020) no. 12, pp. 8525-8542 | DOI | MR | Zbl

[6] Andrea Ferraguti; Giacomo Micheli; Reto Schnyder On sets of irreducible polynomials closed by composition, Arithmetic of finite fields. 6th international workshop, WAIFI 2016 (Lecture Notes in Computer Science), Volume 10064, Springer, 2016, pp. 77-83 | MR | Zbl

[7] Andrea Ferraguti; Giacomo Micheli; Reto Schnyder Irreducible compositions of degree two polynomials over finite fields have regular structure, Q. J. Math., Volume 69 (2018) no. 3, pp. 1089-1099 | DOI | MR | Zbl

[8] Andrea Ferraguti; Carlo Pagano Constraining Images of Quadratic Arboreal Representations, Int. Math. Res. Not., Volume 2020 (2020) no. 22, pp. 8486-8510 | MR | Zbl

[9] Vefa Goksel; Shixiang Xia; Nigel Boston A refined conjecture for factorizations of iterates of quadratic polynomials over finite fields, Exp. Math., Volume 24 (2015) no. 3, pp. 304-311 | DOI | MR | Zbl

[10] Domingo Gomez; Alejandro P. Nicolás An estimate on the number of stable quadratic polynomials, Finite Fields Appl., Volume 16 (2010) no. 6, pp. 401-405 | DOI | MR | Zbl

[11] Domingo Gómez-Pérez; Alina Ostafe; Igor E. Shparlinski On irreducible divisors of iterated polynomials, Rev. Mat. Iberoam., Volume 30 (2014) no. 4, pp. 1123-1134 | DOI | MR | Zbl

[12] David Rodney Heath-Brown; Giacomo Micheli Irreducible polynomials over finite fields produced by composition of quadratics, Rev. Mat. Iberoam., Volume 35 (2019) no. 3, pp. 847-855 | DOI | MR | Zbl

[13] Rafe Jones An iterative construction of irreducible polynomials reducible modulo every prime, J. Algebra, Volume 369 (2012), pp. 114-128 | Zbl | DOI | MR

[14] Rafe Jones Galois representations from pre-image trees: an arboreal survey, Actes de la conférence « Théorie des nombres et applications », CIRM, Luminy, 16-20 janvier 2012 (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres), 2013, pp. 107-136 | Zbl | Numdam | MR

[15] Rafe Jones; Nigel Boston Settled polynomials over finite fields, Proc. Am. Math. Soc., Volume 140 (2012) no. 6, pp. 1849-1863 | DOI | MR | Zbl

[16] Rafe Jones; Nigel Boston Errata to “Settled polynomials over finite fields, Proc. Am. Math. Soc., Volume 148 (2020) no. 2, pp. 913-914 | DOI | MR | Zbl

[17] Alina Ostafe; Igor E. Shparlinski On the length of critical orbits of stable quadratic polynomials, Proc. Am. Math. Soc., Volume 138 (2010) no. 8, pp. 2653-2656 | DOI | MR | Zbl

[18] Lucas Reis On the factorization of iterated polynomials, Rev. Mat. Iberoam., Volume 36 (2020) no. 7, pp. 1957-1978 | DOI | MR | Zbl

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